Dynamic Macroeconomics
Chapter 9: Imperfectly flexible prices
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Overview
1 Introduction and motivation
2 Price-setting under imperfect competition
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Introduction and motivation
Introduction and motivation
• Effects of a monetary impulse in an economy with flexible prices:
−4
2
Responses to a money growth shock
x 10
output
employment
0
−2
−4
−6
−8
0
5
10
15
20
25
2
real int. rate
nominal interest rate
inflation
1.5
1
0.5
0
−0.5
0
5
10
15
20
25
=⇒ Monetary decisions have almost no real effects.
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Introduction and motivation
Introduction and motivation
• Evidence suggests that nominal prices are sticky in the short run:
• Bils and Klenow (2004): Prices in the United States are changed on
average every six months.
• Alvarez et al. (2006): Prices in the euro area are changed on average
every ten to twelve months.
=⇒ Monetary policy has influence on real variables in the short run.
=⇒ Ben Bernanke (Interview, June 2004):
The second issue is that these models [RBC models]
abstract from sticky wages and prices and hence from the
possibility of short-run deviations of output from full
employment. They don’t have any substantial role for
stabilization policies. I believe that we live in a world where
stabilization policy - stabilization of inflation as well as
output - is sometimes needed.
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Introduction and motivation
Introduction and motivation
• Effect of a monetary shock (Christiano, Eichenbaum and Evans, 1997)
Fed Funds Model with M1
NBR Model with M1
MP Shock => Y
NBR/TR Model with M1
MP Shock => Y
0.2
0.1
MP Shock => Y
0.2
0.2
0.1
0.1
-0.0
-0.0
-0.0
-0.1
-0.1
-0.2
-0.1
-0.2
-0.3
-0.2
-0.3
-0.4
-0.3
-0.5
-0.6
-0.4
-0.4
0
3
6
9
12
15
-0.5
0
MP Shock => Price
3
6
9
12
15
0
MP Shock => Price
3
6
9
12
15
MP Shock => Price
=⇒ Monetary shocks have relatively sizeable and persistent real effects.
0.2
0.14
0.1
0.18
0.09
0.07
-0.0
0.00
-0.1
-0.07
-0.2
-0.14
-0.3
-0.21
-0.4
-0.28
-0.5
-0.35
-0.6
-0.42
0
3
6
9
12
15
-0.00
-0.09
-0.18
-0.27
-0.36
-0.45
-0.54
-0.63
0
MP Shock => Pcom
3
6
9
12
15
0.15
6
9
12
15
12
15
0.15
0.10
0.05
0.05
0.05
-0.00
-0.00
-0.05
-0.05
-0.05
-0.10
-0.10
-0.10
-0.15
-0.15
-0.20
-0.25
-0.15
-0.20
0
3
6
9
12
15
0.8
0.6
0.4
0.2
-0.0
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-0.20
0
MP Shock => FF
-0.4
3
MP Shock => Pcom
0.10
0.10
-0.00
-0.2
0
MP Shock => Pcom
3
6
9
12
15
0
MP Shock => FF
0.6
0.5
0.4
0.8
0.3
0.2
0.1
0.4
-0.0
-0.1
-0.2
Dynamic Macroeconomics
3
6
9
MP Shock => FF
0.6
0.2
0.0
-0.2
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Introduction and motivation
Introduction and motivation
• To model price-setting in macro models several approaches have been
developed.
• Leading examples are:
• Calvo pricing (Calvo, 1983)
• Staggered wage setting (Taylor, 1979)
• Quadratic adjustment costs (Rotemberg, 1982)
• Menu-cost models (Akerlof and Yellen, 1985)
=⇒ Most popular: Calvo pricing (and - increasingly - menu-cost
models).
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Price-setting under imperfect competition
Price-setting under imperfect competition
• We consider the supply side of the economy.
• There is one representative firm.
• This firm has monopolistic power.
• The firm produces good Q using input factors X1 , X2 , ..., Xn .
• To produce the good the firm has access to a production function:
Q = F (X1 , X2 , . . . , Xn ) , F 0 > 0, F 00 < 0
(1)
• The cost of production, C , are given by:
n
C=
∑ Wi Xi
(2)
i =1
where Wi denotes the price of input factor Xi .
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Price-setting under imperfect competition
Price-setting under imperfect competition
• The price of input factor Xi increases with Xi , i.e. we have:
Wi = W (Xi ), W 0 (Xi ) > 0
(3)
• The firm faces a downward-sloping demand for its product:
P = P (Q ) , P 0 (Q ) < 0
(4)
• The firm’s maximization problem is given by:
max
Q,X1 ,X2 ,...,Xn
s.t.
PQ − C
P = P (Q )
(5)
(6)
n
C=
∑ Wi Xi
(7)
i =1
Q = F (X1 , X2 , . . . , Xn )
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Price-setting under imperfect competition
Price-setting under imperfect competition
• The associated Lagrange function is given by:
n
L = P (Q )Q − ∑ W (Xi )Xi + λ [F (X1 , X2 , . . . , Xn ) − Q ]
(9)
i =1
• The first-order conditions are:
∂L
= P + P 0 (Q )Q − λ = 0
∂Q
∂L
= −Wi − W 0 (Xi )Xi + λFi0 = 0
∂Xi
• Combining the two first-order conditions yields:
λ = P + P 0 (Q )Q = MR =
Wi + W 0 (Xi )Xi
MCi
=
= MC
0
MPi
Fi
(10)
(11)
(12)
with MC total marginal cost, MR marginal revenue
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Price-setting under imperfect competition
Price-setting under imperfect competition
• MPi is the marginal productivity and MCi is the marginal cost of the
i-th factor.
• So formally we have
∂C
∂C ∂Xi
=
∂Q
∂Xi ∂Q
(13)
∂C
= W (Xi ) + W 0 (Xi )Xi
∂Xi
(14)
∂Q
= Fi0
∂Xi
(15)
MC =
MCi =
MPi =
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Price-setting under imperfect competition
Price-setting under imperfect competition
• For the price P we obtain:
∂P Q
P = MC ⇐⇒
P + P 0 (Q )Q = MC ⇐⇒ P +
∂Q P
∂P Q
1
MC
P 1+
= MC ⇐⇒ P = ∂Q P
1− 1
(16)
εD
• with
εD = −
∂Q P
∂P Q
(17)
=⇒ Interpretation?
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Price-setting under imperfect competition
Price-setting under imperfect competition
• Substituting
ε Xi =
∂Xi W
∂W Xi
(18)
• we get
1 + ε1X Wi
i
P= 1
1 − ε D Fi0
(19)
• So the price P depends upon the unit cost of the factors of
production Wi , their marginal productivities MPi , the elasticity of
their supply ε Xi and the elasticity of the demand for the good ε D .
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