EC202 Handout - London School of Economicsdarp.lse.ac.uk/pdf/EC202/EC202_Handout.pdf · Diagonal...
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03/12/2018 1 EC202 2018/9 MICHAELMAS TERM SLIDE PACK • five sections: Firm / Consumer / General Equilibrium / Risk & Uncertainty / Welfare • first slide of each lecture has green border • full versions at http://darp.lse.ac.uk/ec202 • for more, contact [email protected] THE FIRM: Lectures 1 - 4 Quantities z i amount of input i z = (z 1 , z 2 , , z m ) input vector Z input requirement set q amount of output (single firm) q f output of firm f Prices and profits w i price of input i w = (w 1 , w 2 , , w m ) input-price vector p price of output profits Functions production function C cost function H i conditional demand for input i S supply function D i ordinary demand for input i Other Lagrange multiplier (min cost) elasticity of demand
Transcript of EC202 Handout - London School of Economicsdarp.lse.ac.uk/pdf/EC202/EC202_Handout.pdf · Diagonal...
03/12/2018
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EC202 2018/9MICHAELMAS TERM SLIDE PACK
• five sections: Firm / Consumer / General Equilibrium / Risk & Uncertainty / Welfare• first slide of each lecture has green border• full versions at http://darp.lse.ac.uk/ec202
• for more, contact [email protected]
THE FIRM:Lectures 1 - 4
Quantitieszi amount of input i
z = (z1, z2 , , zm ) input vector
Z input requirement set
q amount of output (single firm)
qf output of firm f
Prices and profitswi price of input i
w = (w1, w2 , , wm) input-price vector
p price of output
profits
Functions production function
C cost function
Hi conditional demand for input i
S supply function
Di ordinary demand for input i
Other Lagrange multiplier (min cost)
elasticity of demand
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Inputs A first step
Characterise the basic constraint facing a firm
Use the production function (z1, z2, , zm)
• written equivalently (z)
• gives maximum output that can be produced from inputs z
Suppose a given amount of q of output is required
Then the basic constraint is (z) q
This basic constraint can be used in several ways
The input requirement set Z
“Inside”: feasible but inefficient
Boundary: feasible and technically efficient
“Outside”: Infeasible
Z(q)
q < (z)
q = (z)
q > (z)
z1
z2
Pick a particular output level q
Find a feasible input vector z
Repeat to find all such vectors for given q Get the input-requirement set: Z(q) := {z: (z) q}
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z1
z2
If Z smooth and strictly convex…
Pick two boundary points
Draw the line between them
Intermediate points lie in the interior of Z
Combination of two techniques may produce more output
What if we changed some of the assumptions?
z
z
Z(q)
q< (z)
q = (z")
q = (z') Important role of convexity
If Z smooth but not convex…
z1
z2
in this region there is an indivisibility
Join two points across the “dent”
Z(q)
Take an intermediate point
Point lies in infeasible zone
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z1
z2
If Z convex but not smooth
Slope of the boundary is undefined at this point
q = (z)
Isoquants
Pick a particular output level q
Find the input requirement set Z(q)
The isoquant is the boundary of Z: { z : (z) = q }
Think of the isoquant as an integral part of the set Z(q)
(z)i(z) := ——zi .
j (z)——i (z)
If the function is differentiable at zthen the marginal rate of technical substitution is the slope at z:
Where appropriate, use subscript to denote partial derivatives. So
Gives rate at which you trade off one input against another along the isoquant, maintaining constant q
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z1
z2
Draw input-requirement set Z(q)
Isoquant, input ratio, MRTS Boundary: contour of the function
{z: (z)=q}
An efficient point
Input ratio describes one production techniquez2°
z1°
z°
Slope of ray: input ratio
z2 / z1= constant
Slope of boundary: Marginal Rate of Technical Substitution
The isoquant is the boundary of Z
Higher slope: increased MRTS
z′
MRTS21=1(z)/2(z)
MRTS21: implicit “price” of input 1 in terms of 2
Higher “price”: smaller relative use of input 1
MRTS and substitution Responsiveness of input ratio to MRTS
z1
z2
low
z1
z2
high
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Homothetic contours Curves: the isoquant map
Oz1
z2
Ray through the origin: a given input ratio
Same MRTS where ray cuts each isoquant
Contours of a homogeneous function
Curves: the isoquant map
Oz1
z2 Point z°: inputs that will produce q
Point tz°: inputs that will produce t rq
tz1°
tz2°
z2°
z1°
tz°
z°
q
trq
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z2
q > (z)
Boundary: feasible and efficient
Inputs and outputq
0
The cone: q ≤ (z1, z2)
Interior: feasible but inefficient
“Outside”: infeasibleq < (z) q = (z) The expansion path through 0
This case: CRTS(tz) = t (z)Double inputs and output exactly doubles
Relationship to isoquants
z2
q
0
Take any production function
Horizontal “slice”: given q level
Project down to get the isoquant
Repeat to get isoquant map
Isoquant map is the projection of the set of technically efficient points
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Marginal products
Measure the marginal change in output w.r.t. this input
(z)MPi = i(z) = ——
zi .
Pick a technically efficient input vector
Keep all but one input constant
• Any z such that q= (z)
• The marginal product
CRTS production function again
z2
q
0
Vertical slice: keep one input constant
Broken line: path for z2 = const
Let’s look at its shape
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Marginal Product for CRTS production function
z1
q
(z)
Shaded area: feasible set
A section of the production function
Boundary: technically efficient points
Input 1 is essential:If z1 = 0 then q = 0
Slope of tangent: MP of input 1
(z)
Slope depends on value of z1…
1(z) falls with z1 (or stays constant) if is concave
The optimisation problemA classic microeconomic task:
• max some objective function (profits?)
• subject to defined constraints
Translate this into a formal problem
Choose q and z to maximise Π ≔ ∑• subject to (z) q
• and q 0, z 0
Q1: what should we assume about p and wi?• constant? (perfect markets)
• depend on quantities q and zi? (monopoly, monopsony)
• something else? (next term’s lectures)
Q2: would it be useful to break the problem down?
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A standard optimisation method Suppose is a differentiable function
Then we can set up a Lagrangian to take care of the constraints
Write down the First Order Conditions (FOC)
Check out second-order conditions
Use FOC to characterise solution
L (... )
L (... ) = 0z
2 L (... ) z2
z* = …
Stage 1 optimisationAssume perfect competition
• this means all prices are exogenously given
• so p and w are fixed
Suppose we take a given target output level
• this fixes q
• so revenue pq is constant and can be ignored in the optimisation
So the stage-1 problem is
• “maximise profits”: constant ∑
• equivalent to “minimise costs”: ∑
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Isocost lines
z1
z2
w1z1 + w2z2 = c
w1z1 + w2z2 = c'
w1z1 + w2z2 = c"
Diagonal line: set of points where cost of input is c, a constant
One such “isocost” line for each value of the constant c
Arrow: indicates the order of the isocost lines
Use this to derive optimum
Cost-minimisation
z1
z2
z*
Arrow shows objective of the firm
Shaded area: constraint on optimisation
These two define the stage-1 problem
Point z*: solution to the problem
minimisem
wizii=1
subject to (z) q
q
Solution depends on the shape of the input-requirement set
What would happen in other cases?
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Convex Z, touching axis
z1
z2
Here MRTS21 > w1 / w2 at the solution.
z* Input 2 is “too expensive”
and so isn’t used: z2* = 0
Non-convex Z
z1
z2
But note that there’s no solution point between z* and z**
z*
z**
There could be multiple solutions.
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z1
z2
Non-smooth Z
z* z* is unique cost-
minimising point for q
True for all positive finite values of w1, w2
q (z) + [q – (z)]
Cost-minimisation: strictly convex Z
1 (z) = w1
2 (z) = w2
… … … m(z) = wm
q = (z )
m
wi zii=1
Use the objective function ...and output constraint
...to build the Lagrangian
Minimise
Differentiate w.r.t. z1, ..., zm ; set equal to 0
Because of strict convexity we have an interior solution
... and w.r.t
A set of m + 1 First-Order Conditions
Denote cost minimising values by *
m
So we find: z∗z∗
• MRTS = input price ratio
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The solution: Solve the FOC to get three important expressions:
1. The cost-minimising value for each input ∗ w,
2. The cost-minimising value for the Lagrange multiplier∗ ∗ w,
3. The minimised value of cost itself
w, ≔ minz
Properties of C
w1
C
C(w, q)
Dark curve: cost of q as function of w1
Cost is non-decreasing in input prices
Cost is increasing in output, if continuous
C(w, q+q)
Cost is concave in input prices°
C(tw+[1–t]w,q) tC(w,q) + [1–t]C(w,q)
C(w,q) ———— = zj
*
wj
Slope illustrates Shephard’s Lemma
z1*
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What happens to cost if w changes to tw
z1
z2q
C(tw,q) = i t wizi* = t iwizi
* = tC(w,q)
Point z*: cost-minimising inputs for w, given q
• z*
Point z*: also cost-minimising inputs for tw, given q
• z*
So we have:
The cost function is homogeneous of degree 1 in prices
Stage 2 optimisation: Average and marginal cost
p
C/q
Green curve: average cost curve
Cq
Marginal cost cuts AC at its minimum
Slope of AC depends on RTS
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Revenue and profits
q qq qqq
p
C/qCq
Horizontal line: a given market price p
Large rectangle: Revenue if output is q
q*
Small rectangle: Cost if output is q
Difference: Profits if output is q
Profits vary with q
Point q*: Maximum profits
price = marginal cost
What happens if price is low...
p
C/qCq
price < average cost
qq* = 0
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Profit maximisation
From FOC if q* > 0:
p = Cq (w, q*)
C(w, q*) p ————
q*
In general:
p Cq (w, q*)
pq* C(w, q*)
“Price equals marginal cost”
“Price covers average cost”
covers both the cases: q* > 0 and q* = 0
Objective: choose q to max
pq – C (w, q) “Revenue minus minimised cost”
The first response function
m
wi zi subject to q (z), z ≥ 0i=1
Review the cost-minimisation problem and its solution
Cost-minimising value for each input:
zi* = Hi(w, q), i=1,2,…,m
The firm’s cost function:
C(w, q) := min wizi{(z) q}
Hi is the conditional input demand function Demand for input i, conditional on given output level q
The “stage 1” problem
The solution function
Choose z to minimise
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Mapping into (z1,w1)-space
z1
z2
z1
w1
Left-hand panel: conventional case of Z the slope of the tangent: value of w1
Repeat for a lower value of w1
…and again to get…
Green curve: conditional demand curve
H1(w,q)
Constraint set is convex, with smooth boundary
Response function is a continuous map:
Another map into (z1,w1)-space
z1
z2
z1
w1
Left-hand panel: nonconvex ZStart with a high value of w1
Repeat for a very low value of w1
Points “nearby” `work the same wayBut what happens in between?
A demand correspondence
Constraint set is nonconvex
Response is discontinuous: jumps in z*
Map multivalued at discontinuity
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Put the cost function to work
Use Shephard’s Lemma , ∗ in first response function: • , , (*)
Differentiate (*) with respect to wj
• , , (**)
Order of differentiation is irrelevant ( ) and so
, ,
Special case: put i = j in (**) to get , , Concavity of C implies , 0andso
, 0
Conditional input demand curve
H1(w,q)
z1
w1 Consider the demand for input 1
Consequence of result 2?
