department of economics eco 204 microeconomic theory ......eco 204, 2012 - 2013, test 2 this test is...

ECO 204, 2012 - 2013, Test 2 This test is copyright material and cannot be used for commercial purposes or posted anywhere without prior permission. Report violations to [email protected]

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S. Ajaz Hussain, Dept. of Economics, University of Toronto (STG)

Department of Economics ● ECO 204 ● Microeconomic Theory for Commerce ● 2012 – 2013 ● Ajaz Hussain

Test 2 Solutions

IMPORTANT NOTES:

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will accompany you to the washroom. You are allowed to go to the washroom ONLY.

You are required to stop writing and turn your exam face down when asked to stop by the instructor or proctor at the end of

the exam

Please note that proctors will take down your name for academic offenses, which will be treated in accordance with the policies as

published by the Faculty of Arts and Sciences

Exam details:

Duration: 1 hour and 50 minutes

Total number of questions: 4

Total number of pages: 26

Total number of points: 100

Please answer all questions. To earn credit you must show all calculations. Please see last page of this exam for the allocation of points

across questions.

Exam aids:

This is a closed note and closed book exam.

You may use a non-programmable calculator. Sharing is not allowed.

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Question 1 [25 POINTS]

Consider an agent in an economy with a single good (corn) and where all agents live for exactly two periods At

the beginning of each period, the agent is endowed with “real” incomes and (in units of corn). Let and

denote the amounts of corn consumed in and respectively. Assume that the inter-temporal consumption

set is {( ) }

At agents can save or borrow corn at the real interest rate . Agents do not save to bequeath savings to

future generations because in this economy there’s no romance and no possibility of amorous encounters between

agents – as such, agents do not have children.

(1.1) [3 POINTS] Suppose an agent tells you that she must consume positive amounts of corn in both periods and that she

perceives and as “imperfect substitutes” with diminishing marginal rate of substitution. Write down the equation

of a utility function that represents this agent’s preferences and show why this utility function represents the agent’s

preferences. Show all calculations and state all assumptions.

Answer

We can represent the agent’s preferences over inter-temporal consumption by the Cobb-Douglas utility function:

Where are positive parameters. Notice that the for we must have and which means she must

consume corn in each period. The marginal rate of substitution (slope of indifference curves) is:

As we see that | | (i.e. diminishing marginal rate of substitution).


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(1.2) [7 POINTS] State and solve this agent’s inter-temporal Utility Maximization Problem (UMP) with a real (Future

Value) inter-temporal budget constraint. To earn credit you must use the appropriate constrained optimization method,

solve for all variables, and interpret any “Lagrange multipliers”. Show all calculations. Hint: If applicable, use a

convenient positive monotonic transformation of the utility function in part (1.1).

Answer

We can represent the agent’s preferences over inter-temporal consumption by the Cobb-Douglas utility function:

Where are positive parameters. For convenience, take the log-linear transformation:

The inter-temporal UMP is:

( ) ( )

Assuming , combined with , implies that at the optimal solution . As such, we can drop the

non-negativity constraints:

( ) ( )

The Lagrangian equation is:

{ ( ) ( ) }

For convenience, denote ( ) so that:

{ ( ) }

The FOCs are (there are no KT conditions – why?):

( )

{ ( ) }

Re-arranging the first two FOCs gives:

( )

The 3rd FOC implies:

( )


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Solving

( ) and ( ) simultaneously yields:

( )

Substitute these in any of the first two FOCS to get:

By the envelope theorem, what is the interpretation of ? The change in due to a small change in arising from a

change in either or but not is:

⏟

{ ( ) }

|

|

At the optimum, so that:

|

That is, is the marginal utility of future value lifetime income due to a small change in either or but not :

|

Notice that this is strictly positive.


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(1.3) [5 POINTS] Suppose the agent is given the opportunity of having either one more unit of corn income at or one

more unit of corn income at . True or false: the agent will prefer to have one more unit of corn income at ?

Show all calculations.

Answer

By the envelope theorem, the change in due to a small change in arising from a change is:

⏟

{ ( ) [ ( ) ⏟

]}

( )

( )

At the optimum, so that:

( )

This is the marginal utility due to a small change in . Applying the envelope theorem to change in yields:

Notice that since :

( )

Thus, the agent prefers to have more corn at . This is the classic finance principle of “a dollar today is worth more

than a dollar tomorrow”.


