Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Revision Lecture 2

EC202: Microeconomic Principles IIEC202: Microeconomic Principles IIFrank Cowell Frank Cowell

May 2008May 2008

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Objectives of the lecture

Think more about method for long questionsThink more about method for long questions Look at a few CfDLook at a few CfD

4.12, 4.134.12, 4.13 5.15.1 (7.8)(7.8) 9.69.6

See how they illustrate methodSee how they illustrate method Connect these to past exam questionsConnect these to past exam questions

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Question Style – three types

1 Principles1 Principles reason on standard results and argumentsreason on standard results and arguments can use verbal and/or mathematical reasoningcan use verbal and/or mathematical reasoning

2 Model solving2 Model solving a standard frameworka standard framework you just turn the wheelsyou just turn the wheels

3 Model building3 Model building usually get guidance in the questionusually get guidance in the question longer question sometimes easier?longer question sometimes easier?

One type not necessarily “easier” or “harder” than another One type not necessarily “easier” or “harder” than another focus here on types 2 and 3focus here on types 2 and 3 examples throughout the “long” questions of recent exam papersexamples throughout the “long” questions of recent exam papers

A simple type 2 question –

Ex 5.1

A simple type 2 question –

Ex 5.1

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(1) Question

purpose: construct a simple model of household supply and examine how it works

method: build model up step-by-step through the question parts

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

0

Ex 5.1(1) Preference mapx2

x1

•0

•

1•

•

k

Shift the origin to (0, k)

Draw ICs homothetic to the shifted origin

indifference curves are “shifted” Cobb-Douglas k is min requirement of other goods. is share of budget of rice after an amount has been set aside to buy the min requirementrice

othe

r go

ods

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(2) Question

method: Work out the budget constraint. Use the utility function to set out the Lagrangean Find the FOCs for an interior solution Find the demand functions Use these to get household supply function

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(2) Budget constraint

Use good 2 as unit of value price of rice (good 1) is p price of all other goods (good 2) is 1

The consumer’s income is therefore:y := pR1 + R2

The budget constraint is

px1 + x2 y where y is given by the above

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(2) Lagrangean method

The Lagrangean is

log(x1) + [1– ]log(x2–k) + [ y – px1 – x2 ]

The FOC for an interior maximum are — – p

x1*

1–—— –

x2*–k

y – px1* – x2

*

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(2) Demand functions

From the FOC: px1

*— 1–

x2*k + ——

Adding these and using the budget constraint, we have

y k + 1/ Eliminating in the above:

x1

*— [y – k] p

x2*k + [1– ] y

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(2) Supply function

Supply of good 1 is given by

S(p) := R1 – x1*

Substituting in for y, we have S(p) [1– ]R1 – — [R2 – k]

p

Supply increases with price if R2 > k

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

0

Ex 5.1(2) Solutionx2

x1

k

Endowment

Budget constraint

rice

othe

r go

ods

x*

x2

p

S

R•

•

Optimal consumption

Supply of rice

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1(3)

Let c be the amount of the ration If k + [1– ] y c nothing changes from previous case Otherwise

px1 + c = y so that R2 – c

x1 = R1 +——— p

c – R2 S(p) ———

p

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

0

Ex 5.1(3) Modified solutionx2

x1

k

Original solution

rice

othe

r go

ods

x*

x2

S

R•

•

Generous ration

Severe ration

x**•

Generous ration has no effect

Severe ration on other goods affects supply of rice

c

c

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 5.1: Points to remember Use diagram to understand features of utility function Model supply as mirror image of demand Use diagram to see effect of ration Connection to exams

see 2005 q3 …it’s the same model!

