Economics 402A
UCLA
Professor Lee Ohanian
One-PeriodGeneral EquilibriumModels: Class
Notes
1 Introduction
These notes analyze a one period general equilibrium economy. Recall that
a general equilibrium model determines equilibrium in all the markets in the
economy. There are N number of consumers. They are all identical. Therefore
we abstract from issues of heterogeneity and distribution among the people in
the economy. Recall that agents will be rational – they will assess the costs and
benefits of their actions, and will make choices that are in their best interest.
We formalize rationality within the model by requiring that the economic actors
in our model optimize – they maximize an objective subject to the constraints
they face.
We will specify the preferences of consumers, the technology available to
produce goods, and the endowments of resources available to consumers and to
firms. This information, combined with optimizing behavior and an equilibrium
definition, allow us to use the model to make predictions for what will happen
if exogenous variables, such as the technology or government policies, change.
We will consider a competitive economy, which means that we will analyze
the competitive equilibrium of the economy, in which all economic agents are
assumed to be price-takers. That is, no individual consumer or firm is large
enough to affect the price of any good.
2 Preferences, Endowments, and Technology
There is one period in the model. Since there are many identical consumers,
we just need to study one of them, rather than all of them, as they will all
make the same decisions.. Preferences for the consumer are given by the utility
function u(c.l), in which c is consumption, and l is leisure. The utility function
is strictly increasing in each argument, which means consumers like to consume
goods and they also like leisure. The function is strictly concave. The function
is also twice continuously differentiable (which means that the first and second
derivatives are continuous functions). Denote the partial derivative of the utility
function with respect to the ith argument as ui. Assume that the function has
the property that uc → ∞ as c → 0. Let’s also assume the same condition for
the partial derivative for leisure, though there will be some situations in which
we will relax that assumption.
1
The consumer is endowed with one unit of time, which will be allocated
between work and leisure. We denote this as 1 = n + l, in which n is the time
spent working and l is time spent in leisure.There is a total of k0 units of capital
in the economy, so each consumer is endowed with k0
N
units of capital that can
be rented to firms for production.
There is also a representative firm, which also is a price-taker, and which has
a technology for producing goods according to y = zf(k, n), in which y is total
output for the economy, n is the amount of labor hired by the firm, and k is the
amount of capital hired by the firm. In this one period model, all the output
that is produced will be consumed. When we develop multi-period models, then
we will study investment, as well as consumption.
The term z shifts the production function, and we will call this term total
factor productivity, and we abbreviate this as TFP.. The inputs in production
are capital (k) and labor (n). We will assume that the technology f, which we
also call the production function, is constant returns to scale. This means that
for any value of χ > 0, we have
χzf(k, n) = zf(χk, χn)
In words, constant returns to scale means that if we double the inputs capital
and labor, then we double the output as well.
We also have some other assumptions about the production function. We
assume that f(0, n) = f(k, 0) = 0. We will assume that the function is strictly
increasing in each argument. We will call the partial derivatives of the function,
which are f1 and f2, the marginal products (also called marginal productivities)
of capital and labor, respectively. The marginal products are always positive.
However, the function also features diminishing marginal productivity of each of
the two factors of production, which means that the second partial derivatives
, f11 and f22 is negative.
Recall our discussion about exogenous and endogenous variables. Recall
that the endogenous variables and exogenous variables will be model-specific.
In this model, the endogenous variables (the variables that we solve for are
labor, output, and consumption. The exogenous variables are the productivity
term, z, and the amount of capital in the economy, k0. Thus, we will be able
to determine how the equilibrium of the model would change in the event there
were exogenous changes in productivity or the amount of capital.
To analyze the equilibrium, we will choose a numeraire good. That is a good
whose price is equal to one. All that this really means is that we pick the units
for the prices in the economy. For example, consider an economy with apples
and oranges. Suppose we define the numeraire to be apples. Then the price of
an apple is one – that is, we trade one apple for another apple. The price of
oranges in this economy in which the numeraire is apples, is the price of oranges
relative to the price of apples. This is how many apples we have to give up to
get one orange. Note that we could also pick the numeraire to be the orange.
