Social Security payments to the elderly are adjusted every year in the following way: The government has in the past determined some average bundle of goods consumed by an average elderly person. Each year, the government then takes a look at changes in the prices of all the goods in that bundle and raises social security payments by the percentage required to allow the hypothetical elderly person to continue consuming that same bundle. This is referred to as a “cost of living adjustment” or COLA.

A. Consider the impact on an average senior’s budget constraint as cost of living adjustments are put in place. Analyze this in a 2-good model where the goods are simply x₁ and x₂.

(a) Begin by drawing such a budget constraint in a graph where you indicate the “average bundle” the government has identified as A and assume that initially this average bundle is indeed the one our average senior would have chosen from his budget.

(b) Suppose the prices of both goods went up by exactly the same proportion. After the government implements the COLA, has anything changed for the average senior? Is behavior likely to change?

(c) Now suppose that the price of x₁ went up but the price of x₂ stayed the same. Illustrate how the government will change the average senior’s budget constraint when it calculates and passes along the COLA. Will the senior alter his behavior? Is he better off, worse off or not affected?

(d) How would your answers change if the price of x₂ increased and the price of x₁ stayed the same?

(e) Suppose the government’s goal in paying COLAs to senior citizens is to insure that seniors become neither better nor worse off from price changes. Is the current policy successful if all price changes come in the form of general “inflation”—i.e. if all prices always change together by the same proportion? What if inflation hits some categories of goods more than others?

(f) If you could “choose” your tastes under this system, would you choose tastes for which goods are highly substitutable or would you choose tastes for which goods are highly complementary?

B. Suppose the average senior has tastes that can be captured by the utility function u(x₁,x₂) =

(a) Suppose the average senior has income from all sources equal to $40,000 per year and suppose that prices are given by p₁ and p₂. How much will our senior consume of x₁ and x₂?

(b) If p₁ = p₂ = 1 initially, how much of each good will the senior consume? Does your answer depend on the elasticity of substitution?

(c) Now suppose that the price of x₁ increases to p₁ = 1.25. How much does the government have to increase the senior’s social security payment in order for the senior to still be able to purchase the same bundle as he purchased prior to the price change?

(d) Assuming the government adjusts the social security payment to allow the senior to continue to purchase the same bundle as before the price increase, how much x₁ and x₂ will the senior actually end up buying if ρ = 0?

(e) How does your answer change if ρ = −0.5 and if ρ = −0.95? What happens as ρ approaches −1?

(f) How does your answer change when ρ = 1 and when ρ = 10? What happens as ρ approaches infinity?

(g) Can you come to a conclusion about the relationship between how much a senior benefit from the way the government calculates COLAs and the elasticity of substitution that the senior’s tastes exhibit? Can you explain intuitively how this makes sense, particularly in light of your answer to A(f )?

(h) Finally, show how COLAs affect consumption decisions by seniors under general inflation that raises all prices simultaneously and in proportion to one another as, for instance, when both p₁ and p₂ increase from1 to 1.25 simultaneously.

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