Consider an individual€™s utility function over two goods, q_mand q_s, where m indicates the primary market in which a policy will have its effect and s is a related secondary market:

whereα, β_m, β_s, and γ are parameters such that β_m>0, and β_s>0, β_m<(1-γq_s)/2q_m, β_s<(1-γq_m)/2q_s, and γ<p_mβ_s/p_s + p_sβ_m/p_m. For purposes of this exercise, assume that α =1, β_m = 0.01, β_s = 0.01, and γ = -0.015. Also assume that the person has a budget of $30,000 and the price of q_m, p_m, is $100 and the price of q_s, p_s, is $100. Imagine that the policy under consideration would reduce p_m to $90.

The provided spreadsheet has two models. Model 1 assumes that the price in the secondary market does not change in response to a price change in the primary market. That is, p_s equals $100 both before and after the reduction in p_m. Step 1 solves for the quantities that maximize utility under the initial p_m. Step 2 solves for the quantities that maximize utility under the new p_m. Step 3 requires you to make guesses of the new budget level that would return the person to her original level of utility prior to the price reduction €” keep guessing until you find the correct budget. (You may wish to use the Tools|Goal Seek function on the spreadsheet instead of iterative guessing.) Step 4 calculates the compensating variation as the difference between the original budget and the new budget. Step 5 calculates the change in the consumer surplus in the primary market.

Model 2 assumes that p_s = a + bq_s. Assume that b=0.25 and a is set so that at the quantity demanded in Step 2 of model 1, p_s=100. As there is no analytical solution for the quantities before the price change, Step 1 requires you to make guesses of the marginal utility of money until you find the one that satisfies the budget constraint for the initial p_m. Step 2 repeats this process for the new value of p_m. Step 3 requires you to guess both a new budget to return the person to the initial level of utility and a marginal utility of money that satisfies the new budget constraint. A block explains how to use Tools|Goal Seek to find the marginal utility consistent with your guess of the new budget needed to return utility to its 

original level. Step 4 calculates the compensating variation. Step 5 calculates the change in the consumer surplus in the primary market and bounds on the change in consumer surplus in the secondary market. 

Use these models to investigate how well the change in social surplus in the primary market approximates compensating variation. Note that as utility depends on consumption of only these two goods, there are substantial income effects. That is, a price reduction in either of the goods substantially increases the individual€™s real income.Getting started: The values in the spreadsheet are set up for a reduction in p_m from $100 to $95. Begin by changing the new primary market price to $90 and resolving the models.

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