Substitution Effects and Social Security Cost of Living Adjustments: In end-of chapter exercise 6.16, you investigated the government’s practice for adjusting social security income for seniors by insuring that the average senior can always afford to buy some average bundle of goods that remains fixed. To simplify the analysis, let us again assume that the average senior consumes only two different goods.
A. Suppose that last year our average senior optimized at the average bundle A identified by the government, and begin by assuming that we denominate the units of x1 and x2 such that last year p1 = p2 = 1.
(a) Suppose that p1 increases. On a graph with x1 on the horizontal and x2 on the vertical axis, illustrate the compensated budget and the bundle B that, given your senior’s tastes, would keep the senior just as well off at the new price.
(b) In your graph, compare the level of income the senior requires to get to bundle B to the income required to get him back to bundle A.
(c) What determines the size of the difference in the income necessary to keep the senior just as well off when the price of good 1 increases as opposed to the income necessary for the senior to still be able to afford bundle A?
(d) Under what condition will the two forms of compensation be identical to one another?
(e) You should recognize the move from A to B as a pure substitution effect as we have defined it in this chapter. Often this substitution effect is referred to as the Hicksian substitution effect — defined as the change in behavior when opportunity costs change but the consumer receives sufficient compensation to remain just as happy. Let B′ be the consumption bundle the average senior would choose when compensated so as to be able to afford the original bundle A. The movement from A to B′ is often called the Slutsky substitution effect—defined as the change in behavior when opportunity costs change but the consumer receives sufficient compensation to be able to afford to stay at the original consumption bundle. True or False: The government could save money by using Hicksian rather than Slutsky substitution principles to determine appropriate cost of living adjustments for social security recipients. Adjustments are calculated —with these proposals attempting to get closer to Hicksian compensation.
(f) True or False: Hicksian and Slutsky compensation get closer to one another the smaller the price changes.
B. Now suppose that the tastes of the average senior can be captured by the Cobb-Douglas utility function u(x1,x2) = x1x2, where x2 is a composite good (with price by definition equAl to p2 = 1). Suppose the average senior currently receives social security income I (and no other income) and with it purchases bundle (x1A ,x2A).
(a) Determine (x1A ,x2A ) in terms of I and p1.
(b) Suppose that p1 is currently $1 and I is currently $2000. Then p1 increases to $2. How much will the government increase the social security check given how it is actuAlly calculating cost of living adjustments? How will this change the senior’s behavior?
(c) How much would the government increase the social security check if it used Hicksian rather than Slutsky compensation? How would the senior’s behavior change?
(d) Can you demonstrate mathematically that Hicksian and Slutsky compensation converge to one another as the price change gets small—and diverge from each other as the price change gets large?
(e) We know that Cobb-Douglas utility functions are part of the CES family of utility functions— with the elasticity of substitution equal to 1. Without doing any math, can you estimate, for an increase in p1 above 1, the range of how much Slutsky compensation can exceed Hicksian compensation with tastes that lie within the CES family? (Hint: Consider the extreme cases of elasticities of substitution.)