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There are two goods, food and clothing, whose quantities are denoted by x and y and prices by px and py, respectively. There is a consumer whose income is denoted by I and utility by U. His utility function is u (x; y) = x 1=2 y 1=2 . (a) Find his Hicksian (compensated) and Marshallian (uncompensated) demand functions. Find the expenditure function and the indirect utility function. (b) Initially I = 100, px = 1, and py = 1. What quantities does the consumer buy, and what is his resulting utility? (c) Now the price of food rises to px = 1:21, while income and the price of clothing are as before. What quantities does the consumer buy and what is his resulting utility? (d) Suppose the increase in the price of food was caused by the government levying a tax of 21 percent on food. What is the governmentís revenue from this tax? (e) If the government wants to compensate the consumer by giving him some extra income, how much extra income would be needed to restore him to the old utility level (i.e., the ìHicksian compensationî or ìcompensation variationî for the price change)? Is the governmentís revenue from the tax on food itself su¢ cient to provide this compensation? What is the economic intuition for your answer? (f) If the government tries to compensate the consumer by giving him enough extra income to enable him to purchase the same quantities as he did at the original income and prices of part (b), how much extra income would the government have to give him (the ìSlutskyî compensation)? With this income and the new prices, what quantities will the consumer actually buy? What will be his resulting utility? (g) In a graph with x on the horizontal axis and px on the vertical axis, show the consumerís choices with the income I = 100 and the two prices px = 1 and px = 1:21. Calculate the reduction of ìconsumer surplusîmeasured by the area under the Marshallian demand curve. Compare this to the Hicksian compensation (your answer to part (e)). If the two are di§erent, explain the di§erence.
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