One-Period Macro Model Households & Firms Competitive Equilibrium Effects of Productivity Shocks Government Sector
Jan 20, 2016
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