“Downward-sloping” conditional demand
In some cases it is also possible that Hi
i = 0
Corresponds to the case where isoquant is kinked: multiple w values consistent with same z*
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The second response function
From the FOC:
p Cq (w, q*), if q* > 0
pq* C(w, q*)
“Price equals marginal cost”
“Price covers average cost”
Review the profit-maximisation problem and its solution
pq – C (w, q)
The “stage 2” problem
q* = S (w, p) S is the supply function
(again it may actually be a correspondence)
Profit-maximising value for output:
Choose q to maximise:
Prices and supply of output Take the FOC from last slide: , ∗
Substitute in the supply function ∗ ,
, , (*)
Differentiate (*) with respect to wj:• , , , , , 0
• ,, ,
, ,
Differentiate (*) with respect to p:• , , , 1
• ,, ,
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The firm’s supply curve
C/q
Cq
p
q
AC (green) and MC (red) curves
For given p read off optimal q*
Continues down to p Check what happens below p
p_ –
q_|
Case illustrated is for with first decreasing AC, then increasing AC, Response is a discontinuous map: jumps in q*
Multivalued at the discontinuity
Supply response given by q=S(w,p)
The third response function
Demand for input i, conditional on output q
zi* = Hi(w,q)
q* = S (w, p) Supply of output
zi* = Hi(w, S(w, p) )
Di(w,p) := Hi(w, S(w, p) )
Now substitute for q* :
Demand for input i (unconditional )
Stages 1 & 2 combined…
Recall the first two response functions:
Use this to analyse further the firm’s response to price changes
Use this to define a new function:
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Demand for i and the price of j
Differentiate w.r.t. wj: Dji(w, p) = Hj
i(w, q*) + Hqi(w, q*)Sj(w, p)
= Hji(w, q*) + Ciq(w, q)Sj(w, p)
Take the basic relationship Di(w, p) = Hi(w, S(w, p))
This and the result on Sj(w, p) give us a decomposition formula:
“substitution effect”
“output effect”Cjq(w, q*) Dj
i(w, p) = Hji(w, q*) Ciq(w, q*)
Cqq(w, q*) .
Substitution effect is just slope of conditional input demand curve
Output effect is [effect of wj on q][effect of q on demand for i]
Results from decomposition formula
The effect wi on demand for input j equals the effect of wj
on demand for input i
Take the general relationship:
Now take the special case where j = i:
If wi increases, the demand for input i cannot rise
Ciq(w, q*)Cjq(w, q*)Dj
i(w, p) = Hji(w, q*)
Cqq(w, q*) .
Ciq(w, q*)2
Dii(w, p) = Hi
i(w, q*) Cqq(w, q*).
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Input-price fall: substitution effect
Change in cost
conditional demand curve
pric
e fa
ll
z1
w1
H1(w,q)
*z1
z1* : initial equilibrium
grey arrow: fall in w1
shaded area: value of price fall
z1
Input-price fall: total effect
pric
e fa
ll
z1
w1
*z1
z1* : initial equilibrium
green line: substitution effect
z1** : new equilibrium
**
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The short-run problem
subject to q (z), q 0, z 0, and zm =zm
:= pq –m
wizii=1
Letq be the q for which zm =zm would be chosen in the unrestricted problem
Choose q and z to maximise
{zm =zm } C(w, q, zm ) := min wi zi The solution function with
the side constraint
~ _
Short-run demand for input i:
Hi(w, q, zm) =Ci(w, q, zm )~ _ ~ _
From Shephard’s Lemma
C(w, q) C(w, q, zm ) ~ _
By definition of cost function
Short-run AC ≥ long-run AC
So, dividing by q:~ _C(w, q) C(w, q, zm )______ _________q q
MC, AC and supply in the short and long run
C/q
Cq
q
p
q
C/q
Cq
~
~
green curve: AC if all inputs variable
q : given output level
red curve: MC if all inputs variable
black curve: AC if input m kept fixed
brown curve: MC if input m kept fixed
LR supply curve follows LRMC
SR supply curve follows SRMC
Supply curve steeper in the short run
SRAC touches LRAC at given output
SRMC cuts LRMC at given output
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Conditional input demand
H1(w,q)
z1
w1 Brown curve: demand for input 1
“Downward-sloping” conditional demand
Conditional demand curve is steeper in the short run
H1(w, q, zm)~ _
Purple curve: demand for input 1 in problem with the side constraint
Key concepts
Basic functional relations
price signals firm input/output responses
Hi(w,q)
S (w,p)
Di(w,p)
Hi(w, S(w,p)) = Di(w,p)
demand for input i, conditional on output
supply of output
demand for input i(unconditional )
And they all hook together like this:
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Market supply
q1
p
low-cost firm 1
q2
p
high-cost firm 2
p
q1+q2
both firms
Supply curve firm 1 follows MC Supply curve firm 2 follows MC Horizontal line: a given price Sum individual firms’ supply of output Repeat… Market supply curve is locus of these points
Market supply (2)
low-cost firm
p'
high-cost firm
p"
p'
both firms
p"
q1 q2 q1+q2
Below p' neither firm is in the market
Between p' and p'' only firm 1 is in the market
Above p'' both firms are in the market
pp p
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Take two identical firms…
p'
p
4 8 12 16
q1
p'
p
4 8 12 16
q2
Sum to get aggregate supply
24 328 16
p'
p
q1 + q2
•
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••• • • •• •• • • ••
Numbers and average supply
p
4 8 12 16
average(qf)
p'
Rescale to get average supply of the firms Compare with S for just one firm Repeat to get average S of 4 firms …average S of 8 firms
… of 16 firms
••
The limiting case
p
4 8 12 16
average(qf)
p'
The limit: continuous “averaged” supply curve
A solution to the non-existence problem?
A well-defined equilibrium
averagesupply
averagedemand
Firms’ outputs in equilibrium
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Industry supply: negative externality
S
S2 (q1=1)
S2 (q1=5)
S1 (q2=1)
S1 (q2=5)
MC1+MC2
q2
firm 2 alone
p
MC1+MC2
both firms
q1+ q2
p
q1
firm 1 alone
p
Each firm’s S-curve (MC) shifted by the other’s output
The result of simple MC at each output level
Industry supply allowing for interaction
Market equilibrium: number of firms price = marginal cost
Entry mechanism: • if p C/q gap is large then another firm could enter• applying this iteratively determines the size of the industry
price average cost
determines output of any one firm
determines number of firms
(0) Assume that firm 1 makes a positive profit
(1) Is pq – C ≤ set-up costs of a new firm?• …if YES then stop. We’ve got the eqm # of firms• …otherwise continue:
(2) Number of firms goes up by 1
(3) Industry output goes up
(4) Price falls (D-curve) and firms adjust output (individual firm’s S-curve)
(5) Back to step 1
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Firm equilibrium with entry
p
q3
marginalcost
average cost
output of firm
p
q1
1
pric
e AC (purple) and MC (red)
Use MC to get supply curve
Use price to find output
Profits in temporary equilibrium
Price-taking temporary equilibrium
nf = 1
Allow new firms to enter
p
q2
234
p
q4
p
qN
In the limit entry ensures profits are competed away
p = C/q
nf = N
Monopoly – model structure
We are given the inverse demand function• p = p(q)• Gives the price that rules if the monopolist delivers q to the market• For obvious reasons, consider it as the average revenue curve (AR)
Clearly, if pq(q) is negative (demand curve is downward sloping), then MR < AR
Differentiate to get monopolist’s marginal revenue (MR):• p(q) + pq(q)q• pq(ꞏ) means dp(ꞏ)/dq
Total revenue is:• p(q)q
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Average and marginal revenue
q
p
AR
AR curve is just the market demand curve
Total revenue: area in the rectangle underneath
Differentiate total revenue to get marginal revenue
MR
Monopoly – optimisation problem Introduce the firm’s cost function C(q)
• Same basic properties as for the competitive firm
From C we derive marginal and average cost:• MC: Cq (q)• AC: C(q) / q
Given C(q) and total revenue p(q)q profits are: • (q) = p(q)q C(q)
The shape of is important:• We assume it to be differentiable• Whether it is concave depends on both C() and p()• Of course (0) = 0
Firm maximises (q) subject to q ≥ 0
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Monopoly – solving the problem
This condition gives the solution• from above get optimal output q*
• put q* in p(ꞏ) to get monopolist’s price• p* = p(q* )
Check this diagrammatically
Evaluating the FOC:• p(q) + pq(q)q Cq(q) = 0 • p(q) + pq(q)q = Cq(q) • MR = MC
First- and second-order conditions for interior maximum:• q(q) = 0 • qq(q) < 0
Problem is “max (q) s.t. q ≥ 0,” where:• (q) = p(q)q C(q)
Monopolist’s optimum
q
p
AR
AR and MR (green)
AC(purple) and MC (red)
q*: optimum where MC=MR
MR
AC
MC
p*: monopolist’s optimum price
q*
p*
: monopolist’s profit
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Monopoly – pricing rule
This gives the monopolist’s pricing rule:
• p(q) =
…can be rewritten as:• p(q) [1+1/] = Cq(q)
First-order condition for a maximum:• p(q) + pq(q)q = Cq(q)
Introduce the elasticity of demand :• := d(log q) / d(log p) = p(q) / qpq(q)
• < 0
Cq(q)———1 +
Monopoly – analysing the optimum
Take the basic pricing rule• p(q) = Cq(q)
———1 + 1/
Clearly as | | decreases:• output decreases• gap between price and marginal cost increases
Use the definition of demand elasticity• p(q) Cq(q) • p(q) > Cq(q) if | | < ∞• “price > marginal cost”
What happens if | | ≤ 1 ( -1)?
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Monopolistic competition: 1
Take linear demand curve (AR)
Standard marginal and average costs
Optimal output for single firm
output of firm
MC AC
MR
AR
p
q1
MR curve derived from AR
Price and profits
outcome is effectively the same as for monopoly
Monopolistic competition: 2
output of firm
p
q1
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THE CONSUMER:Lectures 5 - 8
Quantitiesxi consumption of good i
x = (x1, x2 , , xn ) consumption vector
X consumption set
Ri resource stock of good i
R = (R1,R2 ,,Rn) resource endowment
Prices and incomepi price of good i
p = (p1, p2 , , pn) price vector
y money income
FunctionsU utility function
C cost (expenditure) function
Hi compensated demand for good i
Di ordinary demand for good i
V indirect utility function
Other Lagrange multiplier (min cost)
Lagrange multiplier (max utility)
utility level
The budget constraint
x1
Slope determined by price ratiox2
Two important cases determined by
1. … amount of money income y
2. …vector of resources R
2
p1 – __p2
A typical budget constraint
“Distance out” of budget line fixed by income or resources
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Case 1: fixed nominal income
x1
x2 Budget constraint determined by the two end-points Examine the effect of changing p1
by “swinging” the boundary
y .
.__p1
Budget constraint is
n
pixi ≤ yi=1
x2
Case 2: fixed resource endowment
R
Budget constraint determined by “resources” endowment R
Examine the effect of changing p1
by “swinging” the boundary thus:
Budget constraint is
n n
pixi ≤ piRii=1 i=1
x1
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x1
x2
Revealed Preference
x
Market prices determine a person's budget constraint Suppose the person chooses bundle x Use this to introduce Revealed Preference
x′
Axiom of Rational Choice
Consumer always makes a choice and selects the most preferred bundle that is available
Essential if observations are to have meaning
Weak Axiom of Revealed PreferenceWeak Axiom of Revealed Preference (WARP)
If x RP x' then x' not-RP x
If x was chosen when x' was available then x' can never be chosen whenever x is available
Suppose that x is chosen when prices are p
Now suppose x' is chosen at prices p'
This must mean that x is not affordable at p':
If x' is also affordable at p then:
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x1
x2
WARP in action
Take the original equilibrium
Now let the prices change…
WARP rules out points like x° as possible solutions
x
x′
x°
Clearly WARP induces a kind of negative substitution effect
But could we extend this idea…?
Trying to extend WARP
x1
x2 Take basic idea of revealed preference
Invoke revealed preference again
Invoke revealed preference yet again
Draw the “envelope”
Is this an “indifference curve”…?
No. WARP does not rule out cycles of preference
You need an extra axiom to progress further on this
x''
x
x'
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The (weak) preference relation
The basic weak-preference relation:
x ≽ x'
"Basket x is regarded as at least as good as basket x' …"
…and the strict preference relation…
x ≻ x'
“ x ≽ x' ” and not “ x' ≽ x ”
From this we can derive the
indifference relation
x ~ x'
“ x ≽ x' ” and “ x' ≽ x ”
Fundamental preference axioms
Completeness
Transitivity
Continuity
Greed
(Strict) Quasi-concavity
Smoothness
For every x, x' X either x ≽ x' is true, or x' ≽ x is true, or both statements are true
For all x, x', x" X if x≽x' and x'≽x" then x ≽ x"
For all x' X the not-better-than-x' set and the not-worse-than-x' set are closed in X
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x1
x2
Continuity
Take consumption bundle x°
Construct two bundles, xL with Less than x°, xM
with More
Draw the set of points like xL and the set like xM
Draw a path joining xL , xM
If there’s no “jump” you can draw this blue curve
xL
xM
x°What about the boundary points between the two shaded sets?
