Question

There are two people in an economy, Sayah (S) and Jaki (J), who consume two goods, apples (A) and bananas (B), with utility functions: US(A, B) = 2A0.5B0.5 and UJ(A, B) = A0.4B0.6 The current market prices for A and B are PA=1 and PB=2. Sayah’s income is YS = $60 and Jaki’s is YJ = $70. There are 58 units of A produced and 36 units of B produced

. a. Find the Sayah’s and Jaki’s marginal rates of substitution of A for B. (Remember, first find the marginal utilities of each good, for each person, by taking the derivative of the utility functions with respect to one good holding constant the other good. For example, Sayah’s MU of A is the derivative of US(A, B) = 2A0.5B0.5 treating B as a constant.)

b. Find the amounts of A and B that both people will consume given these prices and incomes.

c. Do Sayah and Jaki consumption amounts achieve an efficient exchange? Explain. d. Do the consumption demands and current supplies achieve and equilibrium in this economy? Explain. If not, what would need to adjust to achieve an equilibrium? e. Are the three efficiency conditions satisfied? Explain.

Answer 1

c) The efficiency in exchange, the marginal rate of substitution between any two products must be the same for every individual who consumes both. But here, the MRS of A for B differ for Jaki and Sayah.

Sayah values good A more and needs more of good B to compensate her for giving up one unit of good A. (2A=B) However, Jaki values good A lesser and needs only 1.33 of good B to compensate him for giving up one unit of good A. To come to an equilibrium their marginal rate of substitutions should adjust and become equal to achieve pareto optimality in efficiency in exchange.

d) As the total demand for good A(by both) = 30+28 = 58 is equal to the output of good A and the total demand for good B(by both) = 15+21 = 36 is equal to the output of good B, the consumption demands and current supplies achieve and equilibrium in this economy.