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(1.4) [3 POINTS] Use your answer to part (1.2) to calculate that value of the real interest at which the agent is

indifferent between saving and borrowing at . Show all calculations.

Answer

Recall that the savings function can be described by the following equation where we substitute for the optimal value of

consumption at time , from part (1.2):

( )( )

The agent is indifferent between saving and borrowing at when . Setting

in the equation above,

solve for the value of the real interest rate

( )( )

( ) ( ) ( )

̂

Therefore, at ̂

the agent is indifferent between saving or borrowing at


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(1.5) [4 POINTS] Suppose the current real interest rate is the value calculated in part (1.4). If the government levies a

small flat income tax (in units of corn) on will the agent become a saver or a borrower at ? Show all calculations.

Answer:

Suppose, ̂

, i.e. the agent is indifferent between saving and borrowing at

If the government levies a small flat income tax at , then the new future value of income, , is:

̂ ( )( ̂)

And savings at are:

̂ ̂

( )( ̂)

( )( ̂) ( )( ̂)

Simplifying the expression above yields:

̂

since by assumption.

Therefore, at the interest rate ̂ from part (1.4) such that the agent indifferent between saving and borrowing, the agent

becomes a saver if the government levies a small flat income tax at .


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(1.6) [3 POINTS] Use your answer to part (1.2) to answer this question. True or false: a small flat income tax (in units of

corn) on has a larger impact on consumption at than consumption at ? Show all calculations.

Answer

With a small flat income tax, the future value of income is given by:

̂ ( )( )

Using the solution from part (1.2) for optimal consumption at and , we can compare the impact of a small

income tax on consumption in two periods:

( )

Impact of a small flat income tax, on consumption at time is

:

( )( )

( )

( )

( )( )

Impact of a small flat income tax, on consumption at time is

:

( )

( )

( )

( )

A small flat income tax has a larger impact on consumption at time if

( )

( )


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(2.1) [3 POINTS] The following table contains the monthly closing price and monthly dividend yields on Sunoco and JP

Morgan stocks for the last 6 months of 2011:

Date Sunoco Price Sunoco Dividend Yield JP Morgan Price JP Morgan Dividend Yield

7/29/2011 40.65 0 40.45 0.006107

8/31/2011 38.14 0.00369 37.56 0

9/30/2011 31.01 0 30.12 0

10/31/2011 37.23 0 34.76 0.008301

11/30/2011 38.81 0.004029 30.97 0

12/30/2011 41.02 0 33.25 0

Source: CRSP through U of T’s CHASS system

● If the last digit of your ID # ends in 0, 2, 4, 6, 8 Calculate Sunoco’s monthly capital gains and monthly dividend

issued and enter these values below. Give a brief explanation on how to calculate capital gains and dividends.

● If the last digit of your ID # ends in 1, 3, 5, 7, 9 Calculate JP Morgan’s monthly capital gains and monthly dividend

issued and enter these values below. Give a brief explanation on how to calculate capital gains and dividends.

Answer:

Numbers in cells correspond to Sunoco/JP Morgan respectively:

Circle the stock you are analyzing:

Sunoco JP Morgan

Date Monthly Capital Gains Dividend Issued

July 2011 NA/NA

NA/NA

August 2011 -0.0617/-0.071

0.1499985/0

September 2011 -0.1869/-0.1981

0/0

October 2011 0.2006/0.1541

0/0.25003

November 2011 0.0424/-0.1090

0.14999967/0

December 2011 0.0569/0.0736

0/0

Brief explanation on how to calculate capital gains and dividends:

To calculate capital gains in period

To calculate dividend issued in period ( )

http://dc1.chass.utoronto.ca/


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(2.2) [3 POINTS] On December 1st, 2011, the monthly interest rate on 3-month US T-Bills and reported on an annualized

basis was 0.01%. Calculate the monthly interest out of 100 on a “de-annualized” basis. Hint: You were asked to do this in

Excel Project #1. Only if you are unable to calculate this number here should you use the value 0.00002 below for the

“risk free rate” (note this is NOT the correct value and you will get a zero in all parts below if you calculate the correct

value here but use 0.00002 below instead).