Do we jump straight from a point marked “better” to one marked “worse"?
The utility function
Representation Theorem:• given completeness, transitivity, continuity
• preference ordering ≽ can be represented by a continuous utility function
In other words there exists some function U such that• x ≽ x' implies U(x) U(x')
• and vice versa
U is purely ordinal• defined up to a monotonic transformation
So we could, for example, replace U(•) by any of the following• log( U(•) )
• ( U(•) )
• φ( U(•) ) where φ is increasing
All these transformed functions have the same shaped contours
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x2
0
Utility function and indifference curves
0 x2
Take a slice at given utility level Project down to get contours
Again take a slice… Project down to get same contours
Draw U* = φ(U)
Draw function U
U*U
The greed axiom
x1
Pick any consumption bundle in X
Gives a clear “North-East” direction of preference
x2
(b) What can happen if consumers are not greedy
(a) Greed implies that these bundles are preferred to x'
Bliss!
x'
x1
x2 (b)(a)
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ICs are smooth…
ICs strictly concaved-contoured
I.e. strictly quasiconcave
Pick two points on the same indifference curve
x1
x2
Draw the line joining them
Any interior point must line on a higher indifference curve
Conventionally shaped indifference curves
(-) Slope is the Marginal Rate of Substitution
U1(x) .—— .U2 (x) .
C
A
B
Slope well-defined everywhere
Other types of IC: Kinks
x1
x2Strictly quasiconcave
C
A
B
But not everywhere smooth
MRS not defined here
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C
A
B
Other types of IC: not strictly quasiconcave
x2
Case 1: Slope well-defined everywhere
Case 1: Not quasiconcave
Case 2: Quasiconcave but not strictly quasiconcave
x1
x2
x1
Case 2Case 1
The problem
Maximise consumer’s utility
U(x)U assumed to satisfy the standard “shape” axioms
Subject to feasibility constraint
x X
and to the budget constraint
n
pixi ≤ yi = 1
Assume consumption set X is the non-negative orthant
The version with fixed money income
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The primal problem
x1
x2
x*
There's another way of looking at this
The consumer aims to maximise utility…
Subject to budget constraint (green set)
max U(x) subject ton
pi xi yi=1
Defines the primal problem
x*: Solution to primal problem
The dual problem
x*
Alternatively the consumer could aim to minimise cost…
Subject to utility constraint
Defines the dual problem
x*: Solution to the problem
minimisen
pixii=1
subject to U(x)
Cost minimisation by the firm
But where have we seen the dual problem before?
x2
x1
z1
z2
z*
q
x*
x2
x1
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A lesson from the firm
z1
z2
z*
q
x1
x2
x*
Compare cost-minimisation for the firm…
…and for the consumer
The difference is only in notation
So their solution functions and response functions must be the same
U(x)+ λ[ – U(x)]
Cost-minimisation: strictly quasiconcave U
U1 (x ) = p1
U2 (x ) = p2
… … … Un (x ) = pn
= U(x )
n
pi xii=1
Use the objective function…and utility constraint
…to build the Lagrangian
Minimise
Differentiate w.r.t. x1, …, xn and set equal to 0
Because of strict quasiconcavity we have an interior solution
… and w.r.t
A set of n + 1 First-Order Conditions
Denote cost-min values with a *
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From the FOC
Ui(x) pi——— = —Uj(x) pj
MRS = price ratio “implicit” price = market price
If both goods i and j are purchased and MRS is defined then…
Ui(x) pi——— —Uj(x) pj
If good i could be zero then…
MRSji price ratio “implicit” price market price
The solution
Solve FOC to get a cost-minimising value for each good…
xi* = Hi(p, )
…for the Lagrange multiplier
* = *(p, )
…and for the minimised value of cost itself
The consumer’s cost function or expenditure function is defined asC(p, ) := min pi xi
{U(x) }
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Main results are immediate
Shephard's Lemma gives demand as a function of prices and utilityHi(p, ) = Ci(p, )
H is the “compensated” or conditional demand function
Properties of the solution function determine behaviour of response functions
Downward-sloping with respect to its own price, etc…
“Short-run” results can be used to model side constraints
For example rationing
The cost function has same properties as for the firm
Same problem as for firm; so results are the same
Comparing firm and consumer
n
minpixix i=1
+ [ – U(x)]
Cost-minimisation by the firm… …and expenditure-minimisation by the consumer …are effectively identical problems So the solution and response functions are the same:
xi* = Hi(p, )
C(p, )
m
minwiziz i=1
+ [q – (z)]
Solution: C(w, q)
zi* = Hi(w, q) Response:
Problem:
Firm Consumer
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n
U(x) + [ y – pi xi ]i=1
The Primal and the Dual…
There’s an attractive symmetry about the two approaches to the problem
…constraint in the primal becomes objective in the dual…
…and vice versa
In both cases the ps are given and you choose the xs. But…
n
pixi+ [ – U(x)]i=1
A useful connection
x1
x2
x*
Compare the primal problem of the consumer…
…with the dual problem
Two aspects of the same problem
So we can link up their solution functions and response functions
x1
x2
x*
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n
y pi xii=1
U(x)
Utility maximisation
U1(x ) = p1
U2(x ) = p2
… … …Un(x ) = pn
n
+ μ[ y – pi xi ]i=1
Use the objective function…and budget constraint
…to build the Lagrangean
Maximise
Differentiate w.r.t. x1, …, xn and set equal to 0
… and w.r.t
A set of n+1 First-Order Conditions
Denote utility maximising values with a *
n
y = pi xii=1
If U is strictly quasiconcave we have an interior solution
From the FOC
Ui(x) pi——— = —Uj(x) pj
MRS = price ratio “implicit” price = market price
If both goods i and j are purchased and MRS is defined then…
Ui(x) pi——— —Uj(x) pj
If good i could be zero then…
MRSji price ratio “implicit” price market price
(same as before)
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The solutionGet U-maximising value for each good and Lagrange multiplier
• xi* = Di(p, y), i = 1,…,n
• * = *(p, y)
Also for the maximised value of utility itself
The indirect utility function is defined as• V(p, y) := max U(x)
pixi y}
• V is non-increasing in every price, decreasing in at least one price• … is increasing in income y• … is quasi-convex in prices p• …is homogeneous of degree zero in (p, y)• …satisfies “Roy's Identity”
Another useful connection Indirect utility function maps
prices and budget into maximal utility: = V(p, y)
The indirect utility function works like an "inverse" to the cost function
Cost function maps prices and utility into minimal budget: y = C(p, )
The two solution functions have to be consistent with each other.
Therefore we have:= V(p, C(p, ))
Odd-looking identities can be useful
0 = Vi(p,C(p,))+Vy(p,C(p,)) Ci(p,)
0 = Vi(p, y) + Vy(p, y) xi* Shephard’s Lemma
Function-of-a-function rule
Rearrange to get Roy’s identity
xi* = – Vi(p, y)/Vy(p, y) The right-hand side is just Di(p, y)
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Utility and expenditure
n
min pixix i=1
+ [ – U(x)]
Utility maximisation …and expenditure-minimisation by the consumer …are effectively two aspects of the same problem
xi* = Hi(p, )
C(p, ) Solution: V(p, y)
xi* = Di(p, y) Response:
Problem:
Primal Dualn
max U(x) + μ[ y – pi xi ]x i=1
So their solution and response functions are closely connected:
The max-utility problem again
n
pixi* = y
i=1
U1(x*) = p1U2(x*) = p2
… … … Un(x*) = pn
x1* = D1(p, y)
x2* = D2(p, y)
… … … xn
* = Dn(p, y)npi Di(p, y) = yi=1
The n + 1 first-order conditions, assuming all goods purchased
Gives a set of demand functions, one for each good: functions of prices and incomes
A restriction on the n equations. Follows from the budget constraint
Solve this set of equations:
The primal problem and its solutionn
max U(x) + [ y – pi xi ]i=1
Lagrangian for the max U problem
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The response function Primal problem response function is
demand for good i:xi
* = Di(p,y)
Should be treated as just one of a set of n equations
The system has an “adding-up” property:
∑ ,
Follows from budget constraint: LHS is total expenditure
Each equation is homogeneous of degree 0 in prices and income. For any t > 0:
xi* = Di(p, y )= Di(tp, ty)
Again follows from the budget constraint
Effect of a change in income y
x1
x*
Take the equilibrium at x*
Suppose income rises
New equilibrium at x**
x**
x2
Demand for each good does not fall if it is “normal”
But could the opposite happen?
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An “inferior” good
x1
x*
Same original budget, different preferences
Again suppose income risesEquilibrium shifts from x** to x**
x2
Demand for good 1 rises, but…
Demand for “inferior” good 2 falls
Can you think of any goods like this?
How might it depend on the categorisation of goods?
x**
Effect of a change in price
x1
x*
Again take the original equilibrium
Allow price of good 1 to fall
Big blue arrow: the effect of the price fall
x**
x2
Small blue arrows: “journey” from x* to x** broken into two parts
°
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A fundamental decomposition
Take the two methods of writing ∗:p, = p,
Two representations of same thing
Use cost function to substitute for y:p, = p, p,
Differentiate with respect to pj:p, = p, p, p,
= p, p, ∗
Implicit relation in prices and utility
Uses (1) y = C(p,) (2) function-of-a-function rule and (3) Shephard’s Lemma
Rearrange to get:
p, p, ∗ p, This is the Slutsky equation
The Slutsky equation
Gives fundamental breakdown of effects of a price change
Income effect: “I'm better off if the price of jelly falls; I’m worse off if the price of jelly rises. The size of the effect depends on how much jelly I am buying…
Substitution effect: “When the price of jelly falls and I’m kept on the same utility level, I prefer to switch from icecream for dessert”
x**
Dji(p,y) = Hj
i(p,) – xj* Dy
i(p,y)
x*
…if the price change makes me better off then I buy more normal goods, such as icecream”
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The Slutsky equation: own-price
Dii(p,y) = Hi
i(p,) – xi* Dy
i(p,y)
Set j = i to get the effect of the price of ice-cream on the demand for ice-cream
Hii (own-price substitution effect)
must be negative
Dyi is non-negative for normal goods
Theorem: if the demand for i does not decrease when y rises, then it must decrease when pi rises
Follows from the results on the firm
Price increase means less disposable income
So the income effect of a price rise must be non-positive for normal goods
Important special case
Price fall: normal good
compensated (Hicksian) demand curve
pric
e fa
ll
x1
p1
H1(p,)
*x1
Initial equilibrium at x*1
first red arrow: price fall, substitution effect
both red arrows: total effect, normal good
For normal good income effect must be positive or zero
ordinary demand curve
x1**
second red arrow: income effect, normal good
D1(p,y)
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Consumer equilibrium: another view
Rx1
x2
x*
Type 2 budget constraint: fixed resource endowmentBudget constraint with endogenous income
Consumer's equilibrium
Its interpretation
Equilibrium is familiar: same FOCs as before
The offer curve
Rx1
x2
x*
x***
x**
Take the consumer's equilibrium
Let the price of good 1 rise
Let the price of good 1 rise a bit more
Draw the locus of points
This path is the offer curve
Amount of good 1 that household supplies to the market
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supply of good 1
p1
Household supply
Flip horizontally , to make supply clearer Rescale the vertical axis to measure price of good 1
Plot p1 against x1
This path is the household’s supply curve of good 1
R supply of good 1
x2
x*
x***
x**
The curve “bends back” on itself
Why?
