Cups of coffee (x1), ounces of milk (x2) and packets of sugar (x3).
A. Suppose each of these goods costs 25 cents and you have an exogenous income of $15.
(a) Illustrate your budget constraint in three dimensions and carefully label all intercepts.
(b) Suppose that the only way you get enjoyment from a cup of coffee is to have at least one ounce of milk and one packet of sugar in the coffee, the only way you get enjoyment from an ounce of milk is to have at least one cup of coffee and one packet of sugar, and the only way you get enjoyment from a packet of sugar is to have at least one cup of coffee and one ounce of milk. What is the optimal consumption bundle on your budget constraint.
(c) What does your optimal indifference curve look like?
(d) If your income falls to $10—what will be your optimal consumption bundle?
(e) If instead of a drop in income the price of coffee goes to 50 cents, how does your optimal bundle change?
(f) Suppose your tastes are less extreme and you are willing to substitute some coffee for milk, some milk for sugar and some sugar for coffee. Suppose that the optimal consumption bundle you identified in (b) is still optimal under these less extreme tastes. Can you picture what the optimal indifference curve might look like in your picture of the budget constraint?
(g) If tastes are still homothetic (but of the less extreme variety discussed in (f)), would your answers to (d) or (e) change?
B. Continue with the assumption of an income of $15 and prices for coffee, milk and sugar of 25 cents each.
(a) Write down the budget constraint.
(b)Write down a utility function that represents the tastes described in A(b).
(c) Suppose that instead your tastes are less extreme and can be represented by the utility function u(x1,x2,x3) = xα1 x β2 x3. Calculate your optimal consumption of x1, x2 and x3 when your economic circumstances are described by the prices p1, p2 and p3 and income is given by I .
(d) What values must α and β take in order for the optimum you identified in A(b) to remain the optimum under these less extreme tastes?
(e) Suppose α and β are as you concluded in part B(d). How does your optimal consumption bundle under these less extreme tastes change if income falls to $10 or if the price of coffee increases to 50 cents? Compare your answers to your answer for the more extreme tastes in A(d) and (e).
(f) True or False: Just as the usual shapes of indifference curves represent two dimensional “slices” of a 3-dimensional utility function, 3-dimensional “indifference bowls” emerge when there are three goods—and these “bowls” represent slices of a 4-dimensional utility function.