Answer

Monthly rate calculated on a “de-annualized” basis is equal to

( )

( )

(2.3) [3 POINTS] The following table shows the average monthly returns and risk of Sunoco and JP Morgan stocks as well

the covariance of Sunoco and JP Morgan monthly returns over the period May 1969 to December 2011:

Sunoco JP Morgan

Average Monthly Return

0.0106 0.0114

Risk 0.0863 0.0917

Covariance 0.0016

Briefly explain: ● how to calculate the “risk” of a stock ● why JP Morgan stocks offer higher returns (on average) than

Sunoco stocks.

Answer:

(1) The “risk” of a stock is the standard deviation of its returns.

(2) JP Morgan offers higher returns on average than Sunoco in order to compensate its investors for the higher risk

associated with holding JP Morgan stocks.


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(2.4) [4 POINTS] Write down an investor’s “mean-variance” utility function (be sure to use the correct notation for the

utility function parameter(s)). True or false: as the asset becomes riskier, the investor demands higher returns but at a

decreasing rate? Show all calculations

Answer:

An investor’s “mean-variance” utility function over the return and risk is:

[ ]

where:

- is the expected return on portfolio

- is the risk-aversion parameter

- is the variance/risk of the portfolio

An investor is facing a trade-off between the risk and return of the portfolio which can be described by the marginal rate

of substitution (MRS) between the risk and return:

It is true that the investor demands higher returns for the riskier asset: as seen from the MRS, to keep investor

indifferent, if risk increases by one unit, the expected return has to increase by (assume that the investor is risk

averse such that the risk-aversion parameter implies that ).

To check whether investor requires higher returns at a decreasing rate, take derivative of MRS with respect to risk,

and see whether MRS is decreasing in

Therefore, investor demands higher returns for riskier asset at an increasing rate (False!).


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(2.4) [5 POINTS] For the month of January 2012, an investor wishes to construct a portfolio consisting of:

● If the last digit of your ID # ends in 0, 2, 4, 6, 8 Sunoco stocks and 3-month US T-Bills.

● If the last digit of your ID # ends in 1, 3, 5, 7, 9 JP Morgan stocks and 3-month US T-Bills.

Using the “risk free rate” calculated in part (2.2), calculate that value of the “mean-variance” utility function parameter

at which the investor will allocate 100% of portfolio funds in the risky asset. You are expected to solve the appropriate

UMP. Show all calculations to 4 decimal places.

Answer:

Let be a fraction of portfolio invested in a risky asset.

Notice that the expected return of the investor’s portfolio is equal to a weighted average of the returns of the risky and

risk free asset:

[ ] [ ] ( ) (1)

The variance of the portfolio is equal to (apply the variance formula from ECO220Y to ):

( )

⏟

( ) ⏟

(2)

The UMP of the investor is:

[ ]

Substituting (1) and (2) into UMP, the investor’s problem becomes:

[ ] ( )

Solving for yields:

[ ]

If the investor allocates 100% of the portfolio funds into the risky assets, then .

The value of the “mean-variance” utility function parameter is:

[ ]

For Sunoco,

For JP Morgan,


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(2.5) [2 POINTS] For the month of January 2012, construct a portfolio consisting of:

● If the last digit of your ID # ends in 0, 2, 4, 6, 8 Sunoco stocks and 3-month US T-Bills by using the “risk free rate”

calculated in part (2.2) and 1.05 times the value of the “mean-variance” utility function parameter calculated in part

(2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor

leveraging her portfolio and if so, indicate how she does this.

● If the last digit of your ID # ends in 1, 3, 5, 7, 9 JP Morgan stocks and 3-month US T-Bills by using the “risk free rate”

calculated in part (2.2) and 0.8 times the value of the “mean-variance” utility function parameter calculated in part (2.4).

What fraction of the portfolio is in the risky asset, what is expected return and risk of this portfolio? Is the investor

leveraging her portfolio and if so, indicate how she does this.

Show all calculations to 4 decimal places.