Decomposition – another look
Function of prices and income
Differentiate with respect to pj : dxi
* dy— = Dj
i(p, y) + Dyi(p, y) —
dpj dpj
= Dji(p, y) + Dy
i(p, y) Rj
Income itself now depends on prices
Now recall the Slutsky relation: Dj
i(p,y) = Hji(p,) – xj
* Dyi(p,y)
The indirect effect uses function-of-a-function rule again
Just the same as on earlier slide
Use this to substitute for Dji:
dxi*
— = Hji(p,) + [Rj – xj
*] Dyi(p,y)
dpj
The modified Slutsky equation
Take ordinary demand for good i:xi
* = Di(p,y)
Substitute in for y :xi
* = Di(p, j pjRj)
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The modified Slutsky equation:
Substitution effect has same interpretation as before
Two terms to consider when interpreting the income effect
The second term (in the income effect) is just the same as before
The first term (in the income effect) makes all the difference:• Negative if the person is a net demander• Positive if the person is a net supplier
dxi*
── = Hji(p, ) + [Rj – xj
*] Dyi(p,y)
dpj
Application: savings
Rx1
x2
x*
Resource endowment is non-interest income profile Slope of budget constraint increases with interest rate, r
Consumer's equilibrium
Its interpretation
Determines time-profile of consumption
What happens to saving when the interest rate changes…?
x1,x2 are consumption “today” and “tomorrow”
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Application: labour supply
R
x1
x2
x*
Endowment: total time & non-labour income Slope of budget constraint is wage rate
Consumer's equilibrium
Determines labour supply
Will people work harder if their wage rate goes up?
x1,x2 are leisure and consumption
The two aspects of the problem
x1
x2
x*
Primal: Max utility subject to the budget constraint
Dual: Min cost subject to a utility constraint
x1
x2
x*
What effect on max-utility of an increase in budget?V(p, y) C(p,)
V
What effect on min-cost of an increase in target utility?
C
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Interpreting the Lagrange multiplier
Differentiate with respect to y:
Vy(p, y) = i Ui(x*)Diy(p, y) + * [1 – i piDi
y(p, y)]
The solution function for the primal:
V(p, y) = U(x*) = U(x*) + * [y – i pixi* ]
At the optimum, either the constraint binds or Lagrange multiplier is zero
Use the ordinary demand functions
Rearrange:Vy(p, y) = i[Ui(x*)–*pi]Di
y(p,y)+* = *
Lagrange multiplier in the primal is MU of income
Differentiate with respect to and rearrange:C(p, ) = i [pi–*Ui(x*)] Hi
(p, )+* = *
The solution function for the dual:C(p, ) = ipi xi
* = ipi xi* – * [U(x*) – ]
Same argument as above
Lagrange multiplier in the dual is MC of utility
We can also show:. * = 1/ * A useful connection between C and V
The problem of valuing utility change
x1
x*
Take the consumer's equilibrium
and allow a price to fall...
Obviously the person is better off.
'
x**
x2
...but how much better off?
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Story number 1 (CV)Price of good 1 changes
• p: original price vector
• p': vector after price change
This causes utility to change • = V(p, y)
• ' = V(p', y)
Value this utility change in money terms:• what change in income would bring a person
back to the starting point?
Define the Compensating Variation:• = V(p', y – CV)
Amount CV is just sufficient to undoeffect of going from p to p'
original utility level restored at new prices p'
original utility level at prices p
new utility level at prices p'
The compensating variation
x1
x**
x*
A fall in price of good 1
Reference point is original utility level
Red line: CV measured in terms of good 2
x2
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Story number 2 (EV)Price of good 1 changes
• p: original price vector
• p': vector after price change
This causes utility to change • = V(p, y)
• ' = V(p', y)
Value this utility change in money terms:• what income change would have been needed
to bring the person to the new utility level?
Define the Equivalent Variation:• ' = V(p, y + EV)
Amount EV is just sufficient to mimiceffect of going from p to p'
new utility level reached at original prices p
original utility level at prices p
new utility level at prices p'
The equivalent variation
x1
x**
x*
' Price fall as before
Reference point is the new utility level '
Red line: EV measured in terms of good 2
x2
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Welfare change as – (cost)
Equivalent Variation as –(cost):
EV(pp') = C(p, ') – C(p', ')
Compensating Variation as –(cost):
CV(pp') = C(p, ) – C(p', )
(–) change in cost of hitting utility level . If positive we have a welfare increase
(–) change in cost of hitting utility level '. If positive we have a welfare increase
Using these definitions we also have
CV(p'p) = C(p', ') – C(p, ')
= – EV(pp')
Looking at welfare change in the reverse direction, starting at p' and moving to p
Cost-of-living indices
An approximation:i p'i xiIL = ———i pi xi
ICV .
An index based on CV:
C(p', )ICV = ————
C(p, )
What's the change in cost of hitting the base utility level ?
What's the change in cost of buying the base consumption bundle x?This is the Laspeyres index (the basis for the Consumer Price Index)
An index based on EV:
C(p', ')IEV = ————
C(p, ') An approximation:
i p'i x'iIP = ———i pi x'iIEV .
What's the change in cost of hitting the new utility level ' ?
What's the change in cost of buying the new consumption bundle x'?This is the Paasche index .
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Another (equivalent) form for CV
Assume that the price of good 1 changes from p1 to p1' while other prices remain unchanged. Then we can rewrite the above as:
CV(pp') = C1(p, ) dp1
Use the cost-difference definition:CV(pp') = C(p, ) – C(p', )
(–) change in cost of hitting utility level . If positive we have a welfare increase
Using definition of a definite integral
Further rewrite as:
CV(pp') = H1(p, ) dp1Using Shephard’s lemma again
CV is an area under the compensated demand curve
p1
p1'
p1
p1'
Compensated demand and the value of a price fall (CV)
CompensatingVariation
compensated (Hicksian) demand curve
pric
e fa
ll
x1
p1
H1(p, )
*x1
The initial equilibrium
price fall: (welfare increase)
shaded area: value of price fall, relative to original utility level
The CV provides an exact welfare measure
But it’s not the only approach
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Compensated demand and the value of a price fall (EV)
x1
EquivalentVariationpr
ice
fall
x1
p1
**
H1(p, )
compensated (Hicksian) demand curve
As before but use new utility level as a reference point
price fall: (welfare increase)
Shaded area: value of price fall, relative to new utility level
The EV provides another exact welfare measure
But based on a different reference point
Other possibilities…
Ordinary demand and the value of a price fall
x1
pric
e fa
ll
x1
p1
***x1
D1(p, y)
Initial equilibrium at x1*
price fall: (welfare increase)
Yellow area: an alternative method of valuing the price fall?
Consumer'ssurplus
ordinary (Marshallian) demand curve
CS provides an approximatewelfare measure
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Three ways of measuring the benefits of a price fall
x1
pric
e fa
ll
x1
p1
**
H1(p, )
*x1
H1(p,)
D1(p, y)
Summary of the three approaches.
Illustrated for normal goods
For normal goods: CV CS EV
For inferior goods: CV > CS > EV
GENERAL EQUILIBRIUM:Lectures 9 - 12
Quantitiesqi aggregate net output of good i
xi aggregate consumption of good i
Ri resource stock of good i
Rih resource holding by h of i
qif net output by f of i
xih consumption by h of i
[x1,x2 , …] allocation across households
[q1,q2 , …] allocation across firms
Prices and incomespi market price of good i
i shadow price of good i
f profits of firm f
yh money income of h
FunctionsUh utility function of h
f production function of firm f
Ei excess demand for good i
OtherN replication factor
h reservation utiity for h
fh share of h in the profits of f
Q technology set
A Attainable set
B “better than” set
K Coalition
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Approaches to outputs and inputs
–z1
–z2
...–zm
+q
=
q1
q2
...qn-1
qn
NET OUTPUTS
q1
q2
qn-1
qn
...
A standard “accounting” approach
An approach using “net outputs”
How the two are related
Outputs: +net additions to the
stock of a good
Inputs: reductions in the
stock of a good
Intermediate goods:
0your output and my input cancel each other out
A simple sign convention
OUTPUT INPUTS
z1
z2
...
zm
q
The technology set Q
q1
0Q
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Tradeoff in inputs
(given q1 = 500)
(given q1 = 750)
q4
q3
high input
q2
q1
Tradeoff between outputs
Again take slices through Q
For low level of inputs
low input
For high level of inputs
CRTS
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q2
q1
The technology set Q and the production function
A view of set Q: production possibilities of two outputs.
The frontier is smooth (many basic techniques)
Feasible but inefficient points in the interior
(q) < 0
Feasible and efficient points on the boundary
(q) = 0Infeasible points outside the boundary
(q) > 0
q Q (q) 0
(q1, q2,…,qn) nondecreasing in each qi
Boundary is the transformation curve
Slope: marginal rate of transformation
MRTij := j (q) / i (q)
The Crusoe problem
max U(x) by choosing x and q subject to...
joint consumption-production decision
• x X logically feasible consumption
• (q) 0 technical feasibility: equivalent to “q Q ”
• x q + R materials balance: can’t consume more than available from net output + resources
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Crusoe’s problem and solution
x2
0x1
Shaded green area: attainable set with R1= R2 = 0
Positive stock of resource 1: stretches to right
More of resources 3,…,n : stretches “outwards
Curves: Crusoe’s preferences
Attainable set derived from technology and materials balance condition
purple line: gives the FOC
MRS = MRT:
U1(x) 1(q)—— = ———U2(x) 2(q)
• x*
x*: the optimum
Profits and income at shadow prices
We know that there is no system of prices Invent some “shadow prices” for accounting purposesUse these to value national income
1 2 ... n profits1q1 + 2q2 +...+ nqn
value ofresource stocks
1R1 + 2R2 +...+ nRn
value ofnational income
1[q1+R1] +...+ n[qn+Rn]
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National income contours
1[q1+ R1] + 2[q2+ R2 ] = const
q1+R1
q2+R2
“National income” of the Island
Dark area: attainable setx2
x10
Using shadow prices we’ve broken down the Crusoe problem into a two-step process:
1.Profit maximisation2.Utility maximisation
Shaded triangle: Island’s “budget set” Use budget set to maximise utility
red lines: Iso-profit – income max
1(x) 1—— = —2(x) 2
U1(x) 1—— = —U2(x) 2
x*
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A separation result
By using “shadow prices” … max U(x) subject tox q + R(q) 0
max U(x) subject ton
i xi yi=1
n
max i [ qiRisubj. toi=1
(q) 0
…a global maximisation problem …is separated into sub-problems:
1. An income-maximisation problem
Maximised income from 1 is used in problem 2
2. A utility maximisation problem
Crusoe problem: another view
A: the attainable set
x2
x10
A = {x: x q + R, (q) 0}
purple line: prices B: the “Better-than-x*” set
A
B
12
B = {x: U(x) U(x*)}
Big arrows: decentralisation
x* maximises income over A
x* minimises expenditure over B
x*
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Optimum cannot be decentralised
x2
x1
Attainable set is nonconvex
x*: the consumer optimum
x°:maximise profits given prices
Implied prices: MRT=MRS
Production responses do not support the consumer optimum In this case the price system “fails”
A
x*
x°
Crusoe's island tradesx2
x1
x**
x1**q1
**
x2**
q2**
x*: equilibrium on the island Price differences: possibility of trade Max income at world prices (top left)
Trade enlarges attainable set (shaded triangle)
q**,x**: equilibrium with trade
x* is Autarkic eqm: x1*= q1
*; x2*=q2
*
World prices: revalue national income
In this equilibrium the gap between x**
and q** is bridged by imports & exports
World prices
q**
x*
Domestic prices
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The nonconvex case with world tradex2
x1x1
**q1**
x2**
q2**
A′A
x*: equilibrium on the island purple line: world prices
x*
Max income at world prices (top left)
q**,x**: equilibrium with trade
Attainable set before trade (A) & after trade (A′)
Trade “convexifies” the attainable set
x**
q**
What is an economy?
Resources (stocks)
U1, U2 ,…
,…
R1 , R2 ,…
nh of these
nf of these
n of these
Households (preferences)
Firms (technologies)
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An allocation
A collection of bundles (one for each of the nh households)
[x] := [x1, x2, x3,… ]
[q] := [q1, q2, q3,… ]
p := (p1, p2, …, pn)
utility-maximising
^
profit-maximising
^
A competitive allocation consists of:
A set of prices (used by households and firms)
A collection of net-output vectors (one for each of the nf firms)
{ } { , h=1,2,…,nh }
How a competitive allocation works
qf: from f’s profit maximisation
p qf(p) xh: from h’s utility maximisation
Firms' behavioural responses map prices into net outputs
{ , f=1,2,…,nf } Hholds’ behavioural responses map prices and incomes into demands
p, yh xh(p) Grey box: the competitive allocation
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What does household h possess?
