Problem 1: ( % = %)

Paul values consumption in the current period (c) and in the future period (c), using the utility function = ln() + 0.8 ln(!). Paul can save or borrow at the real interest rate r=10%. Denote net saving in the current period by (s). Pauls income in the current period (y) is 80, while in the future period (y) is 110. He does not pay any taxes.

1. a) Write down Pauls budget constraint in the current period and in the future period.

2. b) How many units of future period consumption (c) does Paul have to give up to get an extra unit of consumption in the current period (c) (at the margin)? that is what is the cost of one extra unit of (c)?

3. c) How many units of future period consumption (c) is Paul willing to give up to get an extra unit of consumption in the current period (c) (at the margin)? that is what is the value of one extra unit of (c)?

4. d) Explain why the combination (, !) = (, ) is not optimal for Paul. Illustrate with a graph.

5. e) Find Pauls optimal consumption choice.

6. f) CalculatePaulssaving(s),ishealenderoraborrowerinthecurrentperiod?

7. g) Suppose that Pauls income in the future period increases by 11 units. Explain

   why Paul will not increase future period consumption by 11 units. What should

   he do?

8. h) Would an increase in interest rate necessarily lead Paul to save more? Explain,

   why or why not.

Problem 2: ( % = %)

Lisas income in the current period is = 100, and her income in the future period is = 120.Shepaysalump-sumtax = 20inthecurrentperiod,and = 10inthe future period. The real interest rate r = 0.1, or 10%, per period.

1. a) Determine Lisas lifetime wealth ().

2. b) Suppose that current and future consumption are perfect complements for Lisa; Draw her indifference curves.

3. c) DetermineLisasoptimalcurrent-periodandfuture-periodconsumption.

4. d) Calculate Lisas savings (s), is she a lender or a borrower in the current period?

5. e) In a diagram, show Lisas optimal choices, with Lisas endowment point, budget constraint and indifference curves.

6. f) Nowsupposethatinsteadof=100,Lisahasnowanincome=140.Again, determine Lisas optimal consumption in the current and future periods.

7. g) Determine Lisas optimal savings. Is she a lender or a borrower? In a diagram, show Lisas new endowment point, and optimal choices.

8. h) Explain the differences in your results before and after the income change.

Problem 3: ( % = %)

Paul and Lisa value consumption in the current period (c), and in the future period

(c), using the same utility function = ln() + 0.8 ln(!). Pauls income in the

current period = 102, while that in the future period ! = 132. Lisas income in""

the current period = 132, while that in the future period ! = 99. Both Paul and##

Lisa pay 22 units in taxes in the current period and the future period, that is = ! = 22 . Lisa can borrow or save at the real interest rate = 10%. However, everyone knows that Paul is dishonest; As a result, nobody is willing to lend him. Of course, he can still save at the interest rate = 10%.

1. a) Determine Pauls desired consumption choices in the current period and in the future period, if he was able to borrow?

2. b) Determine Pauls actual consumption choices in the current and in the future periods.

3. c) Illustratebothallocationsonagraph.Whatisthecostofthiscreditconstraint?

4. d) Determine Lisas optimal consumption plan.

5. e) Find the value of Lisas savings that would allow her to achieve this plan.

   Suppose that the timing of taxes changes: taxes in the current period are reduced from 22 to 12 units, while taxes in the future period are increased from 22 to 33 units.

6. f) HowdoesthischangeintaxesaffectLisaswealth?DoesitchangeLisasoptimal choices, (c), (c) and (s) from those before the tax change? Explain.

7. g) Illustrate your results in part f) with a graph.

8. h) HowdoesthischangeintaxesaffectPaulswealth?DoesitchangePaulsoptimal

   choices, (c), (c) and (s) from those before the tax change?

9. i) Illustrate your results in part h) with a graph.

10. j) Does this change in the timing of taxes constitute a Pareto improvement? Explain

    why or why not.

Problem 4: ( %)

Consider a two-period economy populated with consumers that have the same income and the same preferences. The governments objective is to spend 60 units in the current period = 60, and 150 units in the future period ! = 150. The government can borrow in the current period by issuing bonds B>0. Each bond pays the real interest rate . Consumers can also borrow at the same real interest rate . Consumers optimal decisions, given , imply that aggregate consumption =

% ( ) + % '(!)*!+. Suppose that =! = 300.

&

a)

b)

& (-./)

(3%) Define the competitive equilibrium of this economy.

(6%) Show that together the conditions in part a) imply that the equilibrium

value of the real interest rate = %((!)1!) 1()1

c) (3%) Calculate Aggregate Demand in the current period.

A major recession begins in the current period. As a result, economic activity falls by 18 units in the current period. Economists expect this recession to continue into the next period, and national income, as a result, is expected to fall by 20 units in the future period. Consumers believe these expectations.

d) (3%) Calculate aggregate Demand in the current period, given the recession.

e) (3%) Use a graph to explain why the equilibrium interest rate falls from 0.25 to 0.20 in the current period because of the recession.

f) (3%) Explain why the government should not increase its expenditure G, to fight the recession.

g) (4%) Would a tax cut in the current period, that is decrease in T, be a better choice than an increase in G to fight this recession? Explain, why or why not.