2
The choice of the numeraire only affects the units in which prices are quoted,
but nothing else. We will follow the convention within the literature by treating
the consumption good as the numeraire. This means that the prices of the other
goods in the economy will be measured in consumption units.
3 Setting up the Competitive Equilibrium Econ-
omy
There are markets in three commodities: (1) consumption, (2) the rental services
of capital, and (3) the rental services of labor.. We will determine equilibrium
in each of these markets. The rental price of capital (in units of consumption) is
denoted as r, and the rental price of labor (in units of consumption) is denoted
as w. The consumer (as well as the firm) takes these prices as given. The
consumer maximizes utility subject to his/her constraints. That is, each solves
the following constrained maximization problem:
max{u(c, l)}
subject to:
w(1− l) + r
k0
N
≥ c
0 ≤ l ≤ 1
c ≥ 0
There are some items to note here. First, we specify that the consumer rents
out their entire endowment of capital for production. They do this because there
is no opportunity to cost to using the capital. That is, there is no alternative use
for capital other than for it to be rented to firms to produce output. If there
was an alternative use, then we would need to specify this in the problem.
Another item to note is that the consumer cannot have negative amount of
leisure, nor can they have negative amount of consumption. Note that these
constraints will never bind because of our assumption that the marginal utilities
of consumption and leisure become infinite at zero. Thus, consumers will choose
strictly positive amounts of consumption and leisure. Finally, we do not need
to worry about leisure exceeding the time constraint. That is, we don’t need
to worry about l > 1. The reason is because if l = 1, then there would be
no production, and therefore no consumption, which is inconsistent with utility
maximization. Thus, the solutions for c and l are known as interior solutions, as
opposed to corner solutions, in which the choice made the consumer is limited
by a binding constraint.
3
张熠华�
C: Consumption
4 Forming and Solving the Lagrangian for the
Economy
Since the assumptions that we have made insure that the inequality constraints
on leisure and consumption will not be binding, the only constraint that we need
to include when we maximize the objective function is the budget constraint.
To maximize, we form the Lagrangian as follows:
L = max{u(c, l) + λ[w(1− l) + r
k0
N
− c]}
We differentiate with respect to the choice variables c and l, and also λ,and
set those derivatives to zero. We obtain the first order necessary conditions for
consumer maximization as follows:
Lc = 0⇒ uc = λ
Ll = 0⇒ ul = λw
Lλ = 0⇒ w(1− l) + r
k0
N
= c
Combining the first two equations, we get:
ul = ucw
This equation has a particular feature that almost all first order conditions
share, which is that they equate the marginal cost of a choice, to the
marginal benefit of a choice. Here, the left hand side of the equation mea-
sures the marginal cost of working, which is the marginal utility of leisure. Thus,
if the consumer works more, the opportunity cost is foregone leisure, and the
value of that foregone leisure is measured by the marginal utility of leisure. The
right-hand side of the equation is the marginal benefit of working, which is the
additional wage earned (w), and that is multiplied by the consumer’s valuation
of the wage, which is the marginal utility of consumption. Thus, the value to
the consumer of receiving an additional w units of resources is those resources
valued at the consumer’s marginal utility: ucw.
In other words, the consumer receives some additional resources (w) to con-
sume, and the value of those resources to the consumer is the marginal utility of
their consumption. Optimization thus means that the marginal cost of working
– the value of foregone leisure – is equal to the marginal benefit of working – the
value of the additional resources valued at their marginal utility of consumption.
Next, note that we have two equations – the first order condition that equates
the marginal benefit of working to the marginal cost of working. But we have
several unknowns, including c, l, r, and w. Therefore we need to also solve the
firm’s optimization problem to be able to solve for the economy’s equilibrium.
We have solved for the optimization conditions for the consumer. Next, we
set up the firm’s maximization problem:
4
max zf(k, n)− wn− rk
First, note that the firm will be hiring labor and capital in order to maximize
their profits. Here, Profits are revenue [zf(k, n)], minus their costs[wn + rk].
Note that revenue is equal to output. If you are wondering where the price of
the output is, then recall that the output is the consumption good, which we
already defined to be the numeraire, and thus it has a price of one.
