0 Lecture 13 Neoclassical Model. 1 Economic Models Real economy is too complicated to understand...
Transcript of 0 Lecture 13 Neoclassical Model. 1 Economic Models Real economy is too complicated to understand...
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Lecture 13
Neoclassical Model
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Economic Models
Real economy is too complicated to understand
Built your own, simple economy Ingredients
PeopleGoods and technologiesInstitutions
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Microfoundations
Use models that explicitly incorporate household and firm decision problems
Allows to capture how decisions adjust when economic environment of policies change
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Using Models
Tools to predict outcomes:OptimizationMarket Clearing
Check whether model matches data:Yes: Likely that model world captures
key features of the real worldNo: Build new model
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A Simple Market Economy
One consumer, one firm Consumer and firm trade in markets Markets for consumption C and labor N
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Market Prices
Prices:Price of consumption normalized to onePrice for N is real wage w
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The Household’s Problem in the Market Economy Utility function U(C,l)
C: Consumption (coconuts)l: Leisure
Budget constraintConsumption expenditure equals income
from capital and labor p is given, capital incomeN is given by time constraint: N=h-l
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sC wN p T
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The Consumer’s Preferences
Utility function U(C,l) Assumptions:
More is better than less: , Diversity is good: Falling MRSConsumption and leisure are normal
goods
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0CU 0lU
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Indifference Curves
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Properties of Indifference Curves
Downward sloping: Follows from positive marginal utilities
Convex: Follows from falling marginal rate of substitution
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Indifference Curves
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Marginal Rate of Substitution
MRS: the minimum # of Coconuts consumer is willing to give up for another unit of leisure
Equal to minus slope of indifference curve
Mathematically:
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l
C
UMRS
U
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The Budget Constraint
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The Optimization Problem
Maximize utility subject to the budget constraint by choosing l and C
s.t.
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max ( , )U C l
s
s
C wN p T
N h l
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Graphical Representation
Draw indifference curves as before Draw budget constraint as a function
of leisure Optimal choice is point in the budget
set that lies on the highest indifference curve
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Graphical Optimization
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Outcome
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Slope of indifference curve equals slope of budget constraint
Slope of budget constraint: wage w Result:
wage = MRS This is a very general result: the MRS
between any two goods is given by the relative price!
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Mathematical Optimization
Substitute constraints into U(C,l)
First-order condition with respect to l:
Result (once again): wage = MRS
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max ( ( ) , )U C w h l p T l
0
C
l
C lwU U
Uw
U
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Example
wage equals 10 coconuts per hour Time: 24 hours Profit and tax: p=30 and T=30
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( , ) log logU C l C l
10 (24 )C l
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Example
Maximization problem:
Solution:
,
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max{log(240 10 ) log }l l
10 10
240 10l l
12l 120C
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Predicting the Reaction to Changes in the Economy
Separate income and substitution effects
Pure income effect: consume more of every (normal) good
Pure substitution effect: consume more of the good that gets cheaper
In practice, often both effects are present
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A Pure Income Effect
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An Increase in the Wage
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The Firm’s Problem in the Market Economy
Production function
Number of coconuts produced with capital and labor input
Assumptions: : both inputs required : positive marginal products : decreasing marginal
products
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( , )dY zF K N
K dN
(0, ) ( ,0) 0dF N F K 0, 0K NF F
0, 0KK NNF F
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Graph of
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( , )dF K N
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The Marginal Product of Labor
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Effect of an Increase in Productivity
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Effect of an Increase in Productivity
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The Firm
The firm maximizes profits subject to the production function
Profit π: output minus cost
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( , )d dzF K N wN
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Graphical Profit Maximization
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Optimization Result
Slope of production function equals slope of cost curve
This is a very general result: the MP
of any factor of production is given by its price!
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NMP wage
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Mathematical Profit Optimization
The maximization problem:
First-order condition:
Wage equals marginal product of labor
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max{ ( , ) }d dzF K N wN
0 N
N
zF w
zF w
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Equilibrium
Requirements for equilibrium:Consumer maximizes utilityFirm maximizes profitsDemand equals supply in every
market Combining firm and household
optimization, we get
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NMRS MP
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What is the Simple Model Good for?