A “meatier” type 2 question – Ex

4.12, 4.13

A “meatier” type 2 question – Ex

4.12, 4.13

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Question

purposepurpose: to derive solution and response functions for quasilinear : to derive solution and response functions for quasilinear preferencespreferences

methodmethod: substitution of budget constraint into utility function and then : substitution of budget constraint into utility function and then simple maximisationsimple maximisation

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Preliminary

First steps are as follows:First steps are as follows:

Sketch indifference curvesSketch indifference curves Straightforward – parabolic contoursStraightforward – parabolic contours

Write down budget constraintWrite down budget constraint Straightforward – fixed-income caseStraightforward – fixed-income case

Set out optimisation problemSet out optimisation problem

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

0

0 1 2

Ex 4.12(1) Indifference curves

x1

x2

Could have x2 = 0

Could have x2 = 0

Slope is vertical here

Slope is vertical here

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Budget constraint, FOC

Budget constraint:Budget constraint: Substitute this into the utility Substitute this into the utility

function:function: We get the objective function:We get the objective function:

FOC for an interior solution:FOC for an interior solution:

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Using the FOC

Remember that person might consume zero of commodity 2Remember that person might consume zero of commodity 2 consider two cases consider two cases

Case 1: Case 1: xx22** > 0 > 0

From the FOC: From the FOC:

But, to make sense this case requires:But, to make sense this case requires:

Case 2: Case 2: xx22** = 0 = 0

We get We get xx11** from the budget constraint from the budget constraint

xx11** = = yy / / pp11

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Demand functions

We can summarise the optimal demands for We can summarise the optimal demands for the two goods thus the two goods thus

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Indirect utility function

Get maximised utility by substituting Get maximised utility by substituting xx** into the utility into the utility function function VV((pp11, , pp22, , yy) = ) = UU((xx11

**, , xx22**) )

= = UU((DD11((pp11, , pp22, , yy), ), DD22((pp11, , pp22, , yy))))

Case 1: Case 1: pp11 > >pp11

Case 2: Case 2: pp11 ≤ ≤pp11

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(1) Cost function

Get cost function (expenditure function) from the indirect Get cost function (expenditure function) from the indirect utility functionutility function maximised utility is maximised utility is = = VV((pp11, , pp22, , yy))

invert this to get invert this to get yy = = CC((pp11, , pp22, , ))

Case 1: Case 1: pp11 > >pp11

Case 2: Case 2: pp11 ≤ ≤pp11

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(2) Question

purposepurpose: to derive standard welfare concept: to derive standard welfare concept methodmethod: use part 1 and manipulate the indirect utility function : use part 1 and manipulate the indirect utility function

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(2) Compute CV Get compensating variation (1) from indirect utility functionGet compensating variation (1) from indirect utility function

before price change: before price change: = = VV((p1, , p2, , y)) after price change: after price change: = = VV((p1', , p2, , y − CV))

Equivalently (2) could use cost function directlyEquivalently (2) could use cost function directly CV = = CC((p1, , p2, , )) − CC((p1', , p2, , ))

In Case 1 above we haveIn Case 1 above we have

Rearranging, we find:Rearranging, we find:

EquivalentlyEquivalently

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12(3)

In case 1 we have In case 1 we have xx11** = [ = [½½ pp2 2 / / pp11]]22

So demand for good 1 has zero income effectSo demand for good 1 has zero income effect Therefore, in this case CV = CS = EVTherefore, in this case CV = CS = EV

Now look at the follow-up

question

Now look at the follow-up

question

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(2) Question

MethodMethod: : Find monopolist’s AR from consumer demand using Find monopolist’s AR from consumer demand using

answer to Ex 4.12. answer to Ex 4.12. Then use standard optimisation procedureThen use standard optimisation procedure

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(2) Monopoly profits

Aggregate demand over Aggregate demand over NN consumers using Exercise 4.12consumers using Exercise 4.12

Rearrange to get AR curve:Rearrange to get AR curve:

Total Revenue is:Total Revenue is: Profits are therefore:Profits are therefore:

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(2) Maximising profits

FOC (MC = MR) yields:FOC (MC = MR) yields:

So monopolist’s optimal output is:So monopolist’s optimal output is:

From AR curve, price at optimum is:From AR curve, price at optimum is:

Simplify this to:Simplify this to: (clearly price > MC)(clearly price > MC)

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(3) Question

MethodMethod: : Aggregate the CV for each consumer to define Aggregate the CV for each consumer to define LL. . Use marginal cost and monopolist’s equilibrium price to evaluate Use marginal cost and monopolist’s equilibrium price to evaluate LL

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(3) Evaluating loss Use definition of CV with

p1' = c:

