UNIVERSITY OF TORONTO Faculty of and April Examinations ... · scores 1 60 2 45 3 45 total score...

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S. Ajaz Hussain, Dept. of Economics, University of Toronto

UNIVERSITY OF TORONTO

Faculty of Arts and Sciences

April Examinations 2011

ECO 204 Y1Y

Duration: 3 hours Total Points: 150 points Examination Aids Allowed: Calculator.

Instructions:

‐ This exam consists of 3 questions in 36 pages, double‐sided.

Please give your name and ID # as it appears in ROSI

Last Name: First Name:

Student ID #

___________________________________________________________________

Question Points Scores 1 602 453 45

Total score (out of 150 points) / 150

YOU MUST STAY SEATED DURING THE LAST 10 MINUTES OF THE FINAL EXAM ‐‐ WAIT UNTIL ALL EXAMS ARE COLLECTED

FOR YOUR CONVENIENCE, THERE ARE TWO WORKSHEETS AT THE END OF THIS EXAM.

KEEP ANSWERS BRIEF. TO EARN CREDIT, SHOW CALCULATIONS AND GIVE ARGUMENTS TO SUPPORT YOUR ANSWERS

Good luck!

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Question 1 [60 Points] A consumer has the following utility function over goods 1 and 2:

She has exogenously given income $ 0, is a price taker with $ 0 and $ 0, and has the consumption set , : , 0 .

(1) [5 Points] Does she have a quasi‐linear utility function? Give a brief explanation. Answer (2) [5 Points] Does she have monotone preferences? If she does (or does not) have monotone preferences then what must be true about her budget constraint? Give a brief explanation and show all calculations. Answer:

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(3) [5 Points] Show that the consumer’s indifference curve for an arbitrary utility level looks like:

Q2

Q1

Calculate the and intercepts, the slope at any bundle , and the slope at the and intercepts. Show all calculations.

Answer:

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(4)[5 Points] Give a graphical and algebraic argument for whether this consumer has convex preferences. Show all calculations. Hint: For the “algebraic” proof, pick any two bundles on an indifference curve and use the definition of convex preferences. Answer:

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(5) [10 Points] Solve the consumer’s Utility Maximization Problem (UMP). You are expected to use the appropriate optimization method (i.e. Lagrange/Kuhn‐Tucker). Show all calculations.

Answer:

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(6) [10 Points] Suppose the consumer’s budget set is given by , , $10, $200, $100 . From the UMP solution, calculate the optimal demands, price elasticity, income elasticity, and cross‐price elasticity of goods 1 and 2 as well as the MU due to a small decrease in income, a small increase in , a small decrease in , and relaxing the constraint that 0. Show all calculations and fill your answers in the table below.

Answer:

Summary of Answers: please fill shaded cells below Good 1 Good 2

Price elasticity Income elasticity

Cross‐price elasticity M g tility due to ar inal U

A small decrease in income Relaxing the constraint

A small increase in A small decrease in

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(7) [10 Points] Suppose the consumer’s budget set is given by , , $10, $20, $100 [This budget constraint is different from the previous part]. From the UMP solution, calculate the optimal demands, price elasticities, income elasticities, and cross‐price elasticities for goods 1 and 2 as well as the MU due to a small decrease in income, a small increase in , a small decrease in , and relaxing the constraint that 0. Show all calculations and fill your answers in the table below.

Answer:

Summary of Answers: please fill shaded cells below Good 1 Good 2

Price elasticity Income elasticity

Cross‐price elasticity M g tility due to ar inal U

A small decrease in income Relaxing the constraint

A small increase in A small decrease in

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(8) [10 Points] What is the impact on the consumer’s utility if the government mandates (requires) that the consumer must consume at least 0.1 units of good 2? Answer this part using the general UMP solution in part (5) above. Show all calculations below.

Answer:

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Question 2 [45 Points] In this question you will calculate the cost of producing the products for the mixed bundling problem in question 3. That said, you don’t have to do question 2 in order to answer question 3 since question 3 gives you the necessary information from question 2.

