Every Thanksgiving, my wife and I debate about how we should prepare the turkey we will serve (and will then have left over). My wife likes preparing turkeys the conventional way — roasted in the oven where it has to cook at 350 degrees for 4 hours or so. I, on the other hand, like to fry turkeys in a big pot of peanut oil heated over a powerful flame outdoors. The two methods have different costs and benefits. The conventional way of cooking turkeys has very little set-up cost (since the oven is already there and just has to be turned on) but a relatively large time cost from then on. (It takes hours to cook.) The frying method, on the other hand, takes some set-up (dragging out the turkey fryer, pouring gallons of peanut oil, etc. — and then later the cleanup associated with it), but turkeys cook predictably quickly in just 3.5 minutes per pound.
A. As a household, we seem to be indifferent between doing it one way or another — sometimes we use the oven, sometimes we use the fryer. But we have noticed that we cook much more turkey — several turkeys, as a matter of fact, when we use the fryer than when we use the oven.
(a) Construct a graph with “pounds of cooked turkeys” on the horizontal and “other consumption” on the vertical. (“Other consumption” here is not denominated in dollars as normally but rather in some consumption index that takes into account the time it takes to engage in such consumption.) Think of the set-up cost for frying turkeys and the waiting cost for cooking them as the main costs that are relevant. Can you illustrate our family’s choice of whether to fry or roast turkeys at Thanksgiving as a choice between two “budget lines”?
(b) Can you explain the fact that we seem to eat more turkey around Thanksgiving whenever we pull out the turkey fryer as opposed to roasting the turkey in the oven?
(c) We have some friends who also struggle each Thanksgiving with the decision of whether to fry or roast—and they, too, seem to be indifferent between the two options. But we have noticed that they only cook a little more turkey when they fry than when they roast. What is different about them?
B. Suppose that, if we did not cook turkeys, we could consume 100 units of “other consumption”— but the time it takes to cook turkeys takes away from that consumption. Setting up the turkey fryer costs c units of consumption and waiting 3.5 minutes (which is how long it takes to cook 1 pound of turkey) costs 1 unit of consumption. Roasting a turkey involves no set-up cost, but it takes 5 times as long to cook per pound. Suppose that tastes can be characterized by the CES utility function u(x1,x2) = (0.5x1−ρ +0.5x2−ρ )−1/ρ where x1 is pounds of turkey and x2 is “other consumption”.
(a)What are the two budget constraints I am facing?
(b) Can you calculate how much turkey someone with these tastes will roast (as a function of ρ)? How much will the same person fry? (Hint: Rather than solving this using the Lagrange method, use the fact that you know the MRS is equal to the slope of the budget line — and recall from chapter 5 that, for a CES utility function of this kind, MRS = −(x2/x1)ρ+1.)
(c) Suppose my family has tastes with ρ = 0 and my friend’s with ρ = 1. If each of us individually roasts turkeys this Thanksgiving, how much will we each roast?
(d) How much utility will each of us get (as measured by the relevant utility function)? (In the case where ρ = 0, the exponent 1/ρ is undefined. Use the fact that you know that when ρ = 0 the CES utility function is Cobb-Douglas.)
(e) Which family is happier?
(f) If we are really indifferent between roasting and frying, what must c be for my family? What must it be for my friend’s family? (Hint: Rather than setting up the usual minimization problem, use your answer to (b) determine c by setting utility equal to what it was for roasting).
(g) Given your answers so far, how much would we each have fried had we chosen to fry instead of roast (and we were truly indifferent between the two because of the different values of c we face)?
(h) Compare the size of the substitution effect you have calculated for my family and that you calculated for my friend’s family and illustrate your answer in a graph with pounds of turkey on the horizontal and other consumption on the vertical. Relate the difference in the size of the substitution effect to the elasticity of substitution.