The first order necessary conditions for profit maximization are obtained by
differentiating with respect to the two factors of production, and setting the
derivative to 0. Doing this, we get:
zf1 = r
zf2 = w
These first order conditions provide us with a theory of how the factor prices
are determined, and how the income that is generated in the economy is dis-
tributed. Specifically, the rental prices of capital and labor – r and k – are equal
to their marginal productivities, or marginal products. That is, the payment to
the factors of production are equated to their contributions to production.
Note also that these equations also have a marginal benefit and marginal
cost interpretation. In particular, the left hand side of each equation is the
marginal benefit to the firm of hiring more of the factors of production, which
is just the additional revenue that is generated – the marginal product. The
right-hand side of the equations is the marginal cost of hiring the factors of
production, which in this case is the rental price of the factors of production.
Note that these equations provide a theory for the distribution of income
to the factors of production. In particular, the income paid to workers is wn,
where w is zfn, and the income paid to capital is rk, where r is zfk. Because
we assumed that the production function is constant returns to scale, it is also
the case that the payments to the factors of production are equal to the output
that is produced:
y = zf(k, n) = wn+ rk = zf1k + zf2n
This is the result of Euler’s theorem. To see this, recall that we have:
χzf(k, n) = zf(χk, χn)
Now, differentiate this expression with respect to χ, and for simplicity assume
that z = 1 :
f(k, n) = f1(χk, χn)k + f2(χk, χn)n
Since we assumed constant returns to scale (χ = 1), we therefore set χ = 1
to get the result
f(k, n) = f1(k, n)k + f2(k, n)n
5
Example: Cobb-Douglas function. The constant returns to scale Cobb-
Douglas production function is given by:
y = kθn1−θ
Suppose we double the inputs capital and labor. This will double output as
follows:
2y = (2k)θ(2n)1−θ
In addition, the share of income paid to capital is equal to θ, and the share of
income paid to labor is 1− θ. To see this, we have the following:
y = fkk + fnn = θ(k
θ−1)n1−θk + (1− θ)(n−θ)kθn = θy + (1− θ)y
Since the payments to labor and capital are equal to output, maximized
economic profits, or economic rents, are zero. Recall that this is economic
profits, or rents – not profits as indicated on an accounting statement in an actual
economy. In particular, what are stated as profits on a business accounting
statement, are represented as payments to capital in our framework.
Now that we have solved for the maximization problem for both of the
economic actors in this economy – consumers and firms – we can now define a
competitive equilibrium for this economy. A definition of equilibrium will be
model specific in terms of details. Broadly speaking, an equilibrium defines what
will happen in the economy when the economic actors make their decisions.
Moreover, there are some aspects of a competitive equilibrium that will be
shared across all models that have perfect competition. These are that (1) the
economic actors take prices as given – this is an implication of a competitive
economy (2) that prices will clear the markets for all of the commodities, which
means that supply equals demand in all of the markets, (3) that the allocations
of commodities will solve the decision-maker optimization problems. That is,
given prices and the constraints, the decision makers are as well off as they can
be.
For this specific economy, the competitive equilibrium is given by the fol-
lowing:
A competitive equilibrium is a set of quantities, c, n, k, and prices w and r
such that:
1. The representative consumer chooses c and l optimally, given the factor
prices w and r:
2. The representative firm chooses n and k optimally given w and r:
3. The markets for c,n, and k clear.
Here, there are three markets: the labor market, the market for consumption
goods, and the market for rental services of capital. In a competitive equilibrium
the following market-clearing conditions hold:
6
N(1− l) = n
y = Nc
k0 = k
Now, we will discuss the concept of excess demand to understand what is
known as Walras Law. Walras Law will help you further understand general
equilibrium, and can be convenient in some situations to analyze general equi-
librium.
