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Problem 2. Consider a consumer has Cobb-Douglas preferences over two goods 21 and 22, given by u (21, 22) = 7 ln 21 + In 22. Let p_ = 5 and p2 = 3 be the prices of the two goods, and suppose the agent has income I = 20. Suppose there is rationing of goods, so that in addition to paying for goods, the agent must have the appropriate number of coupons. Suppose, the agent begins with 10 coupons, and each unit of good 1 requires ci = 2 coupons and each unit of good 2 requires C2 = 1 coupon. Assume goods are perfectly divisible, so the agent can purchase fractional quantities, but that negative quantities are impermissible. 1. Suppose the agent ignores the coupon constraint, and plans to purchase the bundle that maximizes her utility given the budget constraint. Verify that this bundle violates the coupon constraint. 2. Suppose the agent ignores the budget constraint, and plans to purchase the bundle that maximizes her utility given the coupon constraint. Verify that this bundle does not violate the budget constraint. Note: The agent’s budget constraint is satisfied as long as she does not exceed her budget. Argue that this is the optimal bundle. 3. Does the agent exhaust her budget at the optimum? 4. Suppose a black-market arises where the agent can trade coupons for cash (and vice versa) at a fixed rate of 50c per coupon. Let z be the quantity of coupons purchased. Argue that both constraints must bind at the optimum. 5. Find the consumer’s Marshallian demand for each good. Describe her behavior on the black market.
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