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McMaster University

Economics

ECON 2HH3

Marc- Andre Letendre

Winter

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Marc-Andre Letendre January 24, 2011
ECONOMICS 2HH3
Solving for a Competitive equilibrium: An example
Contents
1 Introduction 1
2 Consumer Optimization 2
3 Firm Optimization 4
4 Government 5
5 Solving for the Competitive Equilibrium Outcome 6
6 A Numerical Example 7
6.1 Solving the Model . . . . . . . . . . . . . . . . . . . . . . . . . .7. . . . . . .
6.2 An Increase in Government Expenditures . . . . . . . . . . . . . . . 8 . . . .
6.3 An Increase in Total Factor Productivity (TFP) . . . . . . . . . . .12 . . . .
1 Introduction
This handout explains how to proceed to solve for the general equilibrium of the static
model described in chapters 4 and 5 of Williamson. In general equilibrium all markets clear
(i.e. demand equals supply on all markets). Since the consumers and ﬁrms are behaving
competitively in the model we are solving, we are solving for the model’s competitive
equilibrium.
1 Solving for the equilibrium of the model involves solving for the endogenous variables (vari-
ables determined inside the model) as functions of exogenous variables (variables determined
outside the model) and parameters. Moreover, the solutions we get for the endogenous vari-
ables must satisfy the four conditions of a competitive equilibrium listed on pages 128 and
129 of the book (3rd edition). To simplify, these conditions state that all markets must clear,
consumers and ﬁrms optimize and the government satisﬁes its budget constraint.
In order to solve for the general equilibrium of the model, we need demand and supply func-
tions which are obtained from the representative consumer and representative ﬁrm decision
making processes. Thus, that’s where we start.
2 Consumer Optimization
Note: Also read textbook pages 589-590 (up to equation (A.4)).
Suppose the utility function of the representative consumer is
C 1−a ℓ1−a
U(C,ℓ) = + θ , θ > 0, 0 < a < 1. (1)
1 − a 1 − a
The consumer chooses C and ℓ to maximize utility subject to a time constraint
N s ℓ = h (2)
and a budget constraint given by
C = wN + π − T (3)
which can be combined to eliminate N s
C = wh − wℓ + π − T. (4)
Equation (4) is the consumer’s budget constraint we use in our analysis.
Maximizing utility function (1) taking into account of constraint (4) can be done using the
Lagrange-multiplier method which involves setting up a Lagrangian function. Without going
2 1
into technical details, we set up the Lagrangian function L(C,ℓ,λ) as
1−a 1−a
C ℓ
L(C,ℓ,λ) = + θ + λ[wh − wℓ + π − T − C] (5)
1 − a 1 − a
where λ is a Lagrange multiplier and the arguments of L are the consumer’s choice variables
and the Lagrange multiplier. The term in square brackets in equation (5) corresponds to
the right-hand side of 0 = wh−wℓ+π −T −C which is obtained by subtracting C oﬀ both
sides of (4).
To solve the maximization problem we set the partial derivatives ∂L(−)/∂C, ∂L(−)/∂ℓ, and
∂L(−)/∂λ equal to zero and solve for C, ℓ and λ (although we will not pay attention to the
solution for λ here).
The three partial derivatives are
∂L(C,ℓ,λ) −a
∂C = ▯▯▯▯−λ (6)
MU C
∂L(C,ℓ,λ)
= θℓ−a −wλ (7)
∂ℓ ▯▯▯ ▯
MUℓ
∂L(C,ℓ,λ)
= wh − wℓ + π − T − C (8)
∂λ
Setting the three derivatives above equal to zero and re-arranging yields the ▯rst-order
conditions
λ = C −a (9)
−a
wλ = θℓ (10)
C = wh − wℓ + π − T (11)
Using equations (9) and (10) to eliminate λ we get
−a
−a −a θℓ
wC = θℓ ⇔ w = −a = MRS ℓ,c (12)
C
Note that the optimality condition represented in equation (12) is a special case of equation
(A.4) on page 590 of the book.
We now solve for the optimal consumption function by rewriting (12) as
1
w θ θC a θ C
wC −a = θℓ−a ⇔ = ⇔ wℓ = θC ⇔ ℓ = a ⇔ ℓ = 1 (12 )
ℓa C−a w wa
1For more details see for example the discussion starting on page 372 in Alpha C. Chiang Fundamental
methods of mathematical economics, 3rd edition, 1984, McGraw-Hill, NY.
3 and using the right-most expression above to substitute ℓ out of the budget constraint (11)
1
θ C a 1− a
C = wh − w 1 +π − T ⇔ C = wh − θ w C + π − T (13)
w a
▯ ▯▯ ▯
=ℓ
grouping both terms in C we get
[ 1 1]
C 1 + θ w 1− a = wh + π − T (14)