Answer

Sunoco JP Morgan

̃

̃

̃

[ ]

[ ]

[ ] [ ] ( )

[ ]

[ ] [ ] ( )

[ ]

Note:

In this case, the investor is not leveraging portfolio

In this case, the investor is leveraging portfolio by selling 3-month US T-Bills and

buying JP Morgan stocks


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(2.6) [4 POINTS] For the month of January 2012, an investor wants to create a synthetic risky asset consisting of Sunoco

stocks and JP Morgan stocks. What fraction of the synthetic risky asset is in each risky asset and what is expected return

and risk of this synthetic risky asset? Use the risk free interest rate calculated in part (2.2), and if relevant, feel free to

use the utility parameter value from part (2.5) and the formula below. Show all calculations.

( )

( )

( ) ( )

[ ]

Proctors are NOT allowed to tell you anything about this formula. Please indicate which stocks are “A” and “B” and

show all calculations to 4 decimal places.

Answer

Let “A” be Sunoco stocks and let “B” be JP Morgan stocks. Then is a fraction of the synthetic risky asset in Sunoco

stocks. Using the formula above:

( ) ( )

( ) ( )

Therefore, the fraction of the synthetic risky asset in Sunoco stocks is 51% and the fraction in JP Morgan stocks is 49%.

The expected return and risk of the synthetic risky asset are:

[ ] ( )

( )

( )

√


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(2.7) [3 POINTS] For the month of January 2012, construct a portfolio consisting of:

● If the last digit of your ID # ends in 0, 2, 4, 6, 8 3-month US T-Bills and the synthetic risky asset from part (2.6). Use

the “risk free rate” calculated in part (2.2) and use 0.9 times the value of the “mean-variance” utility function parameter

calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this

portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this.

● If the last digit of your ID # ends in 1, 3, 5, 7, 9 3-month US T-Bills and the synthetic risky asset from part (2.6). Use

the “risk free rate” calculated in part (2.2) and use 1.1 times the value of the “mean-variance” utility function parameter

calculated in part (2.4). What fraction of the portfolio is in the risky asset, what is expected return and risk of this

portfolio? Is the investor leveraging her portfolio and if so, indicate how she does this.

Show all calculations.

Answer

ID # ends in 0, 2, 4, 6, 8 ID # ends in 1, 3, 5, 7, 9

̃

̃

̃

[ ]

[ ]

Note:

218% of this portfolio is invested in the synthetic risky asset. The investor is leveraging by short-selling the risk-free asset – 3 month US T-Bills

187% of this portfolio is invested in the synthetic risky asset. The investor is leveraging by short-selling the risk-free asset – 3 month US T-Bills

[ ] [ ] ( )

[ ]

[ ] [ ] ( )

[ ]


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(2.8) [3 POINTS] Use your answer to part (2.7) to answer this question. An investor has $5,000 to invest. Fill the entries in

the following table and show all calculations.

Answer

● If the last digit of your ID # ends in 0, 2, 4, 6, 8 An investor has $5,000 to invest. Fill the entries in the following table

and show all calculations below:

US 3-Month T-Bills Synthetic Asset Sunoco JP Morgan

Fraction of Portfolio in: -1.1754 2.1754 1.10959 1.0658

Dollars in: -5,876.93 10,876.93 5,547.94 5,328.99

Show calculations:

( )

( )

● If the last digit of your ID # ends in 1, 3, 5, 7, 9 An investor has $8,000 to invest. Fill the entries in the following table

and show all calculations below:

US 3-Month T-Bills Synthetic Asset Sunoco JP Morgan

Fraction of Portfolio in: -0.8687 1.8687 0.9532 0.9156

Dollars in: -6,949.75 14,949.75 7,625.34 7,324.41

Show calculations:

( )

( )


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Phew! You passed ECO 204 and cruised through your third and fourth year courses. But you’re one course shy of

graduation. In desperation, you look for a “bird course” (like ECO 204) and after consulting your brainy friends you settle

on RSM 000 “The Business of Nothingness” (nope, it’s not a marketing course). The trouble is that this course is full (after

all, it’s a bird course) and you must choose one of the following options:

Either offer the RSM 000 professor a $200 bribe to let you in. With a chance ( ) the professor accepts the

bribe (tsk tsk) and immediately enrolls you in the course, for which you’ll have to pay $4,000 in tuition. On the

other hand, with a chance the professor is offended that you tried to bribe him and reports you for academic

violation (life sucks). The school confiscates your bribe, kicks you out, and fines you $2,000.

