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Econ 401A October 9, 2016
1
Solving for the Walrasian Equilibrium: Two examples
Example 1:
Every consumer has the same utility function function 1 1
1 2( )U x x x . There is an initial
endowment of 30 units of commodity 1. Commodity 1 is both consumed and used as an input
in the production of commodity 2. The production function is 4q z .
The utility function is homothetic since 1
( ) ( )U x U x
. Then ( ) ( )U x U y implies that
( ) ( )U x U y .
(a) Solve for the production plan that maximizes the utility of the representative consumer.
(b) What must be the WE price ratio?
Example 2:
Every consumer has the same utility function function 4
1 2( )U x x x . There is an initial
endowment of 81 units of commodity 1. Commodity 2 is a produced using commodity 1 as an
input. The production function is 1/26q z .
(a) Show that the utility function is homothetic.
(b) Solve for the production plan that maximizes the utility of the representative consumer.
(c) What must be the WE price ratio?
(d) What is the equilibrium profit of the firm?
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Example 1
Step 1: Feasible outcomes
No production: 1 2 1 2( , ) ( , ) (30,0)x x x .
Use 1 unit of commodity 1 in production. 4q z . So the maximum output of commodity 2 is 4.
Use z units of commodity 1. The maximum output is 4q z .
Then 1 30x z and 2 4x z .
The maximum output of commodity 2 is 4*30 = 120.
The set of feasible alternatives is therefore as depicted.
Preferences
Instead of maximizing U we maximize 1 2ln 4ln lnu U x x Note that
1
1 1
1 1
44
uMU x
x x
and 1
2 2
2 2
1uMU x
x x
.
These are both positive so utility is strictly increasing.
Note also that
21 21 2
2 1
( , ) ( )MU x
MRS x xMU x
2x
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Consider moving from A to B on the level set depicted below. The ratio 2
1
x
x increases.
Therefore the 1 2( , )MRS x x increases. Thus the steepness of the level set is greater at B . Thus
without worrying about the exact shape, we can draw level sets as shown below.
Superimposing the level sets on the first figure, the optimum for the representative consumer is
the point on the boundary of the feasible set tangential to the indifference curve.
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Step 2: Solve for the optimum
For any z the output must satisfy 4q z . Since utility is increasing this must be an equality for
a maximum. All the output of commodity 2 will be consumed therefore
2 4x q z .
Commodity 1 is used both as an input in the production of commodity 2 and as consumption.
Therefore if z units are used in production,
1 1 30x z z .
Substitute these into the utility function.
1 1 1 114
(30 ) (4 ) (30 )U z z z z .
Look on the margin
2 214 2 2
1 1(30 )
(30 ) 4
dUz z
dz z z
This must be zero if the maximizer, * 0z . Then 2 2(30 ) 4z z . Take the square root.
30 2z z
Therefore * 10z and so * *
1 2( , ) (20,40)x x
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Step 3: Solve for prices that support the optimal production plan.
In the model, firms are price takers. Consider any pair of prices 1 2( , )p p p and a production
plan ( , )z q . That is, the firm purchases z units of commodity 1 and sells q units of commodity
2. The profit is
2 1( , )z q p q p z .
For any z the firm will produce the maximum possible output to maximize profit. In this
example the set of feasible plans is the set {( , ) | 4 }S z q q z so the firm will choose 4q z .
Then profit is
2 14p z p z
The marginal profit to increasing the input is therefore
2 14d
p pdz
.
Supporting prices
The price vector p is said to “support” the optimum if * *( , )z q is profit-maximizing. Then
marginal profit must be zero. Then the price vector is supporting if
1
2
4p
p .
If the relative price of the input is higher
then marginal profit is always negative so
* 10z is not supported.
This is depicted in the figure. The level set
of zero profit,
2 1 0p q p z ,
passes through ( , ) (0,0)z q .
Profit is higher in the direction of the
arrows (more output and less input).
The firm therefore maximizes profit by producing nothing.
