Assume that a consumer has the utility function U(x,y) = (3x+1)y, where x and y represent the quantities of two goods, X and Y. For parts (a)-(h), assume that good X costs pX=$3 per unit and good Y costs pY=$4 per unit. “Wealth” and “income” have the same meaning in this problem.
Now assume that the consumer’s wealth is fixed at W=$31, the price of Y is still fixed at pY=4, but pX is an unknown exogenous variable.
(i) Assuming that the optimal point satisfies the first-order condition, derive the consumer’s demand for X, as a function of pX. Under what conditions (i.e., for which values of pX) is it clear even before calculating any utility values that the consumer must choose a boundary point?
(j) Assuming that the optimal point satisfies the first-order condition, derive the consumer’s demand for Y, as function of pX. Based on this demand function, are X and Y substitutes or complements? For what values of pX should the consumer choose the point that satisfies the FOC? (k) Using your answers to (i) and (j), calculate the consumer’s optimal consumption of X and Y, for each of the following five prices: pX=1, 3, 5, 7, 15. Most of the answers are not integers: you should report the x values to the nearest integer and the y values to the nearest 0.1.
(l) Draw a diagram showing the five budget lines for the five prices of part (k). On the five budget lines, show the five optimal points from part (k). Connect the points to show the price expansion path, for changes in the price of X. (m) As pX rises, does the substitution effect tend to increase or decrease the consumer’s consumption of X and Y? As pX rises, does the income effect tend to increase or decrease the consumer’s consumption of X and Y? For which of these goods, if any, do the income and substitution effects work in the opposite directions, and if so then which effect is stronger?