# Economics -Q55

## Economics -Q55 Online Services

The assignment is out of 20 points. You are welcome to collaborate with your classmates for this assignment. Please do acknowledge with whom you collabo- rate, if anyone. Each person must hand in their own assignment. As described
in the syllabus, there will be a 50 % point penalty for an assignment handed in late.

1 The Household Problem for T = 2 (the Fisher model)
(6 points) Joe Six-pack just learned about the two period household maximization model and fell so deeply in love with it that he decides that he wants to live (i.e. consume) according to it. He is in the rst period of his life and he spends it in college. In college he works in the cafeteria where he makes \$20; 000 after taxes. He knows for sure that he is going to get a job at McKinsey (he is an economics major) and will make \$105; 000 after taxes in the second period of his life (sadly enough he only lives for two periods). He has no initial wealth and he can borrow and lend at an interest rate of 5%: His utility function is

U(c1; c2) = log(c1) + 0:5 log(c2)

1. (2 points) Determine Joe’s optimal consumption in both periods of his life. Is Joe a saver or a borrower? Determine his optimal level of savings (assets to be brought from period 1 to period 2).

2. (0.5 point) How would Joe’s optimal consumption and saving decision change if he had a utility function of the form U(c1; c2) = c1  c0:5

3. (1 point) Back to Joe’s original utility function for the rest of the question. Suppose the interest rate increases to 10%: Now what is Joe’s optimal consumption and sav- ing/borrowing plan?

4. (0.5 point) Suppose the interest rate is back at 5%: The government wants to cut taxes to stimulate the economy. Suppose the government cuts the taxes that Joe has to pay in the rst period by \$2; 000; so that Joe’s after tax income increases from \$20; 000 to \$22; 000: In order to nance this tax cut the government has to increase taxes in the second period by \$2000  (1 + r) = \$2; 100: Hence Joe’s income in the second period goes down from \$105; 000 to \$102; 900: What is Joe’s new optimal consumption and saving plan. Compare to your answer in Part 1.

5. (2 point) Suppose more generally that Joe’s utility is given by

U(c1; c2) =
c1􀀀
1 􀀀 1
1 􀀀 
+
c1􀀀
2 􀀀 1
1 􀀀 

where  > 0 is a xed parameter. He has no initial wealth, and only income y1 in period 1. The interest rate is r. What can you say about the e ect of a change in the interest rate on consumption c1, @c1 @(1+r)? How does this depend on the parameter ?

2 Two-Period Model with Labor Supply(9 points) Here we’ll solve for a representative household’s problem who lives two periods and gets disutility from working. Consider a household that lives for two periods and in each
period works, consumes, and saves/borrows. He solves:

max
c0;c1;l0;l1
log c0 􀀀

l2
0
2
+

log c1 􀀀

l2
1
2

subject to: c0 +
c1
1 + r1
= w0l0 +
w1l1
1 + r1
+ a0(1 + r0)
with
> 0. Here, c0 and c1 are consumption, l0 and l1 are labor e ort, w0 and w1 are wages,and r0 and r1 are the interest rates.

1. (1 point) Here we don’t see assets appearing anywhere. What is given in the problem is the lifetime budget constraint that comes from two budget constraints (one for each period). Write out the budget constraints for each period that combine to generate the lifetime constraint.

2. (3.5 points) Solve the household’s problem, assuming a0 = 0. Here, a solution should be c0, c1, l0, and l1 in terms of only wages, interest rate r1, and parameters. Interpret how a ects your results.

3. (1.25 point) Assume w0 = w1. Find and describe the consumption and labor supply pro les (i.e., how each moves over the household’s lifetime) when

(a) 1
> 1 + r1
(b) 1
< 1 + r1 (c) 1 = 1 + r1 Present some intuition for your ndings.   4. (1.25 point) Assume w0 < w1. Find and describe the consumption and labor supply pro les when (a) 1 > 1 + r1
(b) 1
< 1 + r1 2 (c) 1 = 1 + r1 Present some intuition for your ndings.   You can read more about our case study assignment help services here.

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5. (2 point) Assume now that w0 < w1 and 1 > 1 + r1. What happens with the con-sumption and labor supply pro le if there are borrowing constraints (i.e. c0  w0l0).Compare your answer with the case without constraints.

3 Social Planner Problem and Steady States (5 points) Consider the social planner problem of the neoclassical growth model with loga-rithmic utility and full depreciation

max
fct;kt+1g1
t=0
X1
t=0
t log(ct)
subject to
ct + kt+1 = k
t
ct; kt+1  0 and k0 > 0 given
with ; 2 (0; 1)

1. (2 points) Derive the Euler equation by combining the rst order conditions with respect to ct; ct+1; kt+1.

2. (1 point) Find the steady state capital stock(s) and consumption level(s). How do they depend on the parameters ; ? Interpret your ndings.

3. (1 point) Suppose that k0 > 0. Is the capital stock increasing or decreasing over time?Where does it end up in the long run? How will your answer depend on the concrete value of k0?

4. (1 point) Suppose that the resource constraint becomes
ct + kt+1 = 2  k
t