Billy Bob Buggins (BBB) restaurant in Sudbury Ontario specializes in “Surf and Turf cuisine”1. BBB offers customers the choice of having surf and turf (for example, a lobster tail and steak), or just surf (for example, a lobster tail), or just turf (for example, a steak). Letting “surf” (seafood) be good 1 and “turf” (red meat) be good 2. BBB’s menu is:

Billy Bob Buggins Surf and Turf Restaurant Bringing you the finest fishy and mystery meat cuisine since January 2011

9 Accolades Avenue, Sudbuy, Ontario Call 416 – 555 – 9999 for reservations

Surf and Turf plate PriceBundle

Surf plate Price1

Turf plate Price2

We use the finest canned seafood and blend of “mystery” red meats BBB prepares a target output plates by using labor and seafood as surf complements in a 1:1 ratio.

BBB prepares a target output of turf plates by using labor and “meat” as complements according to the production function:

min 2 Meat,

In turn, “meat” consists of beef ( ) and mutton ( ) combined as “perfect substitutes” where a unit of beef is perfectly substitutable for two units of mutton.

BBB purchases/hires/leases all inputs as a price taker. Denote the price of seafood as , the labor wage rate by , the price of beef by , and the price of mutton by . In this question, assume all inputs are variable.

1 Surf and Turf refers to seafood and meat dishes. “Surf” is seafood (named for ocean surf) and “turf” is red meat (named for grass, or turf, upon which cattle/animals graze).

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(1) [5 Points] Write down BBB’s production function for “surf” (seafood) using labor and seafood as complements in a 1:1 ratio, characterize its returns to scale, and graph some iso‐quants below. Show all calculations below. Answer

S (Seafood)

L (Labor)

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(2) [10 Points] Write down BBB’s production function for “turf” using labor with “meat” as complements according to min 2 Meat, , characterize its returns to scale, and graph some iso‐quants below. Also, write down BBB’s production function for “Meat” using beef and mutton as perfect substitutes (where a unit of beef is perfectly substitutable for two units of mutton), characterize its returns to scale, and graph some iso‐quants below. Show all calculations below.

Answer If necessary use the next page for additional calculations

“Turf” Pay attention to which inputs are on the , axes

Meat

L (Labor)

“Meat” Pay attention to which inputs are on the , axes

M (Mutton)

B (Beef)

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(3) [5 Points] Derive the cost function of “surf” plates and from this characterize the returns to scale. Show all calculations.

Answer

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(4) [10 Points] Derive the cost function of “turf” plates and from this characterize the returns to scale. Show all calculations.

Answer

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(5) [5 Points] What is BBB’s total cost function of producing plates of “surf and turf”? Is it true that the cost of “surf and turf” plates is the sum of the cost of “surf” plates and the cost of “turf” plates? That

is: is ? Show all calculations.

Answer

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(6) [10 Points] Suppose:

$1, $0.50, $2, $2

Show that the cost of a “surf” plate is $1.50 and f “ late is $1.50: the cost o a turf” p

$1.50, $1.50

Show all calculations.

Answer:

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Question 3 [45 Points] NOTE: If you can’t solve a particular part, then try answering later parts which contain all required information

From question 2 (independent of this question) recall that Billy Bob Buggins (BBB) restaurant in Sudbury Ontario specializes in “surf and turf cuisine”2. BBB offers customers the choice of having surf and turf (for example, a lobster tail and steak), or just surf (for example, a lobster tail), or just turf (for example, a steak). Letting “surf” (seafood) be good 1 and “turf” (red meat) be good 2, BBB’s menu is:

Billy Bob Buggins Surf and Turf Restaurant Bringing you the finest fishy and mystery meat cuisine since January 2011

9 Accolade Avenue, Sudbuy, Ontario Call 416 – 555 – 9999 for reservations

Surf and Turf plate PriceBundle

Surf plate Price1

Turf plate Price2

We use the finest canned seafood and blend of “mystery” red meats Suppose that:

S $1.50; T $1.50

BBB’s owner, Mr. Bill Buggins has to price a mixed bundle of a surf plate, a turf plate, and a surf and turf bundle. Assume customers always purchase either a plate of surf, or a plate of turf, or a plate of surf and turf bundle.