Excess demand in a market is the difference between demand and supply in
that market. The value of excess demand is the price in that market, multiplied
by excess demand. .Now, form the value of excess demand across all markets
as follows:
w[n−N(1− l)] + [Nc− y] + r[k − k0]
Now, given that the consumer’s budget constraint holds, and the fact that firm
profits are zero, it is the case that the value of excess demand across all markets
is 0
w[n−N(1− l)] + [Nc− y] + r[k − k0] = 0
To see this, note that since the budget constraint holds with equality, then
we have that Nc = wN(1 − l) + rk0. We also have that y = wn + rk. These
equations guarantee that the value of excess demand is 0.
We are now in a position to apply what is called Walras Law, which states
that if M-1 markets are in equilibrium, then so is the remaining market. There-
fore, given that the value of excess demand across all markets is zero, then all
we need to do when determining equilibrium is to show that M-1 markets are
equilibrated, and the last market is also in equilibrium.
To see this in words, suppose that we have a model in which there are apples
and oranges, and oranges are traded for apples. Suppose that the market for
apples is in equilibrium and the consumer budget constraint holds. Then, the
market for oranges is also in equilibrium. The main implication is that Walras
Law allows us to establish that if we have equilibrium in all markets but one,
then equilibrium in the final market automatically follows as a result. T
To see this, suppose that there are M markets, and that M-1 of the markets
are in equilibrium. Suppose also that the consumer’s budget constraint holds,
which means that consumer income equal consumer expenditure. Then it must
be the case that the “Mth” market is in equilibrium, given that consumers have
exhausted their income on expenditure.
7
4.1 Optimality of the Competitive Equilibrium
A Pareto optimum is defined to be an allocation. An allocation is a production
plan and a distribution of goods across economic agents such that there is no
other allocation which some agents strictly prefer, and which does not make any
agents worse off.. Here, since we have a single agent, we do not have to worry
about the allocation of goods across agents.
To discuss the optimality of an equilibrium, it helps to think in terms of a
fictitious social planner, who can unilaterally choose the inputs for production
by the representative firm, and who can require the consumer to supply the
appropriate quantity of labor and capital, and who then distributes consumption
goods to the consumer, all in a way that makes the consumer as well off as
possible.
Note the large difference in the mechanism for how goods are allocated in
a market equilibrium, and by a social planner. The market equilibrium uses
markets to allocate goods and services, in which the choices within the markets
are being made by utility maximizing consumers, and profit maximizing firms.
The social planner unilaterally chooses allocations – to maximize utility – without
any markets.
This section will establish that the competitive equilibrium allocations are
also Pareto optimal – that is, they are socially effi cient. To establish this, we
will solve for the Pareto optimal allocations that are chosen by a social planner
who maximizes utility subject to the economy resource constraint, and then
show that they are the same as the competitive equilibrium allocations.
The social planner determines a Pareto optimum by solving a very simple
problem, which is to maximize a social welfare function, subject to the econ-
omy’s resource constraint, which simply states that consumption cannot exceed
production. The constrained maximization problem is given by:
maxu(c, l)N
.subject to:
zf(k0, N(1− l))−Nc
One way to solve this is to form a Lagrangian:
L = maxu(c, l)N + λ[f(k0, N(1− l))−Nc]
We differentiate with respect to the choice variables and set the derivatives
to zero to get:
Nuc = λN
Nul = λzf2N
8
Combining these equations, we get
ul = uczf2
Note that this equation is identical to what we derived in the competitive
equilibrium, in which the wage rate was equated to the marginal product of
labor. Thus, the competitive equilibrium share an equation in common. Next,
we show that they share one more – and final – equation in common, which is
the resource constraint.
In the equilibrium problem, we had the consumers budget constraint, which
was
w(1− l) + r
k0
N
= c
Let’s multiply by the population size to get:
w(1− l)N + rk0 = cN
Since the factor prices are equal to marginal products, we can rewrite this as
f2(1− l)N + f1k0 = cN
However, recall that with constant returns to scale, then the payments to the
factors of production is equal to total output:
f2N(1− l) + f1k0 = y
Finally, since Nc = y, then the consumer budget constraint in the equilibrium
problem is the same thing as the resource constraint in the social planner’s prob-
lem. Thus, the equilibrium and the social planning problem are characterized by
the same two equations. Thus, they have the same solutions, which implies that
the competitive equilibrium allocations are also optimal. This is known as the
First Welfare Theorem, which states that the competitive equilibrium alloca-
tions are also Pareto optimal. The requirements for this result is the following:
perfect competition, perfectly functioning markets, and no externalities. (An
externality occurs when a consequence of a choice made by one party affects
other parties without this being reflected in the cost of the goods or services
involved.
Moreover, we can rewrite the marginal cost- marginal benefit first order
condition as
zfn =
uc
ul
Rewritten this way, the equation shows that the price of labor – its marginal
product – is equal to the ratio of the marginal utilities of consumption and
leisure. This ratio of marginal utilities is called the marginal rate of substitution,
which is equated to the price of labor. The marginal rate of substitution tells us
9
how the consumer values in utility terms trading off consumption versus leisure.
The marginal product (wage rate) tells us how the market values the trade-off
between consumption and leisure. Thus, an optimum has the feature that the
consumer’s willingness to trade off consumption and leisure, which comes from
their preferences, is equated to the market’s valuation of using labor to produce
consumption, which comes from the production technology.
5 Example Economy
Consider the following specific functional forms. For the utility function, we
use:
c1−σ
1− σ
+ l
where the parameter σ > 0 measures the degree of curvature in the utility
function with respect to consumption. (This is a commonly used utility function
in economics). The production function is given by:
y = f(k, n) = kαn1−α
Since the allocations for the competitive equilibrium and the social planning
problem are the same, we solve the simpler social planning problem, which we
express as a Langrangian, and in which we subsitute out for n using the fact
that 1 = l+n
L = max{
c1−σ
1− σ
+ l + λ[kα(1− l)1−α − c]}
The first order conditions are
c−σ = λ
1 = λ(1− α)
(
k
1− l
)α
kα(1− l)1−α = c
Using these equations, we can obtain solutions for the endogenous variables
consumption, leisure, (and labor) in the model as follows:
l = 1− [(1− α)(zkα)1−σ]
1
α+(1−α)σ
c = [(1− α)(zkα)1−σ]
1
α+(1−α)σ
10
张熠华�
isoelasticity function
张熠华�
Practice exercise: verify these solutions, and also solve for the factor prices
r and w, and show how these factor prices are increasing in z, and explain why
this is the case.
6 An Economy with Two Types of Consumers
In this section, we analyze an economy with two types of consumers, who differ
in the type of labor they supply. This allows us to study a model with hetero-
geneity, as opposed to the previous model in which all consumers were alike.
There are two types of consumers. We retain the assumption that they have
the same preferences for consumption and leisure. Each consumer supplies a
different type of labor into production of a consumption good. The wage rate
received by the first individual is w1 and that received by the second individual
is w2. There is a competitive firm with a constant returns to scale production
function that hires these two types of labor to produce the consumption good.
There are three steps in solving for a competitive equilibrium problem here..
(1) set up the consumer maximization problems and solve for their first order
conditions, (2) set up the firm’s maximization problem and solve for their first
order conditions, and (3) use the first order conditions and the constraints to
solve for the endogenous variables. The result will be solving n unkowns in n
equations – a standard algebra problem. The solution will be the equilibrium
values for this economy.
The consumer utility functions are increasing and concave in consumption.
Rather than specifying utility over leisure as we did above, let’s alternatively
specify disutility over work, which will be decreasing and convex in labor, and
continuous. Let the subscript i denote the individual, and their preferences and
budget constraint are given by:
maxu(ci)− v(li) (1)
subject to
wili ≥ ci (2)
The Lagrangian for each consumer is:
Li = max{u(ci)− v(li) + λi[wili − ci]} (3)
The first order conditions for the consumer are
11
u′(ci) = λi (4)
v′(li) = λiwi (5)
Substituting the first equation into the second, we get:
v′(li) = u
′(ci)wi (6)
The firm’s maximization problem is
max 2Al
1/2
1 l
1/2
2 − w1l1 − w2l2 (7)
The firm’s first order conditions are:
w1 = A
(
l2
l1
)1/2
, w2 = A
(
l1
l2
)1/2
(8)
Note that this model tells us what determines factor prices – their marginal
productivities.