Or don’t bribe the RSM 000 professor and wait to see if you get into the course. With probability you do

get into RSM 000 in which case your total cost will be $4,000 in tuition and an opportunity cost of $500. On the

other hand, with probability ( ) you don’t get into RSM 000 in which case you will have to take a

course at Ryerson University which costs you $3,000 in tuition and an opportunity cost of $500.

(3.1) [7 POINTS] Assuming you are risk neutral, should you bribe the RSM 000 professor to let you into his course? Show

all calculations and be sure to interpret your calculations. Hint: It is strongly recommended that you draw a decision

tree.

Answer

Draw decision tree here. Show calculations below.

Decision

Bribe

Bribe accepted with probability (1-p) and you must pay the

bribe ($200) + tuition ($4000) = $4,200

You are reported for academic violaitons with probability p and you must pay the bribe ($200) +

a fine ($2000 )= $2,200

Do Not Bribe

You are enrolled into RSM 000 with probability q = 0.1 and you must pay tuition at ($4000) + an

opportunity cost ($500) =$4,500

You taka a course at Ryerson with probability (1-q) = 0.9 and you must pay tuition at Ryerson ($3000) + an opportunity cost

($500) = $3,500


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You have to make a choice between “Bribe” and “Do Not Bribe” based on the expected values of the corresponding

outcomes (see the decision tree).

Expected value of the “Bribe” option is:

( ) ( )( )

Expected value of “Do Not Bribe” option is:

( ) ( )( ) ( ) ( )

The decision rule is to bribe if . For instance, you should bribe if the following inequality

holds:

This implies that you should bribe if the probability that the professor reports you for academic violations (or simply the

probability of being caught) is equal or greater than 0.3


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(3.2) [5 POINTS] Assume that the probability the professor will not take the bribe and report you for academic violations

is . Will you try to bribe the professor and what is the expected value of your decision? Show all calculations.

Answer

From part (3.1) we know that it is optimal to bribe if the probability that the professor will not take the bribe is greater

or equal to 0.3. Let’s show numerically that the expected value of “Do Not Bribe” when probability of being caught is 0.2

is greater than the expected value of “Bribe”:

( )

Since expected value of “Bribe” is smaller than expected value of “Do Not Bribe”, it is optimal not to bribe the professor.


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(3.3) [10 POINTS] Assume the probability that the professor will not take the bribe and report you for academic violations

is Now suppose a friend of yours, as well as the professor’s, tells you that he will for a fee “test” the professor’s

integrity and honesty. Perhaps your friend will bring up the subject of bribes and get a feel for whether the professor will

take the bribe. You need to decide whether to make the decision of whether to bribe or not bribe the professor on the

basis of no test information (as you did in part (3.2)) or on the basis of test information. Since your friend has done this

sort of stuff in the past you ask him for a breakdown of test results and what actually happened. Here’s what your friend

gives you:

Professor Took Bribe Professor Did not Take Bribe Total

Positive (evidence that professor is corrupt) 15 5 20

Negative (evidence that professor is not corrupt) 5 75 80

Total 20 80 100

Based on the information in this table, should you make the decision to bribe versus not bribe the professor by hiring

your friend to do “tests” on the professor? Draw the decision tree below, show all calculations, and indicate what you

will do if the test comes back positive or negative.

Answer

Draw decision tree here. Show calculations below.

Decision

Test result is positive with

probability 0.2

Bribe

Do Not Bribe

Test result is negative with probability 0.8

Bribe

Do Not Bribe

Professor took the bribe with

probability 0.75

You are reported for academic violations with probability 0.25

RMS 000 with probability

q=0.1

Course at Ryerson with

probability 0.9

Professor took the bribe with

probability 0.0625

You are reported for academic violations

with probability 0.9735

RMS 000 with q=0.1

Course at Ryerson with

0.9


Page 21 of 27


First, let’s determine the optimal decision (Bribe or Do Not Bribe) conditional on the result of the test. To do that, we

need to compute four conditional probabilities from the “historical” data:

( | ) ( )

( )

( | ) ( )

( )

( | ) ( )

( )

( | ) ( )

( )

Now, let’s determine the optimal decision for each of the positive or negative result of the test. When the test is

positive:

| ( ) ( )

|

| |

Since the expected value from the “Bribe “option when test is smaller is bigger than the expected value from the “Do

Not Bribe” option, you should not bribe the professor.