dividing both sides by the expression inside the square brackets above we get the consump-
tion demand function
1
C = [wh + π − T] (15)
1 + θ w 1− a
Plugging this consumption function in the rightmost equation in (12’) yields the leisure
demand function
1
θa
ℓ = 1 1 1− 1 [wh + π − T]. (16)
w (1 + θ w a )
Plugging (16) into the time constraint (2) yields the labour supply function
a
s θ
N = h − 1 1 1− 1 [wh + π − T]. (17)
w (1 + θ w a)
3 Firm Optimization
Note: Also read textbook pages 591-592.
Suppose the representative ﬁrm has production function
Y = zK Nα 1−α, 0 < α < 1 (18)
and that it chooses labour N to maximize its proﬁts. Its proﬁt function, written in terms of
2
units of consumption good is
π(K,N) = zK N α 1−α− wN (19)
2
That is the price of the ▯rm’s output is measured in units of the consumption good rather than in
dollars (i.e. the consumption good is theeraire). Since the ▯rm’s actually produces units of the unique
consumption good the unit price of each unit produced is exactly unity. The real wage rate is also measured
in units of the consumption good (it represents the number of units of consumption good paid by the ▯rm
for one unit of labour input.)
4 where we do not subtract the cost of investment in capital since we are in a static framework
where K is taken as given by the ﬁrm. To maximize the ﬁrm’s proﬁt in our static framework
we calculate the partial derivative of the proﬁt function with respect to N and set this
derivative equal to zero. Doing so yields the condition
∂π(K,N) = (1 − α)zK N −α− w = 0 (20)
∂N
which can be re-arranged to show that the ﬁrm’s optimal behaviour implies that the real
wage rate is equal to the marginal product of labour
α
ZK
w = (1 − α) Nα (21)
▯ ▯▯ ▯
MPN
The ﬁrm’s labour demand function can be derived by solving (21) for N
[ ]1 [ ]1
α ZK α ZK α α Z α
N = (1 − α) ⇔ N = (1 − α) ⇔ N = (1 − α) K (22)
w w w
In our graphical analysis of the labour market, the vertical axis measures w and the horizontal
axis measures N. As a result, we use equation (21) to graph the ﬁrm’s labour demand
function. In such a case, equation (21) clearly shows that the graphical representation of the
labour demand function corresponds to the marginal product of labour.
For completeness, note that we get the ﬁrm’s consumption good supply function by
substituting N out of the production function using the rightmost expression in (22)
[ ]1 1−α [ ]1▯α
α Z α Z α
Y = zK (1 − α) K ⇒ Y = zK (1 − α) (23)
w w
4 Government
The government here is simply represented by the budget constraint
T = G (24)
which shows that the government set taxes T (endogenous) equal to its expenditures G
(exogenous). The government’s demand for the consumption good is simply G.
5 5 Solving for the Competitive Equilibrium Outcome
With demand and supply functions on hand we can now see the steps involved in solving
for the equilibrium. The model has two markets: a market for the consumption good and a
market for labour services. From Walras’ law we know that if we ﬁnd the prices that clear
M − 1 markets, then the M th market will also be in equilibrium. Since M = 2 we can focus
on one market to solve for the equilibrium price w. In this document I use the labour market.
1. We solve for the equilibrium wage rate by equating labour supply (17) and labour
demand (22)
1 1
θa [ Z ]α
h − [wh + π − T] = (1 − α) K (25)
w (1 + θ w 1− a) w
Recall that we want to solve for w as a function of parameters and exogenous variables.
Therefore we need to substitute out the endogenous variables π and T from the equation
above and replace them by expressions that involve only parameters and exogenous
variables. For the taxes T, we simply use the government budget constraint to replace
T by G. For proﬁts, we substitute N out of the formula for proﬁts (19) using labour
demand (22) and use the resulting expression to substitute π out of (25) which yields
a [ ]1▯▯ [ ] ▯ [ ] ▯
h− ▯ wh + zK ▯ (1 − ▯) K 1▯▯−w (1 − ▯)Z K − G = (1 − ▯)Z K
w a(1 + ▯ w▯a) w w w
▯ ▯▯ ▯ ▯ ▯▯ ▯
=N 1▯▯ =N
(26)
The above equation shows that, in principle, we can solve for w as a function of
parameters h, a, α and θ as well as exogenous variables Z, K and G. In practice,
w

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