Lecture 10: Partial Equilibrium
Advanced Microeconomics I
Yosuke YASUDA
Osaka University, Department of Economics yasuda@econ.osaka-u.ac.jp
November 4, 2014
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Market Economy
In previous lectures, we studied the behavior of individual consumers and firms, describing optimal behavior when market prices were fixed and beyond the agent’s control.
We begin to explore the consequences of that behavior when consumers and firms come together in markets. First, we consider price and quantity determination in a single market.
In a partial equilibrium model,
individual consumers and firms determine their demands and supplies for the good in question
all prices other than the good in question are fixed the market price is adjusted to clear the market.
⇒In the general equilibrium model, prices of all goods vary and all markets clear at the same time.
Perfectly Competitive Market
In (perfectly) competitive markets, buyers and sellers are sufficiently large in number to ensure that no single one of them, alone, has the power to influence market price.
⇒Market price is outside of their control. (they are price takers) A Consumer’s Problem
maxx,z u(x, z) s.t. px + qz = ω
A Firm’s Problem maxy py − c(y)
where all prices other than p is assumed to remain fixed.
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Q How can we aggregate individual demand or supply?✆
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Market Supply Function
The (individual firm) supply function is straight-forwardly derived from the profit maximization problem. Its first order condition
p = c′(y)
(and second order condition c′′(y) ≥ 0) implies that supply function of a competitive firm coincides with its marginal cost curve.
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Rm For the discussion of short-run and long-run supply functions,✆ see JR, Chapter 4.1 and NS, Chapter 12.
The market (or industry) supply function is simply the sum of the individual firm supply function. If yj(p) is firm j’s supply function in an industry with J firms, the market supply function is
Y (p) =
J
X
j=1
yj(p).
Aggregation across Consumers
Suppose there are I consumers, each of whom has a demand function for n commodities; Consumer i’s demand function is
xi(p, ωi) = (x1i(p, ωi), · · · , xni(p, ωi)).
Then, the aggregate demand function is defined by
X(p, ω1, · · · , ωI) =
I
X
i=1
xi(p, ωi).
The market (or industry) demand function (of good x) is simply the sum of all consumers’ individual demand for the good:
X(p) =
I
X
i=1
xi(p)
assuming all the prices other than p as well as incomes fixed.
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Market Equilibrium
The market supply function measures the total output supplied at any price, and the market demand function measures the total output demanded at any price.
Definition 1
An equilibrium price (in a partial equilibrium) is a price where the amount demanded equals the amount supplied:
I
X
i=1
xi(p) =
J
X
j=1
yj(p).
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Q Why does such a price deserve to be called an equilibrium?✆
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Rm At the equilibrium, no one would desire to change actions,✆ while at any other price some agent would find it in her interest to unilaterally change its behavior.
Properties of Aggregate Demand Function
The aggregate demand function clearly satisfies
homogeneous of degree 0 in all prices and the vector of buyers’ incomes.
It also becomes continuous whenever individual demand functions are continuous.
Unfortunately, the aggregate demand function in general possesses no interesting properties other than homogeneity and continuity.
For example, aggregate version of Slutsky equation or strong axiom of revealed preference cannot hold.
The properties of aggregate demand function are completely different from those of individual demand function.
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Q Under what condition the aggregate demand may look as✆ though it were generated by a single “representative” consumer?
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Representative Consumer (1)
The next theorem shows that the aggregate demand function possesses nice properties when the consumers’ indirect utility functions take a specific functional form.
Theorem 2
The aggregated demand function can be generated by a
representative consumer if and only if indirect utility functions of all individual consumers take the following Gorman form:
vi(p, ωi) = ai(p) + b(p)ωi.
Note that the ai(p) term can differ across consumers, but the b(p) term is assumed to be identical for all consumers.
Gorman form is imposed on an indirect utility function, not on a (direct) utility function.
Representative Consumer (2)
Proof of if part (⇐) By Roy’s identity, the demand function for good j of consumer i will also take the Gorman form
xji(p, ωi) = αji(p) + βj(p)ωi where
αji(p) = −
∂ai(p)
∂pj
b(p) , β
j(p) = −
∂b(p)
∂pj
b(p).
Note that marginal propensity to consume good j
= ∂x
j i(p,ωi)
∂ωi
is independent of the income level and constant across consumers. In other words, income effect is proportional to consumer’s income level, which makes possible to aggregate individual incomes.
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Representative Consumer (3)
The aggregate demand for good j then take the form
Xj(p, ω1, · · · , ωI) = −
I
X
i=1
∂ai(p)
∂pj
b(p) +
∂b(p)
∂pj
b(p)
I
X
i=1
ωi
.
This (market) demand function can be generated by the following representative indirect utility function
v(p, ω) =
I
X
i=1
ai(p) + b(p)ω
where ω =PIi=1ωi shows the aggregate income of consumers.
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Q When does indirect utility function takes Gorman form?✆
⇒A utility function is homothetic or quasilinear.
Aggregation across Goods
When we analyze the demand for a single good (partial equilibrium study), it would be convenient to aggregate “all other goods”.
A Consumer’s Problem (again) maxx,z u(x, z) s.t. px + qz = ω
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Q Under what conditions can we study the demand problem for✆ the z-goods as a group, without worrying about how demand is divided among different components of the z-goods?
Suppose that the relative prices of the z-goods remain constant, so that q = P q0 where P can be interpreted as some price index.
V (P, p, ω) = max
x,z u(x, z) s.t. px + P q0z= ω
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Composite Commodity
Given the indirect utility function V , we can recover the demand function for the z-goods by Roy’s identity (and envelope theorem):
z(P, p, ω) = −
∂V(P,p,ω)
∂P
∂V(P,p,ω)
∂ω
= q0z(p, q, ω).
This shows that z(P, p, ω) is an appropriate quantity index for the z-goods consumption: such z is called composite commodity. Since all prices q move together (by assumption) and the demand function is homogeneous of degree 0, we can write
x = x(P, p, ω) = x(p/P, ω/P ),
which says that the demand for good x depends on the relative price of x to composite good and income divided by price index.