When the test is negative:

| ( ) ( )

|

| |

When the test is negative, you should bribe the professor.

Now, the expected value of the decision to bribe or not to bribe the professor by hiring your friend is equal to:

( ) ( )

( ) ( )

( )

The expected value of decision with test (-$2,580) is bigger than the expected value of the decision without the test (-

$3,600). So, you should hire your friend to do “tests” on professor.


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(3.4) [4 POINTS] [THIS IS INDEPENDENT OF PARTS (3.2) AND (3.3)] Recall that you are risk neutral. From the information above,

derive the equation of your utility function and compute the utilities of the four outcomes in your decision problem.

Show all calculations and briefly explain the steps used to derive the utility function.

Answer

Since the investor is risk-neutral, the utility function is linear and in general form can be written as , where

is a slope and is a constant. Recall that utility is ordinal, i.e. we can assign arbitrarily numbers to utility levels as long as

these numbers satisfy the ranking of preferences. For instance, let’s assign value of 100 to the best/highest outcome (-

$2,200) and 0 to the worst/lowest outcome (-$4,500).

We want to find the parameters of the utility function, and . Consider two equations for two levels of utility which

corresponds to the lowest and highest outcomes:

{ ( )

( )

And the utility function is:

The levels of utility corresponding to the four outcomes are:

-$2,200 100

-$3,500 43.51

-$4,200 13.076

-$4,500 0


Page 23 of 27


(3.5) [4 POINTS] [THIS IS INDEPENDENT OF PARTS (3.2) AND (3.3)] Use your answer to part (3.4) to compute the certainty

equivalence to the uncertainty arising from ● bribing the professor ● not bribing the professor.

Answer

The utility function from part (3.4) is

Since the investor is risk-neutral,

By the definition of the certainty equivalent, :

By the same logic, :


Page 24 of 27



[ALL PARTS ARE INDEPENDENT OF EACH OTHER] The original formula and production process for manufacturing Coca-Cola did

not permit the use of any other sweetener besides sugar.

(4.1) [5 POINTS] Assuming sugar is the only input for producing Coca-Cola, write down the equation of the long run

production function, plot an iso-quant with sugar on the x-axis and fructose on the y-axis, and characterize the returns

to scale.

Answer

The production function with only one input, , and productivity/management parameter is:

This production function exhibits constant returns to scale. That can be easily checked by doubling all the inputs:

( ) ( )

Sugar

Fructose q1

S1

q2

S2


Page 25 of 27


(4.2) [5 POINTS] Assuming sugar, capital and labor are the only complementary inputs for producing Coca-Cola (where

each machine requires 3 units of sugar and 2 workers) write down the equation of the long run production function, plot

an iso-quant with sugar on the x-axis and labor on the y-axis, and characterize the returns to scale.

Answer

Denote machines by , workers by and sugar by . We know that the inputs are combined in the production such that

The long run production function with complementary inputs has the functional form:

( ) {

}

This production function exhibits constant returns to scale and we can check it by doubling all the inputs:

( ) {

}

( ) {

}

( ) ( )

Sugar

Labor


Page 26 of 27


(4.3) [5 POINTS] Eventually Coke was able to reformulate Coca-Cola’s formula and manufacturing process to allow the use

of sugar and/or fructose as imperfect substitutes (with diminishing technical rate of substation) and permitting the use

of no sugar or no fructose. Write down the equation of a long run production function that describes this production

process, plot an iso-quant with sugar on the x-axis and fructose on the y-axis, and characterize the returns to scale.

Answer

There are many examples of production functions that would fit these criteria. One class is quasi-linear production

functions.

For example,

√ √

We can characterize the returns to scale by doubling all the inputs:

( ) √

( ) √ √

This long run production function exhibits decreasing returns to scale as the scaling parameter at √ is smaller than 2.

In other words, doubling all inputs leads to increase in the output less than by two.

The graph of the iso-quant for this production function is described by equation:

√

Fructose

Sugar


Page 27 of 27


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Question Maximum

Possible Points Score

1

2

3

4

Total Points = 100

department of economics eco 204 microeconomic theory ......eco 204, 2012 - 2013, test 2 this test is...

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