3. (100 marks) Consider a two-period consumer's problem where the representative cor sumer's utility is given by U(c,d) = u(c) + Bu(d), where c is current consumption and d' is future consumption, and u() is a strictly increasingly, strictly quasi-convave function. The consumer receives exogenous income endowments y and y' in the cur- rent and future periods respectively, and pays lump sum taxes t and t' in the present and future respectively. The consumer can buy or sell a bond s in the present period that for each unit is a promise to pay 1+r units of consumption in the future period. The consumer begins the current period with no assets, and finishes the future period with no assets (ie it consumes all its wealth in the future period). The government purchases G consumption goods in the present period and G' consumption goods in the future period, levying taxes t and t', and borrowing B in bonds in the present, paying back (1+r)B in the future period. (a) (10 marks) Derive the consumer's Euler equation" and provide an interpreta- tion of this equation. Now assume that u() is given by u(x) = lnr. (b) (10 marks) Find the consumer's optimal level of current consumption in terms of the variables and parameters that it takes as given (ie its consumption func- tion"). (c) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in discounted lifetime wealth we, where 1,">Y, , and show the sign for this if possible. Provide an intuitive explanation of your expression. awe? we = y-t+ = (d) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current income, y, en (ie the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. ac ay (e) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current the real interest rate, r, en in die the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. Now assume that u() is given by u(x) = 77, where o > 0. This utility function is very common in economics and (as you will see), has some very gen- eral properties. (f) (10 marks) Find the consumer's optimal level of current consumption in terms of the variables and parameters that it takes as given (ie its "consumption func- tion"). How does your expression compare to the similar expression you found above? (g) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in discounted lifetime wealth we, we = y-t+, y , and show the sign for this if possible. Provide an intu- itive explanation of your expression. How does your expression compare to the similar expression you found above? duce where (h) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current income, y, and (ie the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. How does your expression compare to the similar expression you found above? (i) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current the real interest rate, r, en in die the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. How does your expression compare to the similar expression you found above? () (10 marks) Show that for u(c) = -1 lim u(c) = Inc. (1) 1-o 7 Hint: Use L'hopital's rule What insight does this offer with respect to your answer above? 3. (100 marks) Consider a two-period consumer's problem where the representative cor sumer's utility is given by U(c,d) = u(c) + Bu(d), where c is current consumption and d' is future consumption, and u() is a strictly increasingly, strictly quasi-convave function. The consumer receives exogenous income endowments y and y' in the cur- rent and future periods respectively, and pays lump sum taxes t and t' in the present and future respectively. The consumer can buy or sell a bond s in the present period that for each unit is a promise to pay 1+r units of consumption in the future period. The consumer begins the current period with no assets, and finishes the future period with no assets (ie it consumes all its wealth in the future period). The government purchases G consumption goods in the present period and G' consumption goods in the future period, levying taxes t and t', and borrowing B in bonds in the present, paying back (1+r)B in the future period. (a) (10 marks) Derive the consumer's Euler equation" and provide an interpreta- tion of this equation. Now assume that u() is given by u(x) = lnr. (b) (10 marks) Find the consumer's optimal level of current consumption in terms of the variables and parameters that it takes as given (ie its consumption func- tion"). (c) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in discounted lifetime wealth we, where 1,">Y, , and show the sign for this if possible. Provide an intuitive explanation of your expression. awe? we = y-t+ = (d) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current income, y, en (ie the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. ac ay (e) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current the real interest rate, r, en in die the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. Now assume that u() is given by u(x) = 77, where o > 0. This utility function is very common in economics and (as you will see), has some very gen- eral properties. (f) (10 marks) Find the consumer's optimal level of current consumption in terms of the variables and parameters that it takes as given (ie its "consumption func- tion"). How does your expression compare to the similar expression you found above? (g) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in discounted lifetime wealth we, we = y-t+, y , and show the sign for this if possible. Provide an intu- itive explanation of your expression. How does your expression compare to the similar expression you found above? duce where (h) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current income, y, and (ie the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. How does your expression compare to the similar expression you found above? (i) (10 marks) Find an expression showing how the consumer's optimal current consumption responds to a change in current the real interest rate, r, en in die the consumer's marginal propensity to consume), and show the sign for this if possible. Provide an intuitive explanation of your expression. How does your expression compare to the similar expression you found above? () (10 marks) Show that for u(c) = -1 lim u(c) = Inc. (1) 1-o 7 Hint: Use L'hopital's rule What insight does this offer with respect to your answer above