Fall 2016

Intermediate Macroeconomics

Econ 3102-002

Intermediate Macroeconomics Fall 2016 Economics 3102-002 University of Minnesota Problem Set 2 Due Date: 17th, October, 2016

Question 1 (10 points total) Chapter 7, Question 1 In the Malthusian model, suppose that the quantity of land increases. Using diagrams, determine what effects this has in the long run steady state and explain your results.

Question 2 (15 points total, 3 each) Country A and country B both have the production function Y=F(K,N)=K0.3N0.7. Assume that both countries experiences population growth of 5% and that 5% of capital depreciates each year. Assume that country A saves 20% of output each year and country B saves 30% of output each year. a) What is the per-worker production function, y=f(k)? b) Calculate the steady-state level of capital per worker for each country. Then find the steady state level of product per worker and consumption per worker for each country. c) Suppose that both countries start off with a capital stock per worker of 3. What are the levels of income per worker and consumption per worker? d) Use a calculator or a computer spreadsheet to show how the capital stock per worker, product per worker and consumption per worker will evolve over time in both countries. How many

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

years will it be before the consumption in country B is higher than the consumption in country A? e) What is the golden rule for each of the countries? Can they improve?

Question 3 (16 points total, 4 points each) Chapter 7, Question 7 Modify the Solow growth model by including government spending as follows. The government purchases G units of consumption goods in the current period, where G = gN and g is a positive constant. The government finances its purchases through lump-sum taxes on consumers, where T denotes total taxes, and the government budget is balanced each period, so that G = T. Consumers consume a constant fraction of disposable income—that is, C = (1 - s)(Y - T), where s is the savings rate, with 0 < s < 1. a) Derive equations similar to Equations (7-18), (7-19), and (7-20), and show in a diagram how the quantity of capital per worker, k*, is determined. b) Show that there can be two steady states, one with high k* and the other with low k*. c) Ignore the steady state with low k* (it can be shown that this steady state is "unstable"). Determine the effects of an increase in g on capital per worker and on output per worker in the steady state. What are the effects on the growth rates of aggregate out-put, aggregate consumption, and aggregate investment? d) Explain your results.

Question 4 (16 points total, 4 points each) Assume that the production function is given by Y=zK αN(1-α), or equivalently y=zkα. a) Country A is initially in a steady-state. Suppose an event leads to a sudden decrease in country A's labor force stock, N. Using the relevant graphs (ignore the consumption curve for this

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

question) and the relevant equation(s), show the short-run impact and long-run impact this event has on k, y, MPK and MPL. Be sure to briefly explain any adjustments. b) What will happen with wages in the short-run and the long-run. Now, assume that country B is identical to country A in all respects (i.e., same savings rate, technology, etc.) EXCEPT for his initial value of k. Specifically, assume that kA > kB with all values bellow kss. c) Which country will have the higher initial MPK? Explain with graph or equation. d) Which country will have the higher growth rate for k? Explain.

Question 5 (16 points total, 4 points each) Assume U(c,c')=c1/2c'1/2 and that initially y = 70, y' =88 , and r = 0.1. There is no government. a) Calculate c, c' and s. b) Looking at the case of a net borrower, what do we have to assume about the substitution and income effects to generate the result that a decrease in r results in an increase in current period consumption. Only with words briefly explain your answer. c) This question requires the use of graphs, no specific numbers involved. c1 - Suppose y decreases and y' increases in such a way as to leave the present value of lifetime income (wealth) unchanged. Using a graph, NO number, show the impact on c and s. c2 - Now suppose y increases and y' decreases in such a way as to leave the present value of lifetime income (wealth) unchanged. Using a graph, NO number, show the impact on c and s.

