Microeconomics–Midterm Solution.

August 21, 2009

1. The elasticity of demand is given by E d =

percentage change in quantity demanded percentage change in price

.

(a) The elasticity of demand for roasted coffee is given by d ERC = 75−70 70 10.35−10.48 10.35

= −5.68681.
∆Q ∆P

The linear demand curve can be written as Q = A − BP where B = Hence, the linear demand curve is Q = 473.077 − 36.4615P . (b) The elasticity of demand for instant coffee is given by d EIC = 820−850 820 4.11−3.76 3.76

=

75−70 10.35−10.48

=

−38.4615. Using (Q, P ) = (75, 10.35) or (Q, P ) = (70, 10.48), we obtain A = 473.077.

= −0.393031.

(c) Roasted coffee is more elastic since it is a luxury item, as can be seen from the high price. 2. Let soft drinks be S and burgers be B. (a) The indifference curves are L-shaped with the corner of the L lying on the line S = 3B. (b) The consumption ratio is
S B

= 3.

(c) The budget constraint is 10B + 5S = 100. (d) We have two equations in two variables: equations, we obtain B = 4 and S = 12. 3. (a) Natasha is risk averse since her utility function is concave. (b) The expected income of the new job is 0.6 × 50 + 0.4 × 27 = 40.8. The expected utility from the new job is √ √ 0.6 10 × 50 + 0.4 10 × 27 = 19.9891.
S B

= 3 and 10B + 5S = 100. Solving the two

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The utility from the existing job is offers a higher expected income.

√

10 ∗ 40 = 20. Since the utility from the existing

job is higher, she should not take up the new job offer despite the fact that the new job (c) The amount that Natasha would be willing to pay is the risk premium. To find the risk premium, we first find the certainty equivalent of the risky income bundle. From (b), we know that the expected utility from the risky job is 19.9891. The certainty equivalent of the risky income bundle is that income level that obtained with certainty gives aa utility equal to utility of the risky income bundle. So, we want I such that √ 10I = 19.9891. Solving this equation, we obtain I = 39.9563 as the certainty equivalent. The risk premium is the expected income of the risky bundle minus the certainty equivalent, or 40.8 − 39.9563 = 0.8437. 4. (a) The expected income from investment A is
2 3

× 400 + 1 100 = 300 > 200. The utility 3

obtained from the investment is min{FS , FH } = min{400, 100} = 100. On the other hand, the utility from the guaranteed income is min{FS , FH } = min{200, 200} = 200. Since the utility obtained from the new investment is lower, the individual will not invest in A. (b) The utility from the new investment B is min{FS , FH } = min{100, 400} = 100 which is less than the utility of 200 obtained from the guaranteed income of 200. Hence, it does not make sense for the investor to invest in B either. (c) Now the consumer invests 100 in both projects. In the sunny state, he obtains a net return of 200 from Investment A and a net return of −100 from Investment B. Hence, his aggregate net return is 100. Hence, FS = 300 (the existing income of 200 plus the net return of 100). In the stormy state, the net return from B is 200 and from A is −100. Hence, aggregate net return is 100 and FH = 200 + 100 = 300. Hence, utility is min{FS , FH } = min{300, 300} = 300. So, with hedging, the investor obtains a guaranteed income, and hence, a utility of 300 which is higher than the existing utility of 200. 5. (a) Since the capital stock is fixed at 9, the short run production function is q = F (L) = √ 2 2 3 L. Hence, L = q9 . So, the cost of labour is wL = 4000 q9 . The cost of capital is rK = 10000 × 9 = 90000. Hence, the short run cost function is C(q) = rK + wL 4000 2 = 90000 + q . 9 The average cost is AC(q) =
90000 4000 q + 9 q.

The marginal cost is M C(q) =

dC(q) dq

=

8000 9 q.

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(b) In order to make 20 motors, the firm should design a new production facility using the cost minimizing capita-labour combination. This is obtained by equating the M RT SLK to the input price ratio w r

=

4 10 .

M RT SLK =

0.5L−0.5 K 0.5 K M PL = = . 0.5 K −0.5 M PK 0.5L L

The firm also needs to produce an output of 20. So, F (L, K) = 20. We therefore have two equations. M RT SLK w r F (L, K) = 20. =

Using the calculated M RT S, and the values of w and r, we solve the two equations to obtain (L = 31.6228, K = 12.6491). This is the cost minimizing capital-labour combination. Hence, in designing the new production facility, the firm needs to increase its capital stock from 9 to 12.6491. 6. (a) The average cost function is AC(Q) = increasing returns to scale. (b) The marginal cost is M C(Q) = dC(Q) dQ C(Q) Q

=

10 . Q0.5

Clearly, this is a decreasing function

of Q. As Q increases, AC(Q) falls. This implies that the industry is characterized by =
5 . Q0.5

A price taking firm would maximize profit

by equating price with marginal cost. Hence, P = M C(Q) which implies 2= 5 , Q0.5

or Q = 6.25. This is the profit maximizing quantity. (c) Profit is given by Π(Q) = (P − AC(Q))Q. At Q = 6.25, AC(Q) = 4. Hence, Π(6.25) = (2 − 4)6.25 = −12.5. Hence, the maximum possible profit for the firm is a loss of 12.5. This implies that firms will inevitably sustain losses in the long run if they behave as price takers. Since no firm can sustain losses in the long run, they cannot continue to behave as price takers in the long run. So, a perfectly competitive structure is not possible in this industry. (Note: This part is not required for the answer. What is going on is that M C(Q) < AC(Q) at all Q. This is the characteristic of all industries with increasing returns to scale. So, P = M C(Q) implies that P < AC(Q). Hence, the firm will suffer losses at all Q. This loss in minimized at Q = 6.25.)

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