The ultimate task of any economic model is to shed light on the real world
The only thing the model could be good for is explaining labor-leisure choice
Does the model explain U.S. data?
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Average Workweek in U.S.
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Average Workweek in U.S.
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How is the Model Evaluated?
Model abstracts from many potential factors
Want to know whether model is sufficient to explain decline in time worked
Need to specify model more precisely
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Making the Model More Precise No capital for simplicity Variables:
C: consumption l: leisure N: labor w: wage z: total factor productivity g: growth rate of z
Productivity grows over time Want to determine N as a function of z38
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Choosing Functional Forms
Production function:
Utility function:
Budget and time constraints:
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1( ) , (1 )t t t t tF N z N z g z
( , ) log logt t t tU C l C l
(1 ), 1t t t t tC w l N l
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Profit Maximization
First order condition:
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max{ }t t t tz N w N
0 t t
t t
z w
w z
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Utility Maximization
The maximization problem:
First-order condition:
Labor constant, independent of wage!
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max{log( (1 )) log }t t tw l l
1 10
1 t tl l
0.5, 0.5t tl N
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What does It Mean?
Model appears to be a complete failure!
Reason: with log utility, income and substitution effects on labor supply cancel (i.e., they have equal size and opposite sign)
Is this realistic in the cross-section?
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Using the Model for Cross-Country Comparision
European countries (France, Germany, Sweden etc.) have higher taxes and higher transfers
Is like a negative substitution effect: income tax lowers the perceived wage
Model predicts less work and more leisure in Europe
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What Else Could Explain the Facts?
There are alternative explanations:Labor-force participationTaxationRelative productivity of “leisure” sector
Try new models in case of failure
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Intertemporal Choice
Most of macroeconomics is about changes over time
So far, have jus considered the decision of work versus leisure
Need to add choice of today versus tomorrow
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Examples
Some intertemporal choices:Borrowing and saving by consumersInvestment by firmsHuman capital investment by studentsFamily decisions
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Important Factors for Intertemporal Choice:
Preferences over time (patience) Expected return on investment Expected future economic conditions
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Modeling Intertemporal Choice
For simplicity:Look at one consumer in isolationTwo periods only
Variables: : consumption today and tomorrow : discount factor (measures patience) : income today and tomorrow : saving : interest rate (return on saving)
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,c c
,y y
sr
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The Setup Utility function:
Budget constraints:
Want to know how and depend on (intertemporal preferences) (economic conditions) (return on investment)
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( , ) ( ) ( )U c c u c u c
(1 )
c y s
c y r s
,c c s
,y y
r
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Mathematical Solution
Substitute constraints into utility function:
Setting derivative wrt. s to zero:
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max{ ( ) ( (1 ) )}u y s u y r s
0 ( ) (1 ) ( )
( )1
( )
u c r u c
u cr
u c
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Outcome
MRS = Interest rate Same as before – Simple Model:
Choice between leisure and laborMRS(l,C) = Relative price (l, C)
Intertemporal model:Choice between today and tomorrow MRS = Relative price
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( , )c c ( , )c c
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The Present-Value Budget Constraint
Present value of x dollars tomorrow:Amount needed to be saved today to
have x dollars tomorrow
Solving period-2 constraint for s:
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( ) / (1 )PV x x r
(1 )
1 1
1 1
c y r s
s c yr r
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The Present-Value Budget Constraint
Plugging the result into the period-1 constraint:
PV(total consumption)=PV(total income)
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1 1
1 11 1
1 1
c y s
c y c yr r
c c y yr r
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Graphical Analysis
Lifetime wealth:
we = PV(total income) Rewriting the budget constraint:
Can now represent choice in standard diagram
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1
1c we c
r
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The Diagram
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Outcome
MRS = Relative price Pure income effect (increase in either
or ) will increase both and Implies that s increases when risesImplies that s falls when rises
Only present value of income matters, distribution irrelevant for consumption
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yy
c c
yy
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Example: Log Utility
FOC for and
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max(log( ) log( (1 ) ))y s y r s 1(1.1) 0.1r
1 10
1.1
2.1
y s y s
y ys
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Computing Consumption
Example I:
Example II:
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1, 0y y
( ) / 2.1 10 / 21
1 10 / 21 11/ 21
(1 ) 1.1 10 / 21 11/ 21
s y y
c y s
c y r s
0, 1.1y y
( ) / 2.1 11/ 21
11/ 21
(1 ) 1.1 (1 11/ 21) 11/ 21
s y y
c y s
c y r s
