department of economics eco 204 2016-2017 …...eco 204, 2016-2017 (ajaz), test 3 this test is...

ECO 204, 2016-2017 (AJAZ), Test 3

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

∎ Department of Economics ∎ ECO 204 ∎ Microeconomic Theory for Commerce ∎ 2016-2017 (Ajaz)

Test 3

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IMPORTANT NOTES

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EXAM DETAILS

Duration: 2 hours

Total number of questions: 6

Total number of pages: 15 (including title page)

Total number of points: 140

Please answer all questions. To earn credit you must show all calculations.

This is a closed note and closed book exam.

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

. KEEP YOUR ANSWERS AS BRIEF AS POSSIBLE AND SHOW ALL NECESSARY CALCULATIONS

GOOD LUCK!

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QUESTION 1 [TOTAL 10 Points] Recall from the Prestige Telephone Company case that:

𝐶𝑝𝑜𝑤𝑒𝑟 = 𝑇𝐹𝐶𝑝𝑜𝑤𝑒𝑟 + 𝑇𝑉𝐶𝑃𝑜𝑤𝑒𝑟 = 179.51 + 4(𝑞𝑖 + 𝑞𝑐 + 𝑞𝑠)

𝐶𝑜𝑝𝑠 = 𝑇𝐹𝐶𝑜𝑝𝑠 + 𝑇𝑉𝐶𝑜𝑝𝑠 = 21,600 + 24(𝑞𝑖 + 𝑞𝑐)

𝑇𝐹𝐶𝑃𝐷𝑆 = 𝑇𝐹𝐶𝑝𝑜𝑤𝑒𝑟⏟

179.51

+ 𝑇𝐹𝐶𝑜𝑝𝑠⏟ 21,600

+ 𝑇𝐹𝐶𝑎𝑙𝑙 𝑜𝑡ℎ𝑒𝑟⏟ 191,200

≈ $212,800

𝑇𝑉𝐶𝑃𝐷𝑆 = 𝑇𝑉𝐶𝑝𝑜𝑤𝑒𝑟 + 𝑇𝑉𝐶𝑜𝑝𝑠 = 28𝑞𝑐 + 28𝑞𝑖 + 4𝑞𝑠

𝐶𝑃𝐷𝑆 = 𝑇𝐹𝐶𝑃𝐷𝑆 + 𝑇𝑉𝐶𝑃𝐷𝑆 ≈ 212,800⏟

𝑇𝐹𝐶𝑃𝐷𝑆

+ 28𝑞𝑐 + 28𝑞𝑖 + 4𝑞𝑠⏟ 𝑇𝑉𝐶𝑃𝐷𝑆

All costs are reported on a monthly basis (in dollars) where 𝑞𝑐 = Commercial hours billed per month, 𝑞𝑖 = Intercompany hours billed per month, and 𝑞𝑠 = “service” hours each month. Denote the amount of the “variable power input” by 𝑃, the amount of the “variable operations input” by 𝐿, and all fixed inputs (including “fixed” power and “fixed” operations) by 𝑘. Suppose that PDS’s short run “commercial output” production function is:

𝑞𝑐 = 𝐴𝑃𝛼𝐿𝛽 𝑘𝛾

Assume 𝛼, 𝛽, 𝛾 > 0. (a) [5 Points] Given the information above, what can we say about the values of the parameters 𝛼, 𝛽 (or any combination thereof)? State all salient assumptions and show all essential calculations. Hint: What is the equation of 𝐶(𝑞𝑐)?


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(b) [5 Points] Suppose that doubling all “commercial” inputs in the long run always reduces long run 𝐴𝐶 by 50%. What is

the value of 𝛾 in the commercial production function 𝑞𝑐 = 𝐴𝑃𝛼𝐿𝛽 𝐾𝛾? State all salient assumptions and show all

essential calculations.


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QUESTION 2 [TOTAL 10 Points] Answer the following questions independently of each other:

(a) [5 Points] Suppose a firm’s short run production function is 𝑄 = 𝐴𝐿𝛼𝑘𝛽 and that 𝑑𝑀𝐶𝑠ℎ𝑜𝑟𝑡 𝑟𝑢𝑛

𝑑𝑞< 0. True or false: the

long run 𝐴𝐶 declines with output? State all salient assumptions and provide a brief explanation. Answer:

(b) [5 Points] Suppose a firm’s short run production function is 𝑄 = 𝐴𝐿𝛼𝑘𝛽 and that 𝑑𝐴𝑉𝐶

𝑑𝑞> 0. True or false: the long run

𝐴𝐶 rises with output? State all salient assumptions and provide a brief explanation.


