ECO 204, Summer 2013, Test 1
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
University of Toronto | Department of Economics | ECO 204 | Summer 2013 | Ajaz Hussain
TEST 1 SOLUTIONS
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 1 [20 POINTS]
[ALL PARTS ARE INDEPENDENT OF EACH OTHER]
(a) [5 POINTS] Consider the following problem:
( )
Under what conditions is a stationary point (i.e. where ( ) ) a (or the) solution to this problem?
Answer
If we’ve found a stationary point then we can be assured it is a solution to the problem if the function is concave, i.e.
( ) , and that it is the solution if the function is strictly concave, i.e. ( ) . In such cases, there’s no need to
check for boundary solutions.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [5 POINTS] Consider the following problem:
( )
At the optimal solution to this problem, why will the optimal value of the Lagrange equation equal ? Show all
calculations and state all assumptions.
Answer
This problem is solved as follows:
( )
( )
( )⏟
⏟
The Lagrange Method
( )
Since we see that:
( )⏟
[ ⏟
]
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(c) [5 POINTS] Consider the following problem:
( ) ⏟
At the optimal solution to this problem, why will the optimal value of the Lagrange equation equal ? Show all
calculations and state all assumptions.
Answer
This problem is solved as follows:
( )
( )
( )
( )⏟
⏟
( )⏟
( )
Since the “animal” we see that:
( )⏟
⏟
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(d) [5 POINTS] Give an example of a single-variable function that is both concave and convex but not strictly concave nor
strictly convex.
Answer
A concave but not strictly concave function is defined as ( ) but not ( ) while a convex but not strictly
convex function is defined as ( ) but not ( ) . Thus, for a function to be concave and convex we require
that ( ) everywhere. The only function with this property is a linear function like .
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 2 [10 POINTS]
[ALL PARTS ARE INDEPENDENT OF EACH OTHER]
(a) [5 POINTS] A business analyst has solved the following problem:
Here
After solving the problem, the analyst finds that the value of the Lagrange multiplier is . Interpret this result
and, if appropriate, make a recommendation to the analyst. Explain your answer.
Answer
To answer this question we need to know what measures. By the envelope theorem the change in due to a small
change in is given by:
Thus:
This means that increasing capacity by 1 unit will decrease revenues by $0.50. You should recommend that capacity be
reduced until
.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [5 POINTS] A business analyst has solved the following problem:
The analyst tells you that the value of the Lagrange multiplier is . What do you recommend the analyst do?
Explain your answer.
Answer
This is an inequality constrained problem and we know that for such problems . The fact that means that the
analyst made a mistake in his calculations.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 3 [15 POINTS]
[ALL PARTS ARE INDEPENDENT OF EACH OTHER]
(a) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom:
Assume neither good can be consumed in negative amounts.
Answer
Here burgers and soda cans must be consumed in integer amounts so that:
{ }
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom:
Assume neither good can be consumed in negative amounts.
Answer
Here burgers must be consumed in integer amounts while gallons of sodas can be consumed in any amount:
{ }
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(c) [5 POINTS] Graph and express mathematically the consumption set of a consumer for whom:
Assume neither good can be consumed in negative amounts.
Answer
Here pounds of burgers and gallons of sodas can be consumed in any amount:
{ }
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 4 [20 POINTS]
[ALL PARTS ARE INDEPENDENT OF EACH OTHER]
(a) [4 POINTS] A consumer perceives goods 1 and 2 to be “good” goods as well as “perfect substitutes” with a marginal
rate of substitution
. Write down two utility functions representing this consumer’s preferences and use one of
these utility functions to state the simplest possible UMP. Do not solve the UMP but do explain how you simplified the
UMP.
Answer
The consumer’s utility function is:
This has:
We are told that:
Hence one utility function is:
Another utility function be obtained by doing any positive monotonic transformation such as:
Now the general linear UMP is:
Since and everywhere in the consumption set, we see that the consumer can choose a
bundle anywhere in the consumption set including the boundaries. As such, we cannot drop the non-negativity
constraints.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [4 POINTS] A consumer perceives goods 1 and 2 to be “good” goods as well as “imperfect substitutes”. She tells you
that she must consume both goods and that she’ll always spend 43% of her income on good 2. Write down two utility
functions representing this consumer’s preferences and use one of these utility functions to state the simplest possible
UMP. Do not solve the UMP but do explain how you simplified the UMP.
Answer
We know that the Cobb-Douglas UMP should be used to model consumers who perceive all goods to be good” goods as
well as “imperfect substitutes”. A property of the Cobb-Douglas model is that the expenditure on any good is always a
constant fraction of income. In fact we know that for:
That:
Now we know that:
Thus:
This implies that:
The utility function is:
Another utility function be obtained by doing any positive monotonic transformation such as:
The UMP is:
Now:
Now notice that:
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{
To see whether there could be a boundary solution we check:
The UMP becomes:
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(c) [4 POINTS] A consumer perceives and to be “good” goods as well as “complements”. A unit of consists of a
combination of and : the consumer perceives 2 units of to be a perfect substitute for 5 units of , and 2
units of to be a perfect substitute for a unit of . Write down a utility function representing this consumer’s
preferences. What is the between and ? Show all calculations.
Answer
Start with and being perceived as “complements”:
( )
Next we know that a unit of is a combination of so that:
Now in the ( ) plane we know that:
This says that 5 units of good 4 are substitutable for 2 units of good 4. Thus: .