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If 1
2
p
pis below 4 it is always strictly profitable to increase output so again the prices are not
supporting. Therefore the only possible WE price ratio is 4. In this case the zero profit line is
the green line (the boundary of the feasible set.)
Step 4: Explain why consumer demand is equal to supply at these prices
Since the endowment of commodity 2 is zero, 2x q . Since the endowment of
commodity 1 is 1 30 , 1 1x z .
Therefore
2 2 1 1 2 1 1( )p x p x p q p z
2 1 1 1p q p z p
1 1( , )z q p
In Step 3, we showed that for profit maximization, equilibrium profit must be zero. Therefore
the budget constraint of the representative consumer is
2 2 1 1 1 1p x p x p
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In the figure below this is the green line through the endowment point
Therefore * (20,40)x is utility maximizing, i.e. the choice of the representative consumer.
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Example 2
Step 1: Feasible outcomes
No production: 1 2 1 2( , ) ( , ) (81,0)x x x .
With an input of z units of commodity 1, the maximum output is 1/26q z .
Then 1 81x z and 1/2
2 6x q z .
The maximum output of commodity 2 is 1/26*(81) 54 .
The set of feasible alternatives is depicted below
Preferences
It is simpler to maximize 1 2ln 4ln lnu U x x . Note that
1
1 1
1 1
44
uMU x
x x
and 1
2 2
2 2
1uMU x
x x
.
These are both positive so utility is strictly increasing.
Note also that
1 21 2
2 1
( , ) 4MU x
MRS x xMU x
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Arguing as in Example 1, moving from A to B on the level set depicted below, the 1 2( , )MRS x x
increases. Thus the steepness of the level set is greater at B . Thus without worrying about the
exact shape, we can draw level sets as shown below.
Superimposing the level sets on the first figure, the optimum for the representative consumer is
the point on the boundary of the feasible set tangential to the indifference curve.
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Step 2: Solve for the optimum
For any z the output must satisfy 1/26q z . Since utility is increasing this must be an equality
for a maximum. All the output of commodity 2 will be consumed therefore
1/2
2 6x q z .
Commodity 1 is both consumed and used as an input in the production of commodity 2.
Therefore if z units are used in production,
1 1 81x z z .
Substitute these into the utility function.
1/2 12
4ln(81 ) ln6 4ln(81 ) ln6 lnu z z z z .
Look on the margin
4 1
81 2
du
dz z z
This must be zero if the maximizer, * 0z . Then 81 8z z .
Therefore * 9z and so * *
1 2( , ) (72,18)x x
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Step 3: Solve for prices that support the optimal production plan.
In the model, firms are price takers. Consider any pair of prices 1 2( , )p p p and a production
plan ( , )z q . That is, the firm purchases z units of commodity 1 and sells q units of commodity
2. The profit is
2 1( , )z q p q p z .
For any z the firm will produce the maximum possible output to maximize profit. In this
example the set of feasible plans is the set 1/2{( , ) | 6 }S z q q z so the firm will choose 1/26q z . Then profit is
1/2
2 16p z p z
The marginal profit to increasing the input is therefore
1/2 22 1 11/2
33
pdp z p p
dz z
.
Supporting prices
The price vector p is said to “support” the optimum if * *( , )z q is profit-maximizing. Then
marginal profit must be zero at * 9z . Then the price vector is supporting if
* 1/2
1
2
( )1
3
p z
p .
The zero profit level set is the green line
through the endowment point.
The maximum profit level set is the
green line,
2 1p q p z ,
tangential to the boundary of the
production set at * *( , )z q
So this must have a slope of 1.
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Step 4: Explain why consumer demand is equal to supply at these prices
Since the endowment of commodity 2 is zero, 2x q . Since the endowment of
commodity 1 is 1 81 , 1 1x z .
The level set for maximized profit is
2 1p q p z
But 2q x and 1 1z x .
Therefore
2 2 1 1 1( )p x p x
Rearranging this equation,
2 2 1 1 1 1p x p x p .
Thus the maximum profit line is also the budget line for the representative consumer.
Therefore * (72,18)x is utility maximizing, i.e. the choice of the representative consumer.
