One-Period Macro Model

Households & FirmsCompetitive Equilibrium

Effects of Productivity ShocksGovernment Sector

Households

• Chooses: Labor Supply (Ns), leisure (l = 1- Ns), and consumption (c) to

subject to

given = real wage rate and a = household wealth (exogenous).

),(max lcu

aNc s 1 lN s

• Optimal values of {c*,l*}, given and a, solves:

*)*,(

*)*,(, lcu

lcuMRS

c

lic

alc *)1(*

• Implications

(i) Changes in wealth creates a pure income effect:

dc*/da > 0

dl*/da > 0 and dN*/da < 0

(ii) Changes in real wages creates both an income and substitution effect:

dc*/d > 0 and dN*/d = ??

Figure 4.12 Real Wage in the United

States, 1980–2003

Figure 4.13 Average Weekly Hours in the

United States, 1980–2003

The workweek and real GDP per person in 36 countries: 1980s

Firms

• Chooses labor demand (Nd) and output Y to maximize profits )

subject to

Assume capital stock K fixed z = Productivity/Technology Shock (“Solow Residual”)

}max{ dNY

),( dNKzFY

)()( dd NfNzFY

Figure 4.20 The Solow Residual for

the United States

• Optimal values of {N*,Y*}, given , solves

• Implications:

(i) dN*/d < 0 (Labor Demand Curve)

(ii) dN*/dz > 0 (Productivity Shock)

)( dN NfMPN

)(* dNfY

Competitive Equilibrium (CE)• Sometimes called “general equilibrium”

• There are many identical “representative” households and firms.

• Households {c*,Ns} given a and • Firms {Y*,Nd} given .

• Households are the owners of firms and takes profits as given: a = Y – N

• Market-Clearing:

Nd = Ns = N* = 1-l* (labor mkt)

Y* = c* (Goods Mkt)

• A competitive equilibrium is {c*,N*,Y* solving:

(utility max)

(profit max)

(prod function)

(market-clearing)

Where l* = 1 – N*

**)*,(*)*,( lcu

lcu

c

l

**)( NfN

*)(* NfY

** cY

Pareto Optimality

• An allocation is Pareto Optimal if no other feasible allocation can improve the welfare of one without reducing the welfare of another.

• PO is a statement about efficiency not necessarily fairness or equality.

• The Welfare Theorem: The competitive equilibrium (CE) is Pareto Optimal (PO).

• Verify – The Social Planner’s (SP) Objective is to choose allocations {c*=Y*, l*} which solves:

subject to

and

Solution – Identical to the CE.

),(max lcu

)(),( NzfNKzFYc

1 lN

• The Welfare Theorem is basically Adam Smith’s Invisible Hand.

• Social planning is difficult to implement. Competitive equilibrium (market system) is easy.

• Exceptions to the theorem:

(i) Externalities not internalized by markets

(ii) Non-competitive markets.

(iii) Government policies (tax distortions).

Productivity Shocks

• Productivity shocks (z): Changes the efficiency of capital and labor (technology, weather, cost of energy, government regulations, ect)

• An increase in z:

Income effect (+) C and (+) l

Substitution Effect (+) C and (-) l

Hence dc*/dz > 0 and dN*/dz = ??• In the case where both effects are roughly equal, Y

and increases..

Figure 5.11 Deviations from Trend in Real GDP and the Solow Residual

Figure 5.12 The Relative Price

of Energy

• Why dN*/dz = ??

Intuition:

(i) (+) z (+) MPN (+) ND (+) (+) NS

(Substitution Effect)

(ii) (+) z (+) firm profits (+) non-labor income (a) (-) NS

(Income Effect)• Consistent with empirical evidence?

One Period CE Model with Government

• Government sector

(i) Collects revenues from taxes (T).

(ii) Purchases goods and services (G)

• Assume balanced budget (G = T)

• Household wealth (a) = Goods Market Clearing: Y = C + G

Labor market Clearing: Nd = Ns

CE Model with Government

• Households {c*,Ns} given a and • Firms {Y*,Nd} given .• Government Sets G = T• Households are the owners of firms and

takes profits as given: a = • Market-Clearing:

Nd = Ns = N* = 1-l* (labor mkt)Y* = c*+G (Goods

Mkt)

• A competitive equilibrium given G is {c*,N*,Y* solving:

**)*,(*)*,( lcu

lcu

c

l

**)( NfN

*)(* NfY

GcY **

Effects of Government Purchases

• Negative Income Effect:

dc*/dG < 0

dl*/dG < 0 dN*/dG and dY*/dG > 0

du(c*,l*)/dG < 0

• G = 0 would maximize welfare.

Effects of Government Purchases

• Stabilization Policy: The government can use government purchases to stabilize output from productivity shocks (dG/dz > 0) but it will lead to a further decrease in economic welfare.

The Growth Rate of U.S. Real Gross Domestic Product since 1870

Figure 5.7 GDP, Consumption, and

Government Expenditures

Comparison with IS Model(Simple Income Determination)

• CE vs IS:

(i) Both Predict dY/dG > 0. Government purchases can be used to stabilize GDP and business cycles.

(ii) Increase in G alone, then dY/dG > 0 and dy/dC > 0 “welfare” increases.

(iii) If G = T, then dY/dG = 1 and dY/dC = 0. dC/dG = 0 “welfare” constant.

(iv) CE dC/dG < 0 and welfare decreases!

Comparison with IS-LM

• CE vs IS-LM: Not entirely comparable since no saving/interest rate in CE model.

(i) Both Predict dY/dG > 0. Government purchases can be used to stabilize GDP and business cycles.

(ii) Increase in G alone, then dY/dG > 0 and dy/dC might be > 0, so “welfare”

ambiguous.

(iii) CE dC/dG < 0 and d (welfare)/dG < 0!

• In basic model the need for government expenditures (G) is exogenous (no direct benefits to private sector).

• Modifications:

(i) Substitutability of public & private consumption:

(ii) Productive Government expenditures:

10, GcCT

0,0 whereGzz

Proportional (Marginal) Taxes

• Most individual taxes in US are collected via marginal income taxes:

(i) Wealth: a = (ii) Consumer’s BC:

(iii) Government’s BC:

sNtc )1(

sNtTG

• Competitive Equilibrium w/ proportional taxes is {c*,N*,Y*} and solving

Where T = tN* = G

)1(*, tu

uMRS

c

llc

**)( NfN

GcNfY **)(*

*)1(* lNNN sd

• Graphical example - Effect of tax rate:

dc*/dt < 0

dl*/dt > 0 dN*/dt < 0• CE w/ proportional taxes is NOT Pareto Optimal

• Laffer Curve: The non-monotonic relationship between tax rates t and tax revenue REV = tN.

Supply Side Economics d(REV)/dt < 0.

MRTMPNtu

uMRS

c

llc )1(*,

• Evidence:

(i) Economic Recovery Act of 1981

* Highest Income Tax Bracket cut from 70% to 50%

* Lowest cut from 14% to 11%

(ii) G.W. Bush Tax Cuts of 2001

* 40%35%

36%33%

31%28%

28%25%

Figure 5.18 Federal Personal Taxes as a Percentage of GDP