Resources R1h, R2
h, …
1h, 2
h, …
Rih 0,
i =1,…,n
Shares in firms’ profits
0 fh 1,
f =1,…,nf
Incomes
Resources
Profits
Rents
Shares in firms
Net outputs
Prices
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The fundamental role of prices
Net output of i by firm f depends on prices p:
qif = qi
f(p)
Supply of net outputs
Thus profits depend on prices:n
f(p):= pi qif(p)
i=1
So incomes can be written as: n nf
yh = pi Rih + f
h f(p)i=1 f=1
Again writing profits as price-weighted sum of net outputs
Income depends on prices : yh = yh(p)
Income = resource rents + profits
yh(•) depends on ownership rights that h possesses
Prices in a competitive allocation
Large box: allocation as a collection of responses
Put the price-income relation into household responses
Gives a simplified relationship for households
p qf(p){ , f=1,2,…,nf }
{ } { , h=1,2,…,nh }p, yh xh(p)p Small box: summarise the relationship
p [q(p)]
[x(p)]
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The price mechanism
d
resource distribution
R1b, R2
b, …
R1a, R2
a, …
…
share ownership
1b, 2
b, …
1a, 2
a, …
…
System takes as given the property distribution
Property distribution consists of two collections Prices then determine incomes
[y]
Prices and incomes determine net outputs and consumptions
[q(p)][x(p)]
Brief summary below…
adistribution
prices
allocation
What is an equilibrium?
What kind of allocation is an equilibrium?
Again we can learn from previous presentations:• must be utility-maximising (consumption) • must be profit-maximising (production)• must satisfy materials balance (the facts of life)
We can do this for the many-person, many-firm case
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Uh(xh), subject ton
pi xih yh
i=1
Competitive equilibrium: basics
Households maximise utility, given prices and incomes
Firms maximise profits, given prices
For each h, maximise
For all goods the materials balance must hold
n
pi qif, subject to f(qf ) 0
i=1
For each f, maximise
For each i:
xi qi + Ri
Consumption and net output
“Obvious” way to aggregate consumption of good i?
nh
xi = xih
h=1
An alternative way to aggregate:
xi = max {xih }
h
Appropriate if i is a rival good Additional resources needed for each additional person consuming a unit of i
Opposite case: a nonrival good Examples: TV, national defence…
Aggregation of net output:nf
qi := qif
f=1
if all qf are feasible will q be feasible? Yes if there are no externalities Counterexample: production with congestion…
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Competitive equilibrium: summary
A set of prices p Everyone maximises at those
prices p
It must be a competitive allocation
Demand cannot exceed supply: x ≤ q + R
The materials balance condition must hold
Alf’s optimisation problem
x1aR1
a
R2a
x2a
Oa
Ra: resource endowment
Preferences (cyan curves)
Prices (red line) & budget constraint (shaded)
Ra
x*a
x*a: equilibrium
Budget constraint is
2 2
pi xia ≤ pi Ri
a
i=1 i=1
Alf sells some of 2 for good 1 by trading with Bill
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Bill’s optimisation problem
x1bR1
b
R2b
x2b
Ob
Rb: resource endowment
Preferences (green curves)
Prices (red line) & budget constraint (shaded)
x*b: equilibrium
Bill sells good 1 in exchange for 2
Budget constraint is
2 2
pi xib ≤ pi Ri
b
i=1 i=1 Rb
x*b
Combine the two problems
Bill’s problem (flipped)
Price-taking trade moves agents from [R]…
Superimpose Alf’s problem
x1b R1
b
R2b
x2b
Ob
…to competitive equilibrium allocation [x*]
This is the Edgeworth box
Width: R1a + R1
b
Height: R2a + R2
b
x1aR1
a
R2a
x2a
Oa
[R]
[x*]
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Response to changes in prices
x1a
R2a
x2a
Oa
Ra •
R1a
Ra: Alf’s endowment
Curve through Ra:Alf’s res. utility Alf’s preference map
No trade if p1 is too high
Trades offered as p1 falls
••
•••••• •
•
Red curve: Alf’s offer curve
x1b
R2b
x2b
ObR1
b
Rb•
x1b
R2b
x2b
Ob
R1b
Rb•
Response to changes in prices (2)
•••
••••
•
••
Ob Bill’s similar situation…
…diagram inverted
No trade if p1 is too low
Trades offered as p1 rises
Bill’s offer curve
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Edgeworth Box and CE
x2b
Ob
x1a
x2a
Oa
x1b
R1b
R1a
R2a
R2b
• [x*]
[x*]: where offers are consistent
By construction [x*] is CE:
Price-taking U-maximising Alf Price-taking U-maximising Bill Satisfies materials balance
[R]: property distribution Draw in the two offer curves
[R]•
Coalitions
K2
K1
K0
Viewed as nh separate individuals
A coalition K…
…is formed by any subgroup
The population…
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A formal approach
An allocation is blocked by a coalition if the coalition members can do better for themselves
Equilibrium conceptUse the idea of blocking to introduce a solution concept
• if allocation is blocked a coalition could stop it happening
• such an allocation could not be a solution to the trading game
So we use the following definition of a solution:• the Core is the set of unblocked, feasible allocations
Let’s apply it in the two-trader case• In a 2-person world there are few coalitions:
{Alf }
{Bill}
{Alf & Bill}
• let’s see what allocations are blocked by them…
• …and what remains unblocked
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Ob
Oa
x1b
x1a
x2a
x2b
x1a
x2a
x2b
[xb]
x1b
b
The 2-person core
Purple line: the contract curve
a: Alf’s reservation IC through [R]
{Bill} blocks allocations below res IC
{Alf, Bill} block allocations off the CC
Blue segment: the resulting core
{Alf} blocks allocations below res IC
[xa]
b: Bill’s reservation IC through [R]
Points on contract curve: can’t be blocked by {Alf,Bill}
If indifference curves are everywhere differentiable……then MRS is everywhere well defined In this case contract curve is locus of common tangencies
a
[R]
[xa]: Bill gets all advantage from trade
[xb]: Alf gets all advantage from trade
•
•
•
Ob
Oa
x1b
x1a
x2a
x2b
The core and CE [R]: the endowment point blue segment: 2-person core
[R]
x*: competitive equilibrium
[x*] A competitive equilibrium must always be a core allocation
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Ob
Oa
x1b
x1a
x2a
x2b
The core and CE (2)Indifference curves yielding multiple equilibria
Endowment point [R] fixes reservation utility[x*]: Equilibrium, low p1/p2
[x**]: Equilibrium, high p1/p2
Dark blue segment: the core
•[x*]
•[x**]b
a
• [R]
A simple result… and a question Every CE allocation must belong to the core
It is possible that no CE exists
What about core allocations which are not CE?• Remember we are dealing with a 2-person model• Will there always be non-CE points in the core?
To find out, let's clone the economy• economy replicated by a factor N, so there are 2N persons• start with N = 2• Alf and twin brother Arthur have same preferences and endowments• likewise the twins Bill and Ben
Now there are more possibilities of forming coalitions• so more blocking!
Core
CE
{Alf & Bill}
{Alf}{Bill}
{Arthur & Ben}
{Arthur}{Ben}
{Alf & Arthur}{Alf, Arthur &Bill}{Alf, Arthur &Ben} {etc, etc}
{Bill&Ben}{Bill, Ben &Alf}
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Effect of cloning on the core
New allocation is not a solution…
But it shows that the core must have become smaller
Dark blue segment: the 2-person core
{Alf,Arthur,Bill} can block [xa] at ½–way mark
The Ben twin is left outside the coalition
[R]
Are the extremes still core allocations in the 4-person economy?
°
[xa], [xb]: the extremes of the two-person core
[xa]
[xb]
How the blocking coalition works
Alf xa = ½[xa+Ra]
Arthur xa = ½[xa+Ra]
Bill [2Ra +Rb – 2xa]——————2Ra + Rb
Consumption within the coalition equals the coalition’s resources
So the allocation is feasible
Big box: consumption in the coalitionSum to get resource requirement
Ben Rb
Small box: consumption out of coalition
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If N is bigger: more blocking coalitions?
numbers of…a-tribe b-tribe
Dark blue segment: the 2-person core
An arbitrary allocation - can it be blocked?
500 250
310360
400
450
We’ve found the blocking coalition
If line is not a tangent this can always be done
Draw a line to the endowmentTake N=500 of each tribe Divide the line for coalition numbers
[R]
[xb]
[xa]
In the limit
If N a coalition can be found dividing the line to [R] in any proportion you want
Only if the line is like this will the allocation be impossible to block
With the large N the core has “shrunk” to the set of CE
[R]
[xa]
[x*]
[xb]
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Aggregates
From household’s demand functionxi
h = Dih(p, yh)= Dih(p, yh(p) )
Because incomes depend on prices
So demands are just functions of pxi
h = xih(p)
“Rival”: extra consumers require additional resources. Same as “consumer: aggregation”
If all goods are private (rival) then aggregate demands can be written:
xi(p) = h xih(p)
xih(•) depends on holdings of
resources and shares
From firm’s supply of net output qi
f = qif(p)
standard supply functions/ demand for inputs
Aggregate:qi = f qi
f(p)valid if there are no externalities.
Derivation of xi(p)
Alf Bill The Market
p1
x1a x1
b x1
p1 p1
panel 1: Alf’s demand curve for good 1 panel 2: Bill’s demand curve for good 1 Horizontal line: one particular price Sum to get consumers’ demand Repeat to get the market demand curve
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Derivation of qi(p)
q1
p
low-costfirm
q2
p
high-cost firm
p
q1+q2
both firms
Supply curve firm 1 (from MC) Supply curve firm 2 Pick any price Sum of individual firms’ supply Repeat… The market supply curve
Subtract q and R from x to get E:
Demand
p1
x1
p1
Res
ourc
e st
ock
R1
1
E1
p1
Ei(p) := xi(p) – qi(p) – Ri
p1
Supplyq1
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Equilibrium in terms of Excess Demand
Equilibrium is characterised by a price vector p* 0 such that:
For every good i:
Ei(p*) 0
For each good i that has a positive price in equilibrium (i.e. if pi
* > 0):Ei(p*) = 0
If this is violated, then somebody, somewhere isn't maximising…
The materials balance condition (dressed up a bit)
You can only have excess supply of a good in equilibrium if the price of that good is 0
Using E to find the equilibrium
Five steps to the equilibrium allocation
1. From technology compute firms’ net output functions and profits
2. From property rights compute household incomes and thus household demands
3. Aggregate the xs and qs and use x, q, R to compute E
4. Find p* as a solution to the system of E functions
5. Plug p* into demand functions and net output functions to get the allocation
But this raises some questions about step 4
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Two fundamental properties…
Walras’ Law. For any price p:n
pi Ei(p) = 0i = 1
You only have to work with n-1 (rather than n) equations
Homogeneity of degree 0. For anyprice p and any t > 0 :
Ei(tp) = Ei(p)
You can normalise the prices by any positive number
Reminder: these hold for any competitive allocation, not just equilibrium
Price normalisation
We may need to convert from n numbers p1, p2,…pn to n1 relative prices The precise method is essentially arbitrary The choice of method depends on the purpose of your model It can be done in a variety of ways:
You could divide by
to give a
a numéraire
standard value system
pn
neat set of n-1 prices
plabourpMarsBar
“Marxian” theory of valueMars bar theory of valueset of prices that sum to 1
This method might seem strange
But it has a nice property
The set of all normalised prices is convex and compact
n
pii=1
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Normalised prices, n = 2
(1,0)
(0,1)
J={p: p0, p1+p2 = 1}
•(0, 0.25)
(0.75, 0) p1
p2 Purple line: set of normalised prices
Point on the line: the price vector (0,75, 0.25)
The existence problem
Imagine a rule that moves prices in direction of excess demand:• “if Ei >0, increase pi” • “if Ei <0 and pi >0, decrease pi”• An example of this under “stability” below
This rule uses the E-functions to map the set of prices into itself
An equilibrium exists if this map has a “fixed point” • a p* that is mapped into itself?