Question 6 (16 points total, 4 points each) Chapter 9, Question 8 Assume a consumer who has current period income y = 200, future-period income y'=150, current and future taxes t = 40 and t' = 50, (respectively, and faces a market real interest rate of r 3

Fall 2016

Intermediate Macroeconomics

Econ 3102-002

= 0.05, or 5% per period. The consumer would like to consume equal amounts in both periods; that is, he or she would like to set c = c', if possible. However, this consumer is faced with a credit market imperfection, in that he or she cannot borrow at all, that is, s ≥ 0. a) Show the consumer's lifetime budget constraint and indifference curves in a diagram. b) Calculate his or her optimal current-period and future-period consumption and optimal saving, and show this in your diagram. c) Suppose that everything remains unchanged, except that now t = 20 and t’ = 71. Calculate the effects on current and future consumption and optimal saving, and show this in your diagram. d) Now, suppose alternatively that y = 100. Repeat parts (a) to (c), and explain any differences.

Question 7 (11 points total) Chapter 9, Question 5 A consumers receives income y in the current period, income y' in the future period, and pays taxes of t and t' in the current and future periods, respectively. The consumer can borrow and lend at the real interest rate r. This consumer faces a constraint on how much he or she can borrow, much like the credit limit typically placed on a credit card account. That is, the consumer cannot borrow more than x, where x < we - y + t, with we denoting lifetime wealth. Use diagrams to determine the effects on the consumer's current consumption, future consumption, and savings of a change in x, and explain your results.

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Intermediate Macroeconomics

Econ 3102-002

Solution Question 1 The amount of land increases, and, at first, the size of the population is unchanged. Therefore, consumption per capita increases. However, the increase in consumption per capita increases the population growth rate, see the figure below. In the steady state, neither c* nor l * are affected by the initial increase in land. This fact can be discerned by noting that there will be no changes in either of the panels of Figure 6.8 in the textbook.

Question 2 a) Y/N=(K0.3N0.7)/N=(K/N) 0.3=k0.3 => y=k0.3=f(k) b) nA=nB=0.05, dA=dB=0.05, sA=0.2 and sB=0.3. At steady-state: (n+d)k*=szf(k*) (n+d)k => 0.1k*=sk*0.3 => k*=(s/0.1)1/0.7 Country A: k*A=(sA/0.1)1/0.7=2.7

y*A=k*A0.3=1.35

Country B: k*B=(sB/0.1)1/0.7=4.8

y*B=k*B0.3=1.6

c) k*=3

y*=k*0.3=1.4= y*B= y*B

c*A=(1-sA)y*A=1.07 c*B=(1-sB)y*B=1.12

c*A=(1-sA)=1.112 5

c*B=(1-sB)=0.97

Fall 2016

Intermediate Macroeconomics

Econ 3102-002

d) Will take 22 years before consumption in country B is higher than consumption in country A.