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QUESTION 3 [TOTAL 10 Points] (a) [5 Points] Show that the 𝑇𝑅𝑆 of the production function 𝑄 = 2𝐴𝐿0.18𝐾0.33 is identical to the 𝑇𝑅𝑆 of the production function 𝑄 = 𝐴𝐿0.18𝐾0.33. State all salient assumptions and provide a brief explanation. (b) [5 Points] True or false: since the 𝑇𝑅𝑆 of the production function 𝑄 = 2𝐴𝐿0.18𝐾0.33 is identical to the 𝑇𝑅𝑆 of the production function 𝑄 = 𝐴𝐿0.18𝐾0.33, these two production functions model the same “input-output” process.


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QUESTION 4 [TOTAL 15 Points] Evaluate the following statement:

“The introduction of the ‘New Coke’ was a stroke of marketing genius. Faced with declining sales of ‘Classic Coke’, Coca-Cola’s executives decided to withdraw ‘Classic Coke’ from the market and introduced the ‘New Coke’ anticipating that the American Public would decry the loss of an American icon and would push Coca-Cola to re-introduce ‘Classic Coke’ and revive sales of ‘Classic Coke’. That’s exactly what happened: there was a backlash when ‘Classic Coke’ was “retired” leading Coca-Cola to re-introduce ‘Class Coke’ which resulted in record high sales. In retrospect, Coca-Cola executives were, to put it mildly, prophetic geniuses.”


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QUESTION 5 [TOTAL 30 Points]

Suppose that a company produces output according to the Cobb-Douglas production function 𝑄 = 𝐴𝐿𝛼𝑘𝛽 where 𝐿 = labor, 𝑘 = fixed capital and the parameters 𝐴, 𝛼, 𝛽 > 0. Denote the wage rate by 𝑃𝐿 > 0 and the price of leasing capital (or, the user cost of owned capital) by 𝑃𝐾 > 0. (a) [20 Points] Setup and solve the following Cost Minimization Problem for the optimal amount of labor needed to produce the target output 𝑞 > 0 and any Lagrange multipliers in this constrained optimization problem:

min𝐿𝑇𝑉𝐶 = 𝑃𝐿𝐿 𝑠. 𝑡. 𝐴𝐿

𝛼𝑘𝛽 = 𝑞

Interpret any Lagrange multipliers in this CMP. State salient assumptions and show essential calculations.


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(b) [10 Points] The following problem was solved in Excel:

min𝐿𝑇𝑉𝐶 = 𝑃𝐿𝐿 𝑠. 𝑡. 𝐴𝐿

𝛼𝑘𝛽 = 𝑞

max𝐿−𝑇𝑉𝐶 = −5𝐿 𝑠. 𝑡. (1)𝐿0.2(10)0.8 = 10

Here is the Solver solution and portions of the “sensitivity” report:

alpha = 0.20

beta = 0.80

k = 10.00

A = 1.00

P_L = $5.00

TVC = $50.00

-TVC = $(50.00) << Objective

Output produced = 10.00

Target output q = 10.00

Optimal Labor = 10.00 << Decision Variable

Constraints

Cell Name Final Value Lagrange Multiplier

$B$9 Output produced 10.00 -25.00

What is the 𝑇𝑉𝐶 equation? State salient assumptions and show essential calculations.


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QUESTION 6 [TOTAL 65 Points]

A risk averse agent must decide whether to invest in “Project A” or in “Project B”. Both projects are characterized by

uncertainty where:

Project 𝐴 = {$7,800,−$300; 0.6, 0.4}

Project 𝐵 = {$9,700,−$200; 0.3, 0.7}

(a) [10 Points] Suppose this decision maker’s utility function is 𝑈($𝑋) = (𝑎 + 𝑋)1

𝑏 where 𝑎, 𝑏 > 0 are parameters and 𝑋 is measured in dollars. Show that 𝑎 = 300 and 𝑏 = 2 and use these parameter values throughout this question (i.e.

assume 𝑈($𝑋) = (300 + 𝑋)1

2) . State all salient assumptions and show all essential calculations.


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(b) [5 Points] True or false: -$200 is the certainty equivalence to the hypothetical “gamble”: {Highest outcome in the decision making problem, Lowest outcome in the decision making problem, ; 0.1, 0.9}?

State all salient assumptions and show all essential calculations.


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(c) [15 Points] Graph the decision tree below and solve the decision making under uncertainty problem to show that Project A is the optimal choice. State all salient assumptions and show all essential calculations. Hint: Remember that

𝑈($𝑋) = (300 + 𝑋)1

2


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(d) [10 Points] Calculate and interpret the “discount due to risk” of Project 𝐴 = {$7,800,−$300; 0.6, 0.4}. State all

salient assumptions and show all essential calculations. Hint: Remember that 𝑈($𝑋) = (300 + 𝑋)1

2


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(e) [25 Points] In part (c) you showed that the agent should choose Project A over Project B. Now suppose she can purchase an actuarially fair insurance policy to eliminate all risk from investing in Project A. Prove that she will purchase full insurance, calculate the insurance premium, and show that she will remove all risk from investing in Project A. State all salient assumptions and show all essential calculations.


department of economics eco 204 2016-2017 …...eco 204, 2016-2017 (ajaz), test 3 this test is...

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