1) A consumer has intertemporal preferences for food given by the utilityfunction: U(C1, C2) = C21C2. The consumer gets a one-time only endowment of food at the beginning of period 1 equal to 60 units.
(a) Solve for the intertemporal consumptions that maximize utility. Illustrateyour results using a carefullty labelled diagram. What is theutility level of the agent at the optimum. Explain why this is likeconsuming a cake on a desert island.
(b) Next, suppose that the agent gains access to a market where shecan borrow or lend units of endowment at an interest rate equal tor. Derive the new optimal consumptions for the agent. Determinewhether the agent benefits from trade in that her utility level is higherat the new optimum than at the old optimum. Illustrate the newoptimum using a carefully labelled diagram
2. Consider a stardard labour supply model of an agent who has an endowment of time equal to R¯. The agent has no endowment of the consumption good. Suppose that the price of the consumption good is p and that the wage rate equals w. Suppose that the government is thinking about taxing workers a per unit rate of t for every hour worked. The government comes to you for advice. They want you to tell them what you think might happen to labour supply as a result of the tax. Provide one answer using a carefully labelled diagram and the appropriate Slutsky equation. Provide asecond answer based upon preferences being given by: U(C, R) = ln(CR)where R is leisure.
3. A consumer has preferences given by U(x1, x2) = x21x2.
(a) Derive the demand curves for x1 and x2 when prices and income aregiven by p1, p2 and m.
(b) Illustrate the equilibrium on a diagram when p1 = p2 = $1 andI = $12.
(c) Calculate the exact income and substitution effects for x1 when p1rises to $3.
(d) Explain your exact results using the appropriate Slutsky equation.Show all steps. Note: no marks will be given for a purely diagrammaticanswer but your results must be illustrated with a carefullylabelled diagram.
4. Suppose there are two goods in an economy. One good is a standard consumptiongood x and the price of this good is 1. An agent gets an initialendowment of the consumption good Y . An agent can also choose to donatepart of her endowment to charity. Denote the charitible contributionby c. We think of c as the second good in the economy. Both goods enterthe utility function of the agent, U(x, c). We suppose that the utilityfunction has contours with the regular shape.
(a) Currently, the government taxes the income (endowment) of theagent. The income tax rate is constant and equal to t. At presentthe govenment only taxes income net of charitible contributions. Thismeans that tax is paid only on the amount Y - c. Using this information,derive the budget constraint of the agent.
(b) Depict the equilibrium of the agent with a carefully labelled diagram.
(c) The government comes to you with the question: What would happento charitible contributions if the income tax rate was raised? Giveyour best answer to them. Assume that the government understandsendowmwent models and Slutsky equations.
5. Respond to the following 3 statements using carefully drawn diagrams.The statements may be true or false or ambiguous.
(a) In a standard endowment model if the agent is a net buyer of good 1 then a decrease in the price of good 1 will make the agent worse off.
(b) In a standard endowment model if the agent is a net seller of good 1then a decrease in the price of good 1 will make the agent worse off.
(c) An agent will prefer a price subsidy for one good to a lump sumsubsidy when both subsidy schemes distribute the same amount ofsubsidy to the agent.