Econ 3: Solutions to Quiz 6

Donald Ribble is a stamp collector. The only things other than stamps that Ribble consumes are Hostess Twinkies. It turns out that Ribble's preferences are represented by the utility function u(s, t) = s + ln t where s is the number of stamps he collects and t is the number of Twinkies he consumes. The price of stamps is p_s and the price of Twinkies is p_t. Donald's income is m.

(a) (2 pts) Write an expression that says that the ratio of Ribble's marginal utility for Twinkies to his marginal utility for stamps is equal to the ratio of the price of Twinkies to the price of stamps. (Hint: The derivative of ln t with respect to t is 1/t, and the derivative of s with respect to s is 1.)

The marginal utility of Twinkies is the derivative of U with respect to t, or 1/t. The marginal utility of stamps is the derivative of U with respect to s or 1. The ratio of the marginal utility of Twinkies to the marginal utility of stamps is this 1/t. The equation is thus 1/t = p_t /p_s.

(b) (2 pts) You can use the equation you found in the last part to show that if he buys both goods, Donald's demand function for Twinkies depends only on the price ratio and not on his income. Donald's demand function for Twinkies is t(p_s, p_t, m) = p_s/p_t. To see this, recall that a demand function is simply the amount of t (Twinkies) that he buys as a function of prices and income, the exogenous variables. You get this by solving the equation above (which is just the optimality condition) for t.

(c) (1 pt) Notice that for this special utility function, if Ribble buys both goods, then the total amount of money that he spends on Twinkies has the peculiar property that it depends on only one of the three variables m, p_t, and p_s, namely the variable p_s. This follows from the above. The amount of money that he spends on stamps is just p_t times the amount of t that he buys, which is given by the demand function above. Multiplying p_t times p_s/p_t, you obtain p_s.

(d) (1 pt) Since there are only two goods, any money that is not spent on Twinkies must be spent on stamps. Use the budget equation and Donald's demand function for Twinkies to find an expression for the number of stamps he will buy if his income is m, the price of stamps is p_s, and the price of Twinkies is p_t.

s = (m/p_s) - 1. To obtain this, simply plug the demand function for t into the budget equation and solve for s.

m = p_s s + p_t t = p_s s + p_s

so

(m - p_s)/p_s = s or s = (m/p_s) - 1

(e) (2 pts) The expression you just wrote down is negative if m < p_s. Surely it makes no sense for him to be demanding negative amounts of postage stamps. If m < p_s., what would Ribble's demand for postage stamps be? It would be 0. He would be consuming at the boundary of the consumption set.

What would his demand for Twinkies be? t = m/p_t (Hint: Recall the discussion of boundary optimum.)

(f) (1 pt) Donald's wife complains that whenever Donald gets an extra dollar, he always spends it all on stamps. Is she right? (Assume that m > p_s) Yes, it's true. Why? Because the marginal utility of Twinkies is 1/t, which is less than 1 for values of t greater than 1. In contrast, the marginal utility of stamps is always 1.

(g) (1 pt) Suppose that the price of Twinkies is $2 and the price of stamps is $1. On the graph below, draw Ribble's Engel curve for Twinkies in red ink and his Engel curve for stamps in blue ink. (Hint: First draw the Engel curves for incomes greater than $1, then draw them for incomes less than $1.) Label the axes.

First, a reminder about the Engel curve. The Engel curve is a graph that relates income and demand for goods. We will put Income on the y axis and quantity demanded for each good on the x axis.

We already know that when income is less than the price of stamps (which is $1 here) ,Donald will buy no stamps and will buy m/2 Twinkies.

For income greater than $1, the demand for s is given by s (m, p_s, p_t) = (m/p_s) - 1. Rearranging this to put m on the lefthand side you obtain m = (s + 1) p_s = or at these prices m = s + 1 This describes an upward sloping straight line with a slope of 1 and a vertical intercept of 1. For income greater than $1, the demand for Twinkies is t (p_s, p_t, m) = p_s/p_t = .5 given these prices. Notice that this does not depend at all on m.