Question

Suppose an individual’s utility derived from goods X and Y is given by

U(X,Y) = 20√(XY) = 20(XY)0.5
a. Show that the law of diminishing marginal utility holds for good X.
b. Sketch the indifference curve for U = 200.
c. Calculate the MRS (of X for Y) for X=4 and for X=5 when U = 200. Does it follow the expected pattern? Explain.

Answer 1

To show that the law of diminishing marginal utility holds for good X, calculate the marginal utility of X by taking the partial derivative of U with respect to X. Sketch the indifference curve for U = 200 by rearranging the utility function and plotting the corresponding values. Calculate the MRS of X for Y by finding the ratio of the marginal utility of X to the marginal utility of Y.

Step-by-step explanation:

a. To show that the law of diminishing marginal utility holds for good X, we need to calculate the marginal utility of X. The marginal utility of X is given by the partial derivative of U with respect to X, denoted as MUX. In this case, U(X, Y) = 20√(XY), so we can find MUX by taking the partial derivative of U with respect to X. MUX = 10√Y/X. Since there is an inverse relationship between X and MUX (as X increases, MUX decreases), the law of diminishing marginal utility holds for good X.

b. To sketch the indifference curve for U = 200, we can use the utility function U(X, Y) = 20√(XY). We can rearrange the equation to isolate Y and plot the corresponding values of Y for different values of X to draw the indifference curve.

c. To calculate the Marginal Rate of Substitution (MRS) of X for Y, we need to find the ratio of the marginal utility of X to the marginal utility of Y. MRS = MUX / MUY. For X = 4 and U = 200, we can substitute these values into the utility function to find MUX. For X = 5 and U = 200, we can do the same to find MUX. Comparing the two values of MRS, we can determine if it follows the expected pattern.