Evaluate L at p1 = 2c:

Firm’s profits are:

Clearly L > profits

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(4) Question

MethodMethod: : Add bonus Add bonus BB into the expression for profits into the expression for profits Again use standard optimisation procedureAgain use standard optimisation procedure

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(4) Evaluating profits (again) Profits including bonus are:

Value of bonus is:

Use demand curve to express this in terms of q:

So profits can now be expressed as:

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.13(4) Evaluating profits (again) Take the expression for profits including bonus FOC for a maximum is again MR = MC:

Rearranging we get the value of optimal output for the regulated monopolist:

Use demand curve to find:

Clearly the regulated price = MC:

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 4.12 & 4.13: Points to note It’s always a good idea to sketch the indifference curvesIt’s always a good idea to sketch the indifference curves

in this case the sketch is revealing…in this case the sketch is revealing… ……because of the possible corner solutionbecause of the possible corner solution

A corner solution can sometimes just be handled as two A corner solution can sometimes just be handled as two separate casesseparate cases

There’s often more than one way of getting to a solutionThere’s often more than one way of getting to a solution in this case two equivalent derivations of CVin this case two equivalent derivations of CV

Aggregate welfare loss is found from individual CV Unregulated monopoly makes profits smaller than losses

to consumer Regulation causes monopoly to behave like competitive

firmNow for a

type 3 “model building” q

Now for a type 3 “model

building” q

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): Question

purposepurpose: to derive equilibrium prices and incomes as a function of endowment. : to derive equilibrium prices and incomes as a function of endowment. To show the limits to redistribution within the GE model for a alternative SWFs To show the limits to redistribution within the GE model for a alternative SWFs

methodmethod: find price-taking optimising demands for each of the two types, use : find price-taking optimising demands for each of the two types, use these to compute the excess demand function and solve for these to compute the excess demand function and solve for

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): budget constraints Use commodity 2 as numéraireUse commodity 2 as numéraire

price of good 1 is price of good 1 is price of good 2 is 1price of good 2 is 1

Evaluate incomes for the two types, given their resources:Evaluate incomes for the two types, given their resources: type type aa has endowment ( has endowment (30, 30, kk)) therefore therefore yyaa = 30= 30 + + kk type type bb has endowment ( has endowment (60, 210 60, 210 kk)) therefore therefore yybb = 60= 60 + [210 + [210 kk]]

Budget constraints for the two types are therefore:Budget constraints for the two types are therefore: xx11

aa + + xx22aa ≤ 30≤ 30 + + kk

xx11bb + + xx22

b b ≤ 60≤ 60 + [210 + [210 kk]]

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): optimisation

We could jump straight to a solutionWe could jump straight to a solution utility functions are simple…utility functions are simple… ……so we can draw on known resultsso we can draw on known results

Cobb-Douglas preferences implyCobb-Douglas preferences imply indifference curves do not touch the origin…indifference curves do not touch the origin… ……so we need consider only interior solutionsso we need consider only interior solutions also demand functions for the two commodities exhibit constant expenditure shares also demand functions for the two commodities exhibit constant expenditure shares

In this case (for type In this case (for type aa) ) coefficients of Cobb-Douglas are 2 and 1coefficients of Cobb-Douglas are 2 and 1 so expenditure shares are so expenditure shares are ⅔ ⅔ and and ⅓⅓ (and for (and for bb they will be they will be ⅓ ⅓ and and ⅔⅔ ) ) gives the optimal demands immediately… gives the optimal demands immediately…

Jump to “equilibrium

price”

Jump to “equilibrium

price”

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): optimisation, type a

The Lagrangean is:The Lagrangean is: 2log 2log xx11

aa + log + log xx22a a + + aa[[yyaa xx11

aa xx22aa ] ]

where where aa is the Lagrange multiplieris the Lagrange multiplier and and yyaa is 30 is 30+ + k k