The economy’s resource constraint states that:
2Al
1/2
1 l
1/2
2 = c1 + c2 (9)
We have 7 equations – the resource constraint, one first order condition for
each of the two consumers, the consumer budget constraints, and the firm’s
two first order conditions, and we have 7 unknowns – the consumption of each
consumer, the labor of each individual, the wage rate paid to each individual,
and the total amount of output in the economy. Given the assumptions we
placed on the utility and production functions, a unique solution exists for the
variables in this economy that boils down to solving this set of equations.
Here is a summary of the solution:
Substitute out for the wage rates in the consumer’s first order conditions
using the fact that wage rates are equal to the marginal products of labor:
v′(l1) = u
′(c1)A
(
l2
l1
)1/2
(10)
v′(l2) = u
′(c2)A
(
l1
l2
)1/2
(11)
12
Next, use the above two equations, the consumer’s budget constraints and
the resource constraint of the economy to solve for the variables.
The consumer budget constraints are:
c1 = w1l1 = Al
1/2
1 l
1/2
2
c2 = w2l2 = Al
1/2
1 l
1/2
2
Thus, each consumer consumes half of the output of the economy. These
equations also imply that each consumer works the same amout.
You should solve these equations yourselves to make sure you completely
understand how this works.
Before we move to the social planning solution for this economy, note how the
two types of consumers “cooperate” with each other through the marketplace.
In particular, each consumer benefits enormously from the other. Specifcally,
suppose that worker 2 did not show up for work. Then output is zero, and
worker 1 receives no income! The same situation occurs if worker 1 doesnt show
up for work, but work 2 does. In that case, there also is no production, and
worker 2 receives no income! Thus, each worker, making a decision solely on
the basis of their own interest, ends up making a decision that is good for the
other person.
Now we compare the equilibrium allocations in this economy to the Pareto
optimal allocations for this economy and we will see that they are the same. This
will deliver the result that the competitive equilibrium allocations are indeed
Pareto optimal.
Setting up an optimization problem to solve for Pareto optimal
allocations.
We wish to maximize social welfare. In this case, social welfare means adding
together the utility functions of the two consumers. Thus our maximization
problem is
max{u(c1)− v (l1) + u(c2)− v (l2) (12)
We maximize this function subject to the resource constraint of the economy.
The resource constraint tells us that the allocations of output to society must
not exceed how much output is produced.
13
2Al
1/2
1 l
1/2
2 = c1 + c2 (13)
The Lagrangian is
L = max{u(c1)− v (l1) + u(c2)− v (l2) + λ[2Al
1/2
1 l
1/2
2 − c1 + c2]} (14)
This is all there is to setting up a problem to determine the Pareto optimal
allocations. We have five first order conditions: one for each consumption, one
for each labor, and the resource constraint. We get:
u′(ci) = λ (15)
v′(li) = λmpli (16)
2Al
1/2
1 l
1/2
2 = c1 + c2 (17)
Note that the first equation tells us that the marginal utility for each con-
sumer is the same. Since both consumers have the same utility function,
and marginal utility only depends on consumption, this implies that the two
consumers will have the same amount of consumption. Example: suppose
u(ci) = log(ci). Then we have marginal utilities being equal means that
1
c1
= 1
c2
.
This means that c1 = c2, which further implies that each consumer receives half
of output for their consumption.
Now, substituting out for the Lagrange multiplier, λi, we get
v′(l1) = u
′(c1)A
(
l2
l1
)1/2
(18)
v′(l2) = u
′(c2)A
(
l1
l2
)1/2
(19)
2Al
1/2
1 l
1/2
2 = c1 + c2 (20)
Substituting in the fact that c1 = c2 = Y/2 means that we have three
equations in three unknowns: l1, l2, and Y in three equations.
Showing that the competitive equilibrium allocations are Pareto
optimal
Now, we can see that the solution to the Pareto optimal allocations will be
the same as in the competitive equilibrium. To see this, note that the above
14
two consumer first order conditions, in which the wage rates are substituted
out using the marginal products, are exactly the same as in the equilibrium
problem. Moreover, the resource constraint is the same.
15