To find the conditions for this, use normalised prices• p J• J is a compact, convex set• So the mapping has a fixed point
We can examine this in the special case n = 2 • In this case normalisation implies that p2 1 p1
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Existence of equilibrium?
E10
1p1
(a) E-functions are: continuous, bounded below
(b) No equilibrium price where E crosses the axis
(c) E never crosses the axis
E0
1p1
E10
p1
(c)
(a)
(b)
p1*
Multiple equilibria
E10
1
p1 (a) Three equilibrium prices
(b) Suppose there were more of resource 1
(c) Suppose there were less of resource 1
E0
1
p1
E10
p1
(c)
(a)
(b)
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Adjustment and stability
Adjust prices according to sign of Ei:• If Ei > 0 then increase pi
• If Ei < 0 and pi > 0 then decrease pi
A linear tâtonnement adjustment mechanism:
Define distance d between p(t) and equilibrium p*
Given WARP, d falls with t under tâtonnement
Globally stable…
0
1
E1
Excess supply
Excess demand
E1(0)
p1(0)
E1(0)
p1(0)
p1
If E satisfies WARP thenthe system must converge…
Start with a very high price
Yields excess supply Under tâtonnement price falls
Start instead with a low price
Under tâtonnement price rises
Yields excess demand
p1*
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Not globally stable…
0
1
E1
Excess supply
Excess demand
p1
Start with a very high price
…now try a (slightly) low price
Start again with very low price
Check the “middle” crossing
…now try a (slightly) high price
Here WARP does not hold
Two locally stable equilibria
One unstable
Decentralisation again
A: The attainable setx2
x10
A = {x: x q+R, (q) 0}
purple line: prices B: The “Better-than-x* ” set
A
B
p1p2
B = {hxh: Uh(xh) Uh(x*h)}
Decentralisation if A, B are convex
x* maximises income over A
x* minimises expenditure over B
x*
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A non-convex technology
inputou
tput
The case with 1 firm
Rescaled case of 2 firms,
… 4 , 8 , 16
A: Limit set of averaging process
B: The “Better-than” set
A
• q*
q'B
Limiting attainable set is convex
Equilibrium q* is sustained by a mixture of firms at q° and q'
“separating” prices and equilibrium
q°
Non-convex preferences
x1
x2
The case with 1 person
Rescaled case of 2 persons,
B: better-than set, continuum of consumers
A: the attainable set
A
“separating” prices and equilibrium
Limiting better-than set is convex
Equilibrium x* is sustained by a mixture of consumers at x° and x'
• x*
x'
x°
B
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UNCERTAINTY AND RISK:Lectures 13 - 15
Quantitiesx scalar payoff under state x vector payoff under state Ri
resource stock of good i
Prices and incomepi price of good i
y money income
certainty-equivalent income
L loss
insurance premium
FunctionsU utility function
u felicity function
Other
state of the world set of all states of the world
P prospect
probability of state proportionate bond holding
r rate of return
E expectation
absolute risk aversion
relative risk aversion
Concepts state-of-the-world pay-off (outcome) x X prospects {x: }
ex ante before the realisation
ex post after the realisation
time
The ex-ante view
The ex-post view
The "moment of truth"
The time line
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The state-space diagram: #
xBLUE
xREDO
Consumption space under uncertainty: 2 states
P0: a prospect in the 1-good 2-state case
P0
payoff if RED occurs
45°
Payoffs: components of a prospect in the 2-state case
Diagonal: no equivalent in choice under certainty
The state-space diagram: #=3
The idea generalises: here we have 3 states
xBLUE
O
= {RED,BLUE,GREEN}
•P0
A prospect in the 1-good 3-state case
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Ranking prospects
xBLUE
xREDO
Greed: Prospect P1 is preferred to P0
Contours of the preference map
P1
P0
Implications of Continuity
xBLUE
xREDO
“Holes” in IC: preferences would violate continuity
P0
Remove holes to impose continuity
P0: an arbitrary prospect
E
Find point E by continuity
Income is the certainty equivalent of P0
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Reinterpret quasiconcavity
xBLUE
xREDO
Take an arbitrary prospect P0
Given continuous indifference curves…
P0
E
…find the certainty-equivalent prospect E
Points in the interior of line P0E represent mixtures of P0 and E
If U strictly quasiconcave P1 is preferred to P0
P1
A change in perception
xBLUE
xREDO
The prospect P0 and certainty-equivalent prospect E (as before)
Suppose RED begins to seem less likely
P0
P1
E
Then prospect P1 (not P0) appears equivalent to E
Green curves: ICs after the change
This alters the slope of the ICs
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The independence axiom
Let P(z) and P′(z) be any two distinct prospects such that the payoff in state-of-the-world is z• x = x′ = z
If U(P(z)) ≥ U(P′(z)) for some z then U(P(z)) ≥ U(P′(z)) for all z
One and only one state-of-the-world can occur• So, assume that the payoff in one state is fixed for all prospects• Level at which payoff is fixed has no bearing on the orderings over
prospects where payoffs differ in other states of the world
We can see this by partitioning the state space for > 2
Independence axiom: illustration
A case with 3 states-of-the-world
Compare prospects with the same payoff under GREEN
Ordering of these prospects should not depend on the size of the payoff under GREEN
xBLUE
O
What if we compare all of these points…?
Or all of these points…?
Or all of these?
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The “revealed likelihood” axiom Let x and x′ be two payoffs such that x is weakly preferred to x′
Let 0 and 1 be any two subsets of
Define two prospects:
• P0 := {x′ if 0 and x if 0}
• P1 := {x′ if 1 and x if 1}
If U(P1) ≥ U(P0) for some such x and x′ then U(P1) ≥ U(P0) for all such x and x′
Induces a consistent pattern over subsets of states-of-the-world
Revealed likelihood: example
1 apple ≽ 1 banana1 cherry ≽ 1 date
apple appleapple
apple
applebanana banana
apple apple appleapple bananabanana
bananaP2:P1:
States of the world (only one colour will occur)
Assume preferences over fruit
Consider these two prospects
Choose a prospect: P1 or P2?
Another two prospects
Is your choice between P3 and P4
the same as between P1 and P2?
cherry cherrycherry
cherry
cherrydate date
cherry cherry cherrycherry datedate
dateP4:P3:
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A key resultA result that is central to the analysis of uncertainty
Introducing the three new axioms:• State irrelevance
• Independence
• Revealed likelihood
…implies that preferences must be representable in the form of a von Neumann-Morgenstern utility function:
ux
Alternatively, write as Eux• “expectation” uses the numbers to weight payoff evaluations ux
Implications of vNM structure (1)
xBLUE
xREDO
Slope where it crosses the 45º ray?
A typical IC
Ratio RED/BLUE : from vNM structure
So all ICs have same slope on 45º ray
RED– _____BLUE
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Implications of vNM structure (2)
xBLUE
xREDO
RED– _____
BLUE
P0: given income prospectvNM structure: slope is given
Ex
Mean income, Ex
P0
P1
P
Extend line through P0 and P to P1
By quasiconcavity U( ) U(P0)
–
April 2018
Risk aversion and concavity of u
Use the interpretation of risk aversion as quasiconcavity
If individual is risk averse then U( ) U(P0)
Given the vNM structure…• u(Ex) REDu(xRED) + BLUEu(xBLUE)• u(REDxRED+BLUExBLUE) REDu(xRED) + BLUEu(xBLUE)
So the function u is concave
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The “felicity” function
u
xxBLUE xRED
If u is strictly concave then person is risk averse
If u is a straight line then person is risk-neutral
xBLUE, xRED: payoffs in states BLUE and RED
Diagram plots utility level (u) against payoffs (x)
If u is strictly convex then person is a risk lover
u of the average of xBLUE
and xRED higher than the expected u of xBLUE and of xRED
u of the average of xBLUE and xRED equals the expected u of xBLUE and of xRED
Attitudes to risk
u(x)
xBLUExxREDEx
Risk-loving
u(x)
xBLUExxREDEx
Risk-neutral
u(x)
xBLUExxREDEx
Risk-averse
Shape of u associated with risk attitude
Neutrality: will just accept a fair gamble
Aversion: rejects some better-than-fair gambles
Loving: accepts some unfair gambles
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Risk premium and risk aversion
xBLUE
xREDO
RED– _____
BLUE
: certainty equivalent income
P0: given income prospect
Slope gives probability ratio
Ex
Ex: mean income
Orange gap: the risk premium
P0
P
Risk premium:
Amount that amount you
would sacrifice to eliminate
the risk
Useful additional way of
characterising risk attitude
–
Risk premium: an example
u
u(x)
xBLUE
xxRED
u(xBLUE)
u(xRED)
Ex
u(Ex)
Eu(x )
Expected payoff and the utility of expected payoff
Expected utility and the certainty-equivalent
The risk premium again
Utility values of two payoffs
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Change the u-function
u
xBLUExxRED
u(xBLUE)
u(xRED)
Ex
Take u-function and distribution of x as before
Now make the u-function “flatter”
u(xBLUE)
Making the u-function less curved reduces the risk premium Ex
…and vice versa
More of this later
Absolute and relative risk aversionDefine absolute and relative risk aversion for scalar payoffs
uxx(x) uxx(x)(x) := ; (x) := x
ux(x) ux(x)• For risk-averse individuals • For risk-neutral individuals
independent of scale and origin of u• can see this from the definitions• are two different ways of capturing “curvature” of u
The definitions are linked:
(x) = x (x)
d(x) d(x) = (x) + x dx dx
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Special cases: CARA and CRRA1. Constant Absolute Risk Aversion
• Assume that (x) = for all x
• Felicity function must take the form
1 u(x) = ex
2. Constant Relative Risk Aversion
• Assume that (x) = for all x
• Felicity function must take the form
1 u(x) = x1
1
Each induces a distinctive pattern of indifference curves…
Constant Absolute Risk AversionCase where = ½Slope of IC is same along 45° ray (standard vNM)
For CARA slope of IC is same along any 45° linexBLUE
xREDO
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Constant Relative Risk AversionCase where = 2Slope of IC is same along 45° ray (standard vNM)For CRRA slope of IC is same along any ray
ICs are homotheticxBLUE
xREDO
Lotteries
Consider lottery as a particular type of uncertain prospect
Take an explicit probability model
Assume a finite number of states-of-the-world
Associated with each state are:• A known payoff x ,• A known probability ≥ 0
Lottery is probability distribution over the “prizes” x, =1,2,…,• The probability distribution is just the vector := (,,…,)
• Of course, + +…+ = 1
What about preferences?
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The probability diagram: #=2
BLUE
RED (1,0)
(0,1)
interior of line: cases where 0 < < 1
Vertical: probability of state BLUEEndpoints: cases of perfect certainty
Horizontal: probability of state RED
Marked point: the case (0.75, 0.25)
•(0, 0.25)
(0.75, 0)
Only points on the purple line make sense
This is an 1-dimensional example of a simplex
The probability diagram: #=3
0
BLUE
RED
Third axis corresponds to probability of state GREEN
(1,0,0)
(0,0,1)
(0,1,0)
Vertices of triangle: three cases of perfect certainty
Interior of triangle: cases where 0 < < 1
•(0, 0, 0.25)
(0.5, 0, 0)
(0, 0.25 , 0)
Marked point: the case (0.5, 0.25, 0.25)
Only points on the purple triangle make sense,
This is a 2-dimensional example of a simplex
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Preferences over lotteries Take probability distributions as objects of choice
• lotteries °, ', ",…
Each lottery has same payoff structure• state-of-the-world has payoff x• probability ° or ' or " … depending on which lottery
Axioms of preference over lotteries • Transitivity over lotteries
• If °≽' and ' ≽ " then °≽"
• Independence of lotteries• If °≽ ' and (0,1) then ° ]" ≽ ' ] "
• Continuity over lotteries• If °≻'≻" then there are numbers and such that• ° ]" ≻ ' and ' ≻ ° ]"
Basic result Take the axioms transitivity, independence, continuity Imply that preferences must be representable in the form of a
von Neumann-Morgenstern utility function:
ux
or equivalently:
where ux
So we can also see the EU model as a weighted sum of s
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-indifference curves
ICs over probabilities: straight lines
Increase in the size of BLUE increases slope
(1,0,0)
(0,0,1)
(0,1,0) .