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Conclusions
Model predicts strong consumption smoothing: timing of income does not matter
Result relies on perfect capital market Even so, evidence for consumption
smoothing is strong
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Consumption Smoothing in Practice
Life-cycle consumption: borrow early in life, then save for retirement
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Informal Capital Markets
Default risk prevents some people from borrowing
Society often finds ways around that problem:Transfers from parents and relativesGift giving and neighborhood helpSocial insurance
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A Neoclassical Growth Model
Overlapping generations:Each consumer lives for two periodsEach year, one old and one young
consumer are alive The young work one unit of time The old are retired and supply capital
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Generational Structure
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The Decision Problem of a Consumer Born at Time t
Utility function:
Budget constraints:
Notice that:There is no income in the old periodSavings are capital in the old period
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1 1( , ) log logt t t tU c c c c
1
1 1 1(1 )t t t
t t t
c w k
c r k
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Solving the Consumer’s Problem Choose to solve:
First-order condition:
Solution:
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1tk
1 1 1max(log( ) log((1 ) ))t t t tw k r k
1 1
1 10
t t tw k k
1 1t tk w
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The Profit-Maximization Problem of the Firm
Firm maximizes production minus cost:
First-order conditions are:
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1max( ( ) )t t t t t t tK z N w N rK
1
1 1
(1 )
( )
t t t t
t t t t
w K z N
r z NK
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Closing the Model
Market clearing for capital and labor:
Assume constant productivity (for now):
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1t
t t
N
K k
tz z
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Working out the Predictions of the Model
Using market-clearing conditions in equations for w and r:
Using wage equation in saving equation of household:
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1
1 1
(1 )t t
t t
w z k
kzr
11 (1 )
1t tk z k
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Using the Law of Motion for Capital
Have derived a law of motion for capital (capital tomorrow depending on capital today)
Starting at any initial capital, can determine how capital will develop in the future
Can compute production and growth rates over time
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Example
Parameter choices:
The law of motion is:
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0.5 1 16z
11
0.51
(1 )1t t
t t
k z k
k k
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Graph of the Law of Motion
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Convergence
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Capital Over Time
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Result
Model predicts convergence across countries with different initial capital
Intuition:Returns to capital are decreasingWage increases less than proportionally
with capitalSavings increase less than proportionally
with capital
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Long-run Predictions
Capital convergence to steady state Solving for capital in steady state:
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1(1 )1SS SSK z K
1
1( (1 ))1SSK z
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What Happens if there is Productivity Growth?
Steady-state level of capital depends on productivity z
Steady state shifts upwards if productivity increases
Assume constant productivity growth g:
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1 (1 )t tz g z
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The Law of Motion after a Change in Productivity
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Implications
In the long run, capital k grows at the same rate as productivity:
What happens to output and the return to capital?
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1 (1 )t tK g K
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Implication for Growth
Output grows at the same rate as capital
Therefore capital/output ratio is constant
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11 1 1 1( )t t t tY K z N
1((1 ) ) ((1 ) )
(1 )t t
t
g K g z
g Y
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Remaining Growth Facts
Labor and capital shares are constant because of Cobb-Douglas technology
Return to capital:
Constant because K and z both grow at rate g
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1 1
1
( )
/ ( )( )
t t t t
t t t
r z N
NK z
K
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Convergence from Different Initial Conditions
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Catching-up after a destruction of Capital
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Two Countries with Different Discount Factors
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Summary
The model explains all the growth facts
Driving force is exogenous, constant productivity growth combined with decreasing returns to capital
Explains catch-up of Germany of Japan after the war
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Revisiting the Asian Miracle
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Unraveling the Puzzle
Asian Tigers started with low capital stock after World War II
Rapid growth through capital accumulation is exactly what model predicts
There is no Asian miracle!
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Log of GDP per capita in the Asian Tigers
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Limits of the Neoclassical Growth Model
Technological progress is just assumed, not explained
Model does not offer a perspective on stagnation throughout history and in poor countries
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