An agent\'s utility\' function is written as U(X, Y) = 2X + 20 Y. The price of g

Question

An agent's utility' function is written as U(X, Y) = 2X + 20 Y. The price of good A' is 2, the price of good I" is 10, and income is 500. Which special case of the utility function applies here? Provide the name of this case or describe the shape of the indifference curves. What is the marginal utility of X? What is the marginal utility of Y? What is the marginal rate of substitution of X for Y? Give indication as to how you are finding your answers. What are the optimal amounts of X and Y? (In other words, what are the quantities of X and That maximize utility?) How do you know? Suppose instead that the agent's utility function is written as U(X, Y) = X^0.75 Y^0.23. Which case of the utility function applies here? Provide the name of this case or describe the shape of the indifference curves. [this subpart is worth 1 point, others are 0.5] What are the optimal amounts of X and Y if the agent's utility function is as given in part d above? (Assume that prices and income are as given in the setup of the problem.) Show your work.

Explanation / Answer

U = 2X + 20 Y, PX = 2, PY = 10, M = 500

1. This is the case of perfect substitutes. Indifference curve are straight lines.

2. U(X,Y) = 2X + 20Y

a. MUX ( marginal utility of X ) = dU / dX = 2

b. MUY ( marginal utility of Y ) = dU/dY = 20

c. MRSXY ( marginal rate of substitution of X for Y) = - MUX / MUY

= - 2/20 = -1/10

3. Here, MRS = -1/10 (i.e. slope of indifference curve) and slope of budget line = -PX / PY = -2/10 = -1/5

1/10 < 1/5 i.e. slope of indifference curve is lower than that of budget line. This suggests that consumer's optimal bundle would be ( 0, m/py) = (0, 500 / 10) = (0, 50)

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