The marketing firm “3 C’s, 4 P’s & 7 M’s” has identified three segments of customers:

Segment Incom o a omer e f cust A $100B $150C $200

Suppose each segment has the following type of utility function over “surf” (good 1) and “everything else” (toothpaste, shampoo, clubbing etc):

,

Ignore “turf” being a component of “everything else”. The parameter , , , varies by segment (will be given below).

Suppose each segment has the following type of utility function over “turf” (good 2) and “everything else” (toothpaste, shampoo, clubbing etc):

2 Surf and Turf refers to seafood and red meat. “Surf” is seafood (named for ocean surf) and “turf” is red meat (named for grass, turf, on which cattle/animals graze).

, 2

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Ignore “surf” being a component of “everything else”. The parameter , , , varies by segment (will be given below).

(1) [5 Points] Suppose that “surf and everything else” and “turf and everything else” can be consumed in integer amounts only. Graph the integer consumption sets below:

Answer:

Integer consumption set of surf and everything else

Everything Else

Q1(Surf)

Integer consumption set of turf and everything else

Everything Else

Q2(Turf)

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(2) [5 Points] Graph an indifference curve for “surf and everything else” and “turf and everything” defined over an integer consumption set. You need only give a rough sketch of the indifference curves. Hint: What does the

indifference curve of , look like in the “standard” consumption set?

Answer:

Indifference curve of surf and everything else (Integer consumption set)

Everything Else

Q1(Surf)

Indifference curve of turf and everything else (Integer consumption set)

Everything Else

Q2(Turf)

Explain how you got your answers below:

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(3) [5 Points] Letting price of a surf plate and price of a unit of everything else, calculate the reservation price (i.e. “willingness to pay”) of one surf plate

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for a “surf” customer with income and utility function:

,

Does the reservation price depend on income? Show all calculations below and if necessary use the graph below:

Answer:

Everything Else

Q1

(Surf)

(3) [5 Points] Letting price of a turf plate and price of a unit of everything else, calculate the reservation price (i.e. “willingness to pay”) of one turf plate

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of a “turf” customer with income and utility function:

, 2

Does the reservation price depend on income? Show all calculations below and if necessary use the graph below:

Answer:

Everything Else

Q2

(Turf)

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(4) [5 Points] Treat “everything else” as the “base good” so that price of a unit of everything else = $1. Suppose a surf customer in segments A, B and C respectively has the following utility function over surf and everything else:

,

1, 2, 4

Suppose a turf customer in segments A, B and C respectively has the following utility function over turf and everything else:

, 2

3, 2, 12

Based on your answers above show that the reservation price (i.e. “willingness to pay”) for a surf plate (good 1) and turf plate (good 2) are:

Segment Cus m me to er Inco A $100 $1 $6 B $150 $2 $4 C $200 $4 $1

Cost of a plate of: $1.50 $1.50 Calculate the reservation price for a bundle of surf and turf. Be sure to state any assumptions about reservation price of a “surf and turf” bundle and indicate whether your assumption is always true.

Answer:

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(5) [5 Points] If the optimal price of a surf plate sold individually equals the reservation price of a customer in

segment C (i.e. show there must be 3 (rounded up) stomers in segment C. cu

Segment # custom r gment e s in se Cus m me to er IncoA 20 $100 $1 $6 B 10 $150 $2 $4

C Show that

(rounded up) $200 $4 $1

Cost of a plate of: $1.50 $1.50 Show all calculations below:

Answer

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(6) [20 Points] Calculate the optimal mixed bundle prices of a surf plate, a turf plate, and a surf and turf bundle plate for the following data. Show all calculations below and summarize your answer below.

Segment # custom r gment e s in se Cus m me to er IncoA 20 $100 $1 $6 B 10 $150 $2 $4 C 8 $200 $4 $1

Cost of a plate of: $1.50 $1.50 Answer:

Optimal Price Which segment(s) will purchase? Good 1, Surf Plate Good 2, Turf Plate

Bundle of Goods 1 & 2, Surf and Turf plate

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UNIVERSITY OF TORONTO Faculty of and April Examinations ... · scores 1 60 2 45 3 45 total score...

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