FOC for an interior solutionFOC for an interior solution 2/2/xx11

aa aa = 0 = 0

1/1/xx22aa aa = 0 = 0

yyaa xx11aa xx22

aa= 0= 0

Eliminating Eliminating aa from these three equations, demands are from these three equations, demands are xx11

aa = = ⅔ ⅔ yya a / / xx22

aa = = ⅓ ⅓ yya a

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): optimisation, type b The Lagrangean is:The Lagrangean is:

log log xx11bb + 2log + 2log xx22

b b + + bb[[yybb xx11bb xx22

bb ] ] where where bb is the Lagrange multiplieris the Lagrange multiplier and and yybb is 60 is 60 + 210 + 210 kk

FOC for an interior solutionFOC for an interior solution 1/1/xx11

bb bb = 0 = 0

2/2/xx22bb bb = 0 = 0

yybb xx11bb xx22

bb= 0= 0

Eliminating Eliminating bb from these three equations, demands are from these three equations, demands are xx11

bb = = ⅓ ⅓ yyb b / / xx22

bb = = ⅔ ⅔yybb

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(1): equilibrium price Take demand equations for the two typesTake demand equations for the two types

substitute in the values for incomesubstitute in the values for income type-type-a a demand becomes demand becomes

type-type-b b demand becomes demand becomes

Excess demand for commodity 2:Excess demand for commodity 2: [10[10 ++ ⅓⅓kk]+[40]+[40+140+140 −− ⅔ ⅔kk]] −− 210210 which simplifies to 5which simplifies to 500 −− ⅓ ⅓kk −− 7700

Set excess demand to 0 for equilibrium:Set excess demand to 0 for equilibrium: equilibrium price must be:equilibrium price must be: = [= [212100 + + kk] / 15] / 1500

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(2): Question and solution

Incomes for the two types are resources:Incomes for the two types are resources: yyaa = 30= 30 + + kk yybb = 60= 60 + [210 + [210 kk]]

The The equilibrium price is:equilibrium price is: = [= [212100 + + kk] / 15] / 1500

So we can solve for incomes as:So we can solve for incomes as: yyaa = = [[212100 + 6 + 6kk] / 5] / 5 yybb = = [1[1474700 3 3kk] / 5 ] / 5

Equivalently we can write Equivalently we can write yyaa and and yybb in terms of in terms of as as yyaa = = 180180 210 210 yybb = = 420 420 90 90

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(3): Question

purposepurpose: to use the outcome of the GE model to plot the “income-possibility” set: to use the outcome of the GE model to plot the “income-possibility” set methodmethod: plot incomes corresponding to extremes of allocating commodity 2, : plot incomes corresponding to extremes of allocating commodity 2,

namely namely k k = 0 and = 0 and k k = 210. Then fill in the gaps. = 210. Then fill in the gaps.

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Income possibility set

ya

yb

0

200

300

100 200 300

100

• (42, 294)

• (294, 168)

incomes for k = 0incomes for k = 210

incomes for intermediate values of k

attainable set if income can be thrown away

yb = 315 ½ya

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(4): Question

purposepurpose: find a welfare optimum subject to the “income-possibility” set: find a welfare optimum subject to the “income-possibility” set methodmethod: plot contours for the function : plot contours for the function WW on the previous diagram. on the previous diagram.

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Welfare optimum: first case

ya

yb

0

200

300

100 200 300

100

income possibility setContours of W = log ya + log yb

Maximisation of W over income-possibility set

• W is maximised at

corner

incomes are (294, 168)

here k = 210

so optimum is where all of resource 2 is allocated to type a

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6(5): Question

purposepurpose: as in part 4: as in part 4 methodmethod: as in part 4: as in part 4

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Welfare optimum: second case

ya

yb

0

200

300

100 200 300

100

income possibility setContours of W = ya + yb

Maximisation of W over income-possibility set

• again W is maximised

at corner

…where k = 210

so optimum is where all of resource 2 is allocated to type a

Frank C

owell:

Frank C

owell: E

C202 M

icroeconomics

EC

202 Microeconom

ics

Ex 9.6: Points to note

Applying GE methods gives the feasible set Limits to redistribution

natural bounds on k asymmetric attainable set

Must take account of corners Get the same W-maximising solution

where society is averse to inequality where society is indifferent to inequality

Link to exam very similar to 2007 q5. but also check out 2005 q4… …same model, but you are asked to different tricks with it

More practice with GE – see CfD 7.8