Trade
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Contingent goods: equilibrium trade
Oa
Ob
xREDa
xREDb
xBLUEb
xBLUEa
1st diagonal: certainty line for Alf Alf's indifference curves (Cyan)
2nd diagonal: certainty line for Bill
Bill's indifference curves (Green)
•
• Endowment point (top left)
Red line: eqm prices & allocation
Trade: problems
Do all these markets exist? • If there are states-of-the-world…
• …there are n of contingent goods
• Could be a huge number
Consider introduction of financial assets
Take a particularly simple form of asset:• a “contingent security”
• pays $1 if state occurs
Can we use this to simplify the problem?
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Attainable set: buying a risky asset
xBLUE
xRED
P
P0
y
y_
_ _
A
P : endowment
P0: if all resources put into bonds
Green area: all points belong to A
Can you sell bonds to others?
Can you borrow to buy bonds?
Bottom right: If loan shark willing to finance
[1+rº]y_
[1+r' ]y_
y+r′, y+r_ _
[1+r′ ]y, [1+r]y_ _
Attainable set: insurance
xBLUE
xRED
Py
y_
_ _
A
P0: endowment
P : Full insurance at premium
Green area: all points belong to A
Can you overinsure?
Can you bet on your loss?
P0y0 – L
y0
L –
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A
1
5
7
4
6
3
2
Consumer choice with a variety of financial assets
xBLUE
xRED
Payoff if all in cash (1)
Payoff if all in bond 2
Payoff if all in bond 3, 4, 5,…
Lines joining: payoffs from mixtures
Green area: attainable set
P*: the optimum
5
4
P*
only bonds 4 and 5 used at the optimum
Problem and its solution
But corner solutions may also make sense…
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A
Consumer choice: safe and risky assets
xBLUE
xRED
y
y_
_
P*
P0
A: attainable set, portfolio problem
_ P
P : equilibrium -- playing safe
P0: equilibrium, "plunging"
P*: equilibrium, mixed portfolio
Results (1)Will the agent take a risk?
Can we rule out playing safe?
Consider utility in the neighbourhood of = 0
Eu( + r) ———— | = uy( ) E r
|
uy is positive
So, if expected return on bonds is positive, agent will increase utility by moving away from = 0
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Results (2) Examine the effect on * of changing a parameter
• take the FOC for an interior solution E (ruy( + *r)) = 0
• differentiate this equation w.r.t. the parameter• for example differentiate w.r.t. endowment (initial wealth)
E (ruyy( + *r)) + E (r2 uyy( + *r)) */ = 0
* E (ruyy( + *r)) —— = – ———————— E (r2 uyy( + * r))
• denominator is unambiguously negative
• to sign the numerator we need to impose more structure
• if ARA is decreasing then numerator is positive
Theorem: If an individual has a vNM utility function with DARA and holds a positive amount of the risky asset then the amount invested in the risky asset will increase as initial wealth increases
A
A: Attainable set, portfolio problem
An increase in endowment
P*
xBLUE
xRED
y
y_
_ P**
o
y+_
y+_
ICs: DARA Preferences
P*: Equilibrium
Arrow: increase in endowment Light dotted line: locus of constant
P**: New equilibrium
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xRED
A
A rightward shift in the distribution
xBLUE
y
y_
_
P**
P*
P0
o
A; Attainable set, portfolio problem
_ P
ICs: DARA Preferences
P*: Equilibrium
Arrow: change in distribution
Light dotted line: locus of constant
P**: new equilibrium
A
An increase in spread
xBLUE
xRED
y
y_
_
P*
P0
A; Attainable set, portfolio problem
_ P
P*: equilibrium, given preferences Arrow: Increase r′, reduce r
y+*r′, y+*r_ _
P* stays put So must have reducedYou don’t need DARA for this
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WELFARE:Lectures 16 - 19
Quantitiesqi
f net output by f of i
xih consumption by h of i
Rih ownership by h of resource i
Prices and incomepi price of good i
yh money income of h
Th tax revenue raised from h
loss
Functions constitution
Ch cost function of h
Uh utility function of h
Vh indirect utility function of h
vh utility of h as function of f production function of firm f
W social welfare function
social evaluation function
Other social state
set of all social states
utility possibility set
Social objectives Two dimensions of social objectives
objective 1
Set of feasible social states
A social preference map?
Assume we know the set of all social states
How can we draw a social preference map?
Can it be related to individual preferences?
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Elements of a constitution
Social states • can incorporate all sorts of information: • economic allocations, • political rights, etc
Individual (extended) preferences over • ≽ ' means that person h thinks state is at least as good
as state '
An aggregation rule for the preferences so as to underpin the constitution• A function defined on individual (extended) preferences
The social ordering and the constitutionWhere does this ordering come from?
Presumably from individuals' orderings over • assumes that social values are individualistic
Define a profile of preferences as• a list of orderings, one for each member of society
• (≽ , ≽ , ≽ , …)
The constitution is an aggregation function • defined on a set of profiles
• yields an ordering ≽
So the social ordering is ≽ = (≽ , ≽ , ≽ ,…)
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Axioms and a resultUniversality
• should be defined for all profiles of preferences
Pareto Unanimity• if all consider that is better than ', then the social ordering should rank
as better than '
Independence of Irrelevant Alternatives• if two profiles are identical over a subset of then the derived social
orderings should also be identical over this subset
Non-Dictatorship• no one person alone can determine the social ordering
Kenneth Arrow’s Theorem:
There is no constitution satisfying these axioms
Relaxing universalityCould it be that the universal domain criterion is just
too demanding?Should we insist on coping with any and every set of
preferences, no matter how bizarre?Perhaps imposing restrictions on admissible
preferences might avoid the Arrow impossibility resultHowever, we run into trouble even with very simple
versions of social states
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Alf, Bill, Charlie decide
' "
pref
eren
ce
defence spending
AlfBill
Charlie
1-dimensional social statesScaling of axes is arbitraryThree possible states
Views about defence spending
Each individual has dramatically different views
But all three sets of preferences are “single peaked”
How do they decide?
′ ≻ ?
Alf Bill Charlie Verdict
′′ ≻ ′?
≻ ′′?
Bill
Alf, Bill, Charlie decide (2)
pref
eren
ce
defence spending
Bill
' "
Alf
Charlie
Same states as before
Same preferences as before
Now Bill changes his mind
Now one set of preferences is no longer “single peaked”
How do they decide?
′ ≻ ?
Alf Bill Charlie Verdict
′′ ≻ ′?
≻ ′′?
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Alternative voting systems…Relaxing IIA involves an approach that modifies the
type of “aggregation rule”Simple majority voting may make too little use of
information about individual orderings or preferences Here are some alternatives:
• de Borda (weighted voting)• Single transferable vote• Simple elimination voting
None of these is intrinsically ideal • Consider the results produced by third example
The IOC Decision Process
An elimination process 1997: Appears to give an orderly convergence
• Athens is preferred to Rome irrespective of the presence of other alternatives
1993: Violates IIA• Ordering of Sydney, Peking depends on whether other alternatives are
present
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A definition of efficiency
The basis for evaluating social states: vh()
the utility level enjoyed by person h under social state
A social state is Pareto superior to state ' if:1. For all h: vh() vh(')2. For some h: vh() > vh(')
Note the similarity with the concept of blocking by a coalition
“feasibility” could be determined in terms of the usual economic criteria
A social state is Paretoefficient if:1. It is feasible 2. No other feasible state
is Pareto superior to
Derive the utility possibility set From the attainable set A
AA
(x1a, x2
a)(x1
b, x2b)
a)
2 )a=Ua(x1
a, x2a)
b=Ub(x1b, x2
b)
…take an allocation
Evaluate utility for each agent
Repeat to get utility possibility set
a
b
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Finding an efficient allocation
max L( [x ], [q], ) :=
U1(x1) + hh [Uh(x h) h] f f f (q f) + i i[qi + Ri xi]
where xi = h xih, qi = f qi
f
Differentiate L w.r.t xih. If xi
h, xjh positive at the optimum:
hUih(xh) = i hUj
h(xh) = j
Differentiate L w.r.t qif. If qi
f , qjf nonzero at the optimum:
f if (qf) = i f j
f (qf) = j
Interpreting the FOC
Uih(xh) i
———— = —Uj
h (xh) j
for every firm:MRT = shadow price ratio
for every household: MRS = shadow price ratio
From the FOCs for any household h and goods i and j:
if(qf) i
———— = —j
f (qf) j
From the FOCs for any firm fand goods i and j:
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Ob
Oa
x1b
x1a
x2a
x2b
Efficiency in an Exchange Economy
Purple: the contract curve
Cyan: Alf’s indifference curves
Green: Bill’s indifference curves
Set of efficient allocations is the contract curve
Includes cases where Alf or Bill is very poor
Allocations where MRS12
a = MRS12b
253
Efficiency with production
Contours: h’s indifference curves
0
slope of dashed line: MRS
h’s consumption in the efficient allocation
MRS = MRT at efficient point
x2h
x1h
xh^
Dark green: firm f’s technology set
f’s net output in the efficient allocation
q2f
q1f
qf^
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Two welfare theorems
Welfare theorem 1• Assume a competitive equilibrium
• What is its efficiency property?
THEOREM: if all consumers are greedy and there are no externalities then a competitive equilibrium is efficient
Welfare theorem 2• Pick any Pareto-efficient allocation
• Can we find a property distribution d so that this allocation is a CE for d?
THEOREM: if, in addition to conditions for theorem 1, there are no non-convexities then an arbitrary PE allocation be supported by a competitive equilibrium
255
Ob
Oa
x1b
x1a
x2a
x2b
Supporting a PE allocation
purple curve: contract curve
Support allocation by a CE
This needs adjustment of the initial endowment
Lump-sum transfers may be tricky to implement
Allocations where MRS12
a = MRS12b
red dot: an efficient allocation
pink line: supporting prices
[x]^
[R]
green dot: property distribution
red arrow: lump-sum transfer
p1p2
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Individual household behaviour
Household h’s indifference curves
0
Supporting price ratio = MRS
:h’s consumption in the efficient allocation
h’s consumption in the allocation is utility-maximising for h
x2h
x1h
h’s consumption in the allocation is cost-minimising for h
xh^
p1p2
Supporting a PE allocation (production)
Green area: irm f’s technology set
0
Supporting price ratio = MRT
: f’s net output in the efficient allocation
p1p2
f’s net output in the allocation is profit-maximising for f
q2f
q1f
qf^
258
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Firm f makes “wrong” choice
q2f
0q1
f
Firm f ’s production function violates second theorem
Suppose we want to allocate to f
Introduce prices
f's chooses at those prices
p1p2
qf~
qf^
Big fixed-cost component to producing good 1
“market failure” once again
PE allocations – two issues
good
2
0good 1
Same production function
Implicit prices for MRS=MRT
Competitive outcome
Issue 1 – what characterises the PE?
Issue 2 – how to implement the PE
or at second red dot?
Is PE at first red dot…?
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Indecisiveness of PE
a
b Construct set as before
Dots on boundary: 2 efficient points
Boundary points cannot be compared on efficiency grounds
First shading: Points superior to
v
v
Second shading: points superior to '
and ' cannot be compared on efficiency grounds
261
“Potential” Pareto superiority
Define to be potentially superior to ' if :• there is a * which is actually Pareto superior to '
• * is “accessible” from
To make use of this concept we need to define accessibility• use a tool from the theory of consumer welfare
CVh(' ): the monetary value the welfare change for person h…• …of a change from state ' to state • …valued in terms of the prices at
CVh > 0 means a welfare gain; CVh < 0 a welfare loss
THEOREM: a necessary and sufficient condition for to be potentially superior to ' is
h CVh(' ) > 0
262
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Applying potential superiority
a
is not superior to ' and ' is not superior to
* is superior to
There could be lots of points accessible by lump-sum transfers
' is potentially superior to
Blue shading: points accessible from '
v
b
v
v*
263
Re-examine potential superiority
a
b
v
v
points accessible from
points accessible from
Blue shading: process from to ', as before
Yellow shading: process in reverse from ' to
Combine the two
is potentially superior to and …
is potentially superior to !
