Download One-Period Macro Model

One-Period Macro Model
Households & Firms
Competitive Equilibrium
Effects of Productivity Shocks
Government Sector
Households
• Chooses: Labor Supply (Ns), leisure (l = 1- Ns),
and consumption (c) to
max u (c, l )
subject to c  wN s  a
N l 1
s
given w = real wage rate and a = household
wealth (exogenous).
• Optimal values of {c*,l*}, given w and a,
solves:
ul (c*, l*)
MRS c ,i 
w
uc (c*, l*)
c*  w (1  l*)  a
• 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*/dw > 0 and dN*/dw = ??
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 (P):
max{ P  Y  wN }
d
subject to
Y  zF ( K , N )
d
d
d
Y

zF
(
N
)

f
(
N
)
Assume capital stock K fixed 
z = Productivity/Technology Shock (“Solow
Residual”)
Figure 4.20 The Solow Residual for
the United States
• Optimal values of {N*,Y*}, given w, solves
MPN  f N ( N )  w
d
Y*  f (N )
d
• Implications:
(i) dN*/dw < 0
(ii) dN*/dz > 0
(Labor Demand Curve)
(Productivity Shock)
Competitive Equilibrium (CE)
•
•
•
•
•
•
Sometimes called “general equilibrium”
There are many identical “representative”
households and firms.
Households  {c*,Ns} given a and w.
Firms  {Y*,Nd} given w.
Households are the owners of firms and takes
profits as given: a = P  Y – wN
Market-Clearing:
Nd = Ns = N* = 1-l*
(labor mkt)
Y* = c*
(Goods Mkt)
• A competitive equilibrium is {c*,N*,Y*,w*} solving:
ul ( c *,l *)
uc ( c *,l *)
w*
f N ( N *)  w *
(utility max)
(profit max)
Y *  f ( N *)
(prod function)
Y*  c *
(market-clearing)
Where l* = 1 – N*
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:
max u (c, l )
subject to
c  Y  zF ( K , N )  zf ( N )
and
N  l 1
Solution – Identical to the CE.
• 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 w 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  (+) w  (+) 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) = P  T
Goods Market Clearing: Y = C + G
Labor market Clearing: Nd = Ns
CE Model with Government
•
•
•
•
•
Households  {c*,Ns} given a and w.
Firms  {Y*,Nd} given w.
Government  Sets G = T
Households are the owners of firms and takes
profits as given: a = PT
Market-Clearing:
Nd = Ns = N* = 1-l*
(labor mkt)
Y* = c*+G
(Goods Mkt)
• A competitive equilibrium given G is
{c*,N*,Y*,w*} solving:
ul ( c *,l *)
uc ( c *,l *)
w*
f N ( N *)  w *
Y *  f ( N *)
Y *  c * G
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:
CT  c  G, 0    1
(ii) Productive Government expenditures:
z  z0  G, where   0
Proportional (Marginal) Taxes
• Most individual taxes in US are collected
via marginal income taxes:
(i) Wealth:
a=P
(ii) Consumer’s BC:
c  w (1  t ) N  P
s
(iii)Government’s BC:
G  T  twN
s
• Competitive Equilibrium w/ proportional taxes is
{c*,N*,Y*} and w* solving
MRS c ,l
ul
  w * (1  t )
uc
f N ( N *)  w *
Y *  f ( N *)  c * G
N *  N d  N s  (1  l*)
Where T = tw*N* = G
• Graphical example - Effect of tax rate:
dc*/dt < 0
dl*/dt > 0  dN*/dt < 0
• CE w/ proportional taxes is NOT Pareto Optimal
MRS c ,l
ul
  w * (1  t )  MPN  MRT
uc
• Laffer Curve: The non-monotonic relationship between
tax rates t and tax revenue REV = twN.
Supply Side Economics  d(REV)/dt < 0.
• 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