Now in the ( ) plane we know that:
This says that a unit of good 5 is substitutable for 2 units of good 4. Thus: . To reconcile this with the fact
that we had earlier we could do:
This still says that a unit of good 5 is substitutable for 2 units of good 4. Thus: .
Combining these we have:
Notice that in the ( ) plane:
This says that 4 units of good 3 are substitutable for 5 units of good 5.
The utility function is:
( ) ( ) ( )
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(d) [4 POINTS] A consumer perceives as a “bad good” and as a “neutral” good. Write down a utility function
representing this consumer’s preferences and graph the indifference curve for an arbitrary level of utility . Show
all calculations.
Answer
The utility function defined over { }
Notice that:
The slope of the indifference curve is:
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(e) [4 POINTS] For the consumer in part (d), could the optimal choice be the bundle ( ) ( )? What about
( ) ( )? Explain briefly. Hint: Feel free to use a graphical argument.
Answer
Since good 1 is a bad good, the optimal choice will have and since good 2 is a neutral good we can have any
quantity where
. Thus, it is possible for ( ) ( ) to be “optimal” so long as
.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 5 [15 POINTS]
(a) [5 POINTS] Consider a UMP where the utility function is defined on the consumption set {( ) }.
Prove that if the consumer has monotone preferences then her marginal utility of income must be strictly positive.
Answer
Consider a general UMP:
( )
From the envelope theorem we know that:
We need to show that if the consumer has monotone preferences then
This is indeed the case from the
first two FOCs where noting that and that :
⏟
⏞
⏟
⏞
⏟
[ ⏟
]
Of course this also implies that expenditure = income.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(b) [10 POINTS] Consider a general UMP where the utility function is defined on the consumption set
{( ) }. Prove that if the optimal choice is in the interior of the consumption set then at the optimal bundle the
indifference curve must be tangent to the budget line. Show all calculations.
Answer
Once again consider a general UMP:
( )
⏟
⏟
If the optimal choice is in the interior then:
The KT conditions, especially the “animals” imply that As such the FOCs become:
⏟
⏞
⏟
⏞
Equating yields:
⏞
⏞
This says that at the interior solution, the indifference curve must be tangent to the budget line.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
Question 6 [20 POINTS]
(a) [5 POINTS] Solve the following problem in two separate ways:
You are expected to use the appropriate constrained optimization methods. Show key calculations and state
assumptions.
Answer
The UMP is:
Method #1
It’s more convenient to take a positive monotonic transformation and work with:
Now:
Now notice that:
{
To see whether there could be a boundary solution we check:
The UMP is:
The FOCs are:
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⏟
⏟
⏟
⏟
⟩
Now:
Sub this in the budget constraint:
(
)
(
)
(
)
Assume so that:
Notice that expenditure on good 1 is a constant fraction of income. Next, from:
Notice expenditures on goods 2 and 3 are also constant fractions of income.
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Finally, let’s solve for .We know that:
(
)
Method # 2
Now:
Now notice that:
{
To see whether there could be a boundary solution we check:
The UMP is:
The FOCs are:
⏟
⏟
⏟
⏟
⟩
Now:
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Sub this in the budget constraint:
(
)
(
)
(
)
Assume so that:
Notice that expenditure on good 1 is a constant fraction of income. Next, from:
Notice expenditures on goods 2 and 3 are also constant fractions of income.
Finally, let’s solve for .We know that:
(
)
(
)
(
) (
)
(
) (
)
We assumed that so that:
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Hate
(b) [5 POINTS] Suppose Without re-solving the problem, calculate the
impact on optimal demands due to a 1% income tax. Show key calculations and state assumptions.
Answer
To use the expressions above we have to re-scale so that . Re-define . Before the income
tax, the consumer’s demands are:
We also know that for either good:
Thus, a 1% income tax will reduce demands of both goods by so that:
One would get the same answer by subbing in the new income into the demand expressions:
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(c) [5 POINTS] Suppose Without re-solving the problem, calculate the impact
on optimal utility due to an 1% income tax in two separate ways. Show key calculations and state assumptions.
Answer
We can compute the change in due to an income tax in two ways: by the envelope theorem and the value function
approach.
The Envelope Theorem Approach
First, write down the objective in terms of parameters (we use the more convenient log linear Cobb-Douglas UMP):
Second, differentiate with respect to the parameter, which in this case is :
Third, evaluate at the optimal solution:
Noting that:
⏟
[
⏟
]
Implies:
The Value Function Approach
First, write down the objective in terms of parameters (we use the more convenient log linear Cobb-Douglas UMP):
Second, sub in the optimal solutions expressed in terms of parameters:
⏟
[
⏟
]
Third, differentiate with respect to the parameter, which in this case is :
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[
]
Nice.
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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)
(d) [5 POINTS] Suppose Suppose the government imposes an excise tax on
good 2 (in dollars per unit) that is designed to raise the same amount of tax revenue as a 1% income tax. Calculate this
excise tax rate on good 2 (dollars per unit). Which tax scheme “hurts” consumers the least? Show key calculations and
state assumptions.
Answer
Revenues from the 1% income tax are:
Now, revenues from an excise tax on good 2 will be:
Now we want:
⏟
Thus:
Post excise tax price of good 2 where
( )
( )
( )
Let’s check if this is right: