Question

Edward consumes goods x and y. His utility function is U(x, y) = xły?. He is endowed with 46 units of x and 10 units of y. The price of x is $2 and the price of y is $4. a. Edward will buy fewer than 20 units of x b. Edward will buy at least 20 units, but fewer than 30 units, of x c. Edward will buy at least 30 units of x d. Edward will sell at least 30 units of x e. Edward will sell fewer than 30 units of x

Answer 1

Ans. Option e

Income = endowment of x * price of x + endowment of y * price of y

=> Income = 46*2 + 10*4 = $132

=> The budget constraint,

2x + 4y = 132 --> Eq1

The utility function of the consumer, U = x*y^2

=> Marginal utility of x, MUx = dU/dx = y^2

=> Marginal utility of y, MUy = dU/dy = 2xy

=> Marginal rate of substitution, MRS = MUx/MUy = y^2 / 2xy = y/2x

At utility maximizing level,

MRS = Price of x/ Price of y

=> y/2x = 2/4

=> x = y

Substituting this in Eq1,

2x + 4x = 132

=> x = y = 22 units

So, the amount of x sold is = 46 - 22 = 24 units

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