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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

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

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

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

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

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!

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

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

Example

Maximization problem:

Solution:

,

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max{log(240 10 ) log }l l

10 10

240 10l l

12l 120C

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

Graph of

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( , )dF K N

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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0.51

(1 )1t t

t t

k z k

k k

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

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

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

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

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

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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