264
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Production externalityOne firm influences another’s production conditions
• affects other firms’ cost curves
• if firm f’s output produces an externality
• production & cost function of firm g has f’s output as a parameter
When f produces good 1 it causes pollution• could affect other firms g = 1, 2, …, f – 1, f + 1, …, nf
• the more f produces good 1, the greater the damage to g
How much damage?• consider the impact of pollution on firm g
• will enter the production function g()Use the firm’s transformation curve
Externality: Production possibilities
low emissions by firm f
q1g
q2g
high emissions by firm f
Firm g affected by others' output of good 1 Φ ;
If g() = 0 an increase in negative externality results in g() > 0
Production possibilities, firm g
g() < 0
g() = 0g() > 0 Production possibilities, if firm
f’s emissions increase
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Find conditions for efficiency Problem is to maximise U1(x1) subject to:
, 1, … ,
Φ ; 0, 1, … ,, 1, …
The LagrangianΣ · Σ Φ · Σ
FOCs for an interior maximum. For all h, f: , 1, 2, … ,
ΦΦ ·
Φ , 2, … ,
Interpretation
Evaluate marginal impact of f’s output using good 2 as numeraire:
≔1
Φ
Φ ·
From first of the FOCs:
=
Use the definition of . Then the other FOCs giveΦ
Φ
This is the efficiency criterion:• instead of the condition “MRT=shadow price ratio” • we have a modified marginal rule
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Efficiency with production externality
1 1— = —2 2
1 1— = — + externality 2 2
q1`
q2f
f
qf^
qf~
Production possibilities
Taking account of externality
If externality is ignored
Produce less of good 1 for efficiency
Production externality: policy
Take the modified FOC
Rearrange:
Introduce market prices:
A tax/subsidy:
The term t “corrects” the market prices• for a negative externality we have t > 0 (a tax)
• for a positive externality we have t < 0 (a subsidy)
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Externality: a private solution? Efficient outcome through individual initiative?
Assume (1) just two firms (2) just two goods• assumption (1) may be important
• assumptions (2) is unimportant
Firm 1’s output of good 1 imposes costs on firm 2
Full information:• each firm knows the other’s production function
• externality is common knowledge
• activity can be monitored
• communication is costless
Firm 2 (victim) has an interest in communicating• does this by setting up a financial incentive for firm 1
• how should this be structured?
The victim’s problem and solutionFirm 2 (the victim) offers firm 1 a side-payment
• accounted for in the computation of profits
• a is a control variable for firm 2 in the maximisation problem:
max,
Φ q2 ;
FOCs: Φ q2 ; 0
1dΦ q2 ;
ddd
0
Using the definition of the externality and rearranging:
1 Φ q2 ;dd
0
dd
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The perpetrator’s problem and solutionFor firm 2’s plan to work, firm 1 has to know about it
• realises that bribe is conditional on a variable under its own control
• has the maximisation problem:
max Φ q1
FOCs: dd
Φ q1 0
Φ q1 0, 2, … ,
Substituting in from firm 2’s solution and rearranging:Φ q1
Φ q1
This is exactly the condition for efficiency!
Private solution: result Bribe function has internalised the externality
• Firm 2 conditions side-payment on observable output of good 1
• Firm 1’s responds rationally to the side-payment
FOC conditions same as before• Private solution induces an efficient allocation
• Implements the same allocation as the Pigovian tax
• But no external guidance is required
It should be independent of where the law places the responsibility for the pollution (Coase’s result)
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x1 0
x2
Price distortion: efficiency loss
Production possibilities
An efficient allocation
Some other inefficient allocation
•x
•x*
p*
At x* producers and consumers face same prices At x producers and consumers face different prices
Price "wedge" forced by the distortion
Waste measurement: a method
To measure loss we use a reference point
Take this as competitive equilibrium• defines a set of reference prices
Quantify the effect of a notional price change:• pi := pi – pi*• This is [actual price of i] – [reference price of i]
Evaluate the equivalent variation for household h :• EVh = Ch(p*,h) – Ch(p, h) – [y*h – yh]• This is (consumer costs) – (income)
Aggregate over agents to get a measure of loss,
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x1 0
x2
If producer prices constant…
Production possibilities
Reference allocation and prices
Actual allocation and prices
•x
•x*
p*
Measure cost in terms of good 2
Losses to consumers are C(p*, ) C(p, )
Cost of at prices p
C(p, )
Cost of at prices p*
C(p*, )
Change in valuation of output
p
is difference between |C(p*, ) C(p, )| and
Efficiency loss: policy
p1
compensateddemand curve
p1
p1*
x1h
x1h
x1*
Equilibrium price and quantityThe tax raises consumer price…
…and reduces demand
Gain to the government
Loss to the consumer
Waste
A model of a commodity tax
Waste given by size of triangle
Sum over h to get total waste
Known as deadweight loss of tax
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Tax: computation of waste
The tax imposed on good 1 forces a price wedge
• p1 = tp1* > 0 where p1
* is the untaxed price of the good
h’s demand for good 1 is lower with the tax:
• x1** = x1
* x1h and x1
h < 0
Revenue raised by government from h:
• Th = tp1* x1
**= x1**p1 > 0
Absolute size of loss to h is
• h= ∫ x1h dp1 ≈ x1
** p1 − ½ x1hp1 = Th − ½ t p1
* x1h > Th
Use the definition of elasticity
• := p1x1h / x1
hp1< 0
Net loss from tax (for h) is
• h = h− Th = − ½tp1* x1
h = − ½tp1x1** = − ½t Th
Overall net loss from tax is ½ |tT
p1
compensateddemand curve
p1
p1*
x1h
x1h
Size of waste depends upon elasticity
low: relatively small waste
high: relatively large waste
Redraw previous example
p1
p1
p1*
x1h
x1h
p1
p1
p1*
x1h
x1h
x1h
p1
p1
x1h
p1*
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Using a SWF
a
b
: the utility-possibility set
Red dot: social-welfare optimum?
Social welfare contours
W defined on utility levels
Not on orderings
Imposes several restrictions…
..and raises several questions
W(a, b,... )
•
“Veil of ignorance”: formalisation
Individualistic welfare:W(1, 2, 3, ...)
use theory of choice under uncertainty to find shape of W
vN-M form of utility function: u(x)
Equivalently:
probability assigned to u : cardinal utility function,
independent of utility payoff in state
A suitable assumption about “probabilities”?
nh1
W = — hnh h=1
welfare is expected utility from a "lottery on identity“
Replace by set of identities {1,2,..., nh}:
h hh
An additive form of the welfare function
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Questions about “equal ignorance”
h
identity
|
nhh|
1
|
2
|
3
|
Construct a lottery on identity
Light blue: “equal ignorance” assumption
Pink: people know their identity with certainty
Dark blue: Intermediate case
The “equal ignorance” assumption: h = 1/nh
But is this appropriate?
Or should we assume that people know their identities with certainty?
Or is the "truth" somewhere between?
From an allocation to social welfare
From the attainable set...
AA
(x1a, x2
a)(x1
b, x2b)
...take an allocation
Evaluate utility for each agent
Plug into W to get social welfare
a)
2 )a=Ua(x1
a, x2a)
b=Ub(x1b, x2
b)
W(a, b)
Take the individualistic welfare modelW(1, 2, 3, ...)
Assume everyone is selfish: h = Uh(xh) , h=1,2,..., nh
Substitute in the above:W(U1(x1), U2(x2), U3(x3), ...)
What happens to welfare if we vary the allocation in A?
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Varying the allocation Differentiate w.r.t. xi
h : dh = Ui
h(xh) dxih
The effect on h if commodity i is changed
Sum over i: n
dh = Uih(xh) dxi
hi=1
The effect on h if all commodities are changed
Differentiate W with respect to h:nh
dW = Wh dh
h=1
Changes in utility change social welfare
Substitute for dh in the above:nh n
dW = Wh Uih(xh) dxi
h
h=1 i=1
So changes in allocation change welfare
The SWF maximum problem
First component of the problem: W(U1(x1), U2(x2), U3(x3), ...)
The objective function
Second component of the problem: nh(x) 0, xi = xi
hh=1
Feasibility constraint
The Social-welfare Lagrangian:nhW(U1(x1), U2(x2),...) − ( xh )
h=1
Constraint subsumes technological feasibility and materials balance
FOCs for an interior maximum:Wh (...) Ui
h(xh) − i(x) = 0From differentiating Lagrangianwith respect to xi
h
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Solution to SWF maximum problem
From FOCs: Ui
h(xh) Uiℓ(xℓ)
——— = ———Uj
h(xh) Ujℓ(xℓ)
MRS equated across all h
We’ve met this condition before - Pareto efficiency
Also from the FOCs: Wh Ui
h(xh) = Wℓ Uiℓ(xℓ)
social marginal utility of ice cream equated across all h
Relate marginal utility to prices:Ui
h(xh) = Vyhpi
This is valid if all consumers optimise
Substituting into the above:Wh Vy
h = Wℓ Vyℓ
At optimum the welfare value of $1 is equated across all h. Call this common value M
Differentiate the SWF w.r.t. {yh}:nh
dW = Wh dh
h=1
Social welfare, income, expenditure
nh
= M dyh
h=1
nh
= WhVyh dyh
h=1
Social welfare can be expressed as:W(U1(x1), U2(x2),...) = W(V1(p,y1), V2(p,y2),...)
SWF in terms of direct utility and in terms of indirect utility
Changes in utility and change social welfare related to income
Differentiate the SWF w.r.t. pi :nh
dW = WhVihdpi
h=1
.
Changes in utility and change social welfare related to pricesnh
= –WhVyh xi
hdpih=1
nh
= – M xihdpi
h=1
THEOREM: in the neighbourhood of a welfare optimum welfare changes are measured by changes in national income / national expenditure
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Income-distribution space: nh=2
Bill's
income
Alf'sincome
O
The income space: 2 persons
y: An income distribution
y
45°
Note the similarity with diagrams used in the analysis of uncertainty
Alf'sincome
Welfare contours
Eyya
yb
Ey
y
y: An arbitrary income distribution Contours of W Mirror image: swap identities
Pink diagonal: distributions with same mean
Anonymity implies symmetry of W
: Equally-distributed-equivalent
Ey is mean income
Principle of Transfers: Richer-to-poorer transfers increase welfare
Quasi-concavity of W implies that social welfare respects this principle
is income that, if received by all, would yield same level of soc welf as y
≔ is prop of income society
would give up to eliminate inequality
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Using the inequality-sensitive SWF Rearrange the last equation:
• = Ey [1 ]• “welfare = mean income [1 inequality]”
•
It makes sense to write W in the special additive formnh1W = — yh= E yh
nh h=1
• function is the social evaluation function
Also makes sense to take constant relative-inequality aversion:1y = —— y1 –
1 –
• is the index of inequality aversion• the larger is , the larger is I for any given income distribution
Social views: inequality aversion
½
yb
yaO
yb
yaO
yb
yaO
= 0: Indifference to inequality
= ½: Mild inequality aversionyb
yaO
= : Strong inequality aversion
= : Priority to poorest
“Benthamite” case ( = 0): nh
W= yh
h=1
Atkinson SWF, general case ( ):
nh
W = [yh]1 / [1 ]h=1
“Rawlsian” case ( ): W = min yh
h
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Social values and welfare optimum
ya
yb The income-possibility set Y
Welfare contours ( = ½)
Welfare contours ( = 0)
Welfare contours ( = )
Y derived from set A
Nonconvexity, asymmetry come from heterogeneity of households
y* maximises total income irrespective of distribution
y*** gives priority to equality; then maximises income subject to that
Y
y*
y***
y** y** trades off some income for greater equality
=
( = 0
= ½
Microeconomics in practice: Social welfare in the US
• Source: Current Population Surveys• Equivalised incomes, 2015 USD
35,000
40,000
45,000
50,000
55,000
60,000
65,000
70,000
75,000
80,000
1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015
EDE Income US: 1967-2015
