Macroeconomics - Spring Semester - QEM 1 Eleni Iliopulos Spring 2015
Perfect Competition and Fixed Prices
1
A Simple Model with Perfect Competition
Static model with three goods, a …nal good, labor and money, three types of agents, producers, consumers and a government, and perfect competition, i.e. all agents are price-takers and prices are ‡exible.
1.1
Production Sector
Assuming a representative …rm, the production is given by Y = F (N ), with Y output and N employment. Assumption 1 F (N ) is a continuous function on R+ , with F (0) = 0, twotimes di¤erentiable on R++ , with F 0 (N ) > 0, F 00 (N ) 0, limN !0 F 0 (N ) = +1 and limN !+1 F 0 (N ) = 0. Pro…ts are de…ned by = P Y and W the nominal wage.
W N , with P the price of the …nal good
Maximizing pro…ts, one obtains the labor demand: W=P = F 0d = F 0 1 (W=P ), decreasing in the real wage W=P (dN=d(W=P ) = 1=F 00 (N )). We deduce the supply of …nal good: Y = F [F 0 1 (W=P )]. Example: Using F (N ) = N a , with a 2 (0; 1], W=P = aN a (aP=W )1=(1 a) and Y = (aP=W )a=(1 a) .
1.2
1
, N =
Consumers
The preferences of a representative household are summarized by the utility function: U (C; M=P; H
N) =
C b=(b + d) 1
b
M=P d=(b + d)
d
Ne e
(1)
with C the consumption, M=P the money demand (in real terms), H the time endowment, > 0, e 1, b; d 2 [0; 1], and b + d 1. The representative household maximizes his utility function facing the budget constraint: P C + M = M0 +
(2)
T + WN
with M0 > 0 the stock of money and T 0 lump-sum taxes. Taking as given the real income I = M0 =P + ( T )=P + (W=P )N , we obtain: C=
b M d I and = I b+d P b+d
We further note that when b + d = 1, C = bI and M=P = (1 Then, labor supply is de…ned by: max I b+d
N e =e
N
One obtains:
W = N e 1I 1 P b+d
(3)
b d
(4)
1=(e 1)
When b + d = 1, W=P = N e 1 , N = 1 W , i.e. 1=(e P the elasticity of labor supply with respect to the real wage.
1.3
1) represents
Government
Public expenditures (in real terms) are given by G balanced P G = T .
1.4
b)I.
0 and the budget is
Equilibrium
Using =P = Y (W=P )N and the balanced-budget rule G = T =P , the real income can be rewritten: I=
M0 +Y P
(5)
G
Therefore, the equilibrium on labor market is given by: W = aN a P
1
=
b+d
Ne
2
1
M0 +Y P
1 b d
G
(6)
with Y = N a , and the equilibrium on the product market by: C=Y
G=
b b+d
M0 +Y P
G (7)
b M0 ,P = dY G Equilibrium on money market is ensured by Walras law.
1.5
Case b + d = 1 a
1=(e a)
a
a=(e a)
N pc = Y
pc
=
;
W P
; P
pc
pc
e 1 a
= ae
M0
b
=
1 a e a
(8)
1 b Y pc G Money and public spending are neutral. M0 > 0 and G > 0 only pc pc pc imply P > 0, without e¤ect on Y , N , and on the relative prices.
1.6
Case b + d < 1
Substituting (7) into (6), we get:
a(b + d)
N
e a
b+d a (N b
1 b d
=1
(9)
>0
(10)
2 (0; 1)
(11)
G)
Di¤erentiating, we obtain: dN = a dG N [e
(1 b d)N a(b + d)] G(e
a)
Using Y = N a , dY (1 b d)aY = a dG N [e a(b + d)] G(e
a)
The economic mechanism is based on an income e¤ect that a¤ects the labor supply when b + d < 1. G ") T ") income # ) labor supply increases for all level of W=P . Therefore, at equilibrium, W=P decreases. Since I = b+d (Y G), we get dI=dG < 0, i.e. an increase of G reduces b C and M=P . Using (7) and (9), one may also conclude that monetary policy is neutral. One attempt to have keynesian results in macroeconomic models with micro-foundations is based on price rigidities: the …xed-price approach. 3
2
The Model with Fixed Price and Wage
Preliminaries: Consider a single good market characterized by a demand Y d = D(P ) and a supply Y s = S(P ), with D0 (P ) < 0 and S 0 (P ) > 0. A competitive equilibrium is de…ned by (Y ; P ) satisfying Y = D(P ) = S(P ). Under a …xed price P generically di¤erent from P , we have Y s 6= Y d . In such a case, the quantity is rationed and is given by Y = minfY s ; Y s g, i.e. Y = Y s if P < P and Y = Y d if P > P . Considering now the model of the previous section with b + d = 1, we have: d
1=(1 a)
N = (aP=W ) b
d
Y =
1
s
; N =
M0 +G; Ys = b P
1W P P a W
1=(e 1)
(12)
a=(1 a)
where the perfectly competitive equilibrium (Y pc ; N pc ; P pc ; W pc ) is given by (8). Assuming that W and P are …xed and (P; W ) is generically di¤erent from (P pc ; W pc ), quantities are rationed. Then, at equilibrium, the level of employment and product are determined by: N = minfN d ; N s g ; Y = minfY d ; Y s g
(13)
Therefore, we are able to de…ne the following typology of equilibria: Y d < Y s and N d < N s : keynesian unemployment; Y d > Y s and N d < N s : classical unemployment; Y d > Y s and N d > N s : repressed in‡ation; Y d < Y s and N d > N s : not relevant.
Keynesian Unemployment (Y d < Y s and N d < N s )
2.1
We have Y = 1 b b MP0 + G and N = F One need Y d < Y s , i.e. b 1
M0 +G< b P
a=(1 a)
P a W
1
(Y ).
W ,
P pc +G< b P 1 b P pc 4
(15)
A decrease in W has no e¤ect. A decrease in P stimulates output and employment. Keynesian multiplier, dY =dG = 1.
2.2
Classical Unemployment (Y d > Y s and N d < N s )
The producer does not perceive any quantity rationing, i.e. F 0 (N ) = W=P . This requires Y d > Y s , i.e. W >a P
b 1
M0 +G b P
(1 a)=a
(16)
and also N d < N pc (< N s ) which is equivalent to: F 0d ) > F 0pc ) ,
W > P
W P
pc
(17)
Increasing demand (G) has no e¤ect on output and employment. Since C = Y G, this only decreases private consumption. Only a decrease of W=P can increase the level of output Y S and restore full employment.
2.3
Repressed In‡ation (Y d > Y s and N d > N s )
In this regime, we have N d > N pc (> N s ), i.e. F 0d ) < F 0pc ) ,
W < P
W P
pc
(18)
and Y d > Y pc (> Y s ), which is equivalent to: b 1
2.4
M0 b M0 +G> + G , P < P pc b P 1 b P pc
(19)
Concluding Remarks One obtains a classi…cation with clear-cut policy recommendations. However, such type of models have a crucial weakness: the absence of a satisfactory theory of price and wage formation. Therefore, another attempt to have economic models with market failures and (perhaps!) keynesian features: macroeconomic models with imperfect competition ! explicit price and wage formation. 5