Country A, s=0.2

Country B, s=0.3

Year

k

y= k0.3

c=(1-s)y

k

y=k0.3

c=(1-s)y

1

3.000

1.390

1.112

3.000

1.390

0.973

2

2.979

1.387

1.110

3.112

1.406

0.984

3

2.960

1.385

1.108

3.217

1.420

0.994

4

2.942

1.382

1.106

3.316

1.433

1.003

5

2.925

1.380

1.104

3.410

1.445

1.011

6

2.909

1.378

1.102

3.498

1.456

1.019

7

2.894

1.376

1.100

3.581

1.466

1.026

8

2.881

1.374

1.099

3.659

1.476

1.033

9

2.868

1.372

1.097

3.732

1.484

1.039

10

2.856

1.370

1.096

3.800

1.493

1.045

11

2.845

1.368

1.095

3.865

1.500

1.050

12

2.835

1.367

1.094

3.925

1.507

1.055

13

2.825

1.366

1.092

3.982

1.514

1.060

14

2.816

1.364

1.091

4.035

1.520

1.064

15

2.808

1.363

1.090

4.085

1.525

1.068

16

2.800

1.362

1.090

4.132

1.531

1.071

17

2.793

1.361

1.089

4.176

1.535

1.075

18

2.786

1.360

1.088

4.217

1.540

1.078

19

2.780

1.359

1.087

4.255

1.544

1.081

20

2.774

1.358

1.086

4.291

1.548

1.084

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Intermediate Macroeconomics

Econ 3102-002

21

2.768

1.357

1.086

4.325

1.552

1.086

22

2.763

1.357

1.085

4.356

1.555

1.089

23

2.758

1.356

1.085

4.386

1.558

1.091

24

2.754

1.355

1.084

4.413

1.561

1.093

25

2.750

1.355

1.084

4.439

1.564

1.095

26

2.746

1.354

1.083

4.463

1.566

1.096

27

2.742

1.353

1.083

4.485

1.569

1.098

28

2.739

1.353

1.082

4.506

1.571

1.100

29

2.736

1.352

1.082

4.526

1.573

1.101

30

2.733

1.352

1.082

4.544

1.575

1.102

e) Golden Rule: MPk=n+d MPk = 0.3k-0.7= 0.1 => kgold = (0.3/0.1)1/0.7 At steady-state: (n+d)k*=szf(k*) => 0.1k*=sk*0.3 => k*=(s/0.1)1/0.7 k*=(s/0.1)1/0.7= kgold = (0.3/0.1)1/0.7 => s/0.1 = 0.3/0.1 => sgold = 0.3 Both countries should have a saving rate equal to 0.3. Country A need to increase the savings rate and country B already have the optimal saving rate.

Question 3 Government spending in the Solow model. a) By assumption, we know that T = G, and so we may write: K'  s(Y  G )  (1  d )K  sY  gN  (1  d ) K

Now divide by N and rearrange as: 7

Fall 2016

Intermediate Macroeconomics

Econ 3102-002

k' (1  n)  szf (k )  sg  (1  d )k

Divide by (1 + n)) to obtain: k' 

szf (k ) sg (1  d )k   (1  n) (1  n) (1  n)

Setting k = k,, we find that: szf (k * )  sg  (n  d )k *.

This equilibrium condition ion is depicted in the figure below.

b)

The two steady states are also depicted in the figure above.

c)

The effects of an increase in g are depicted in the bottom panel of the figure above.

Capital per capita declines in the steady state. Steady Steady-state growth owth rates of aggregate output, aggregate consumption, and investment are all unchanged. The reduction in capital per capita is accomplished through a temporary reduction in the growth rate of capital.

Question 4 a,b)

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

Short-run: N ↓ => k = K/N ↑ , y increases to y1. By MPK = αzK(α-1)N(1-α) = αz/k(1- α), if k ↑ => MPK ↓. By MPL = (1- α)zK αN(-α) = (1- α)zk α, if k ↑ => MPL ↑. Because w=MPL then the wages increases in the short-run. run. Long-run: run: Adjustments. Since szkα<(n+d)k

at k1, then k will decrease in the long-run. long

Therefore, actual investment < required investment. So y decreases back to s-s, s, MPK increases back to s-s, s, and MPL decrease back to ss-s. s. In the long run wages decreases and go back to the original level. c) MPK = αzK(α-1)N(1-α) = αz/k(1- α), if k ↓ => MPK ↑. So since kA > kB, then MPKA < MPKB.

α) d) k' = k + sy -(n+d)k => k'-kk = ∆k =sf(k)-(n+d)k, => (k'-k)/k = ∆k/k =sz/k(1-α) -(n+d) since kA

> k B, then ∆k A /k A < ∆k B /k B.

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

Question 5 a) we=

+

= 70 +

.

= 150. At optimum Uc / Uc' = c' / c = 1+r. Therefore, +

150 => 2c = 150=> c*=75 and c'*=(1+r)c*=82.5.

(

)

=

s = y - c = -5. Borrower.

b) Substitution effect: r↓ => c↑ due to substitution effect the opportunity cost of future consumption value today goes down. Income effect: Since r↓ the total wealth increases (assuming a borrower), so to smooth out consumption over both periods the worker must consume more today and tomorrow: c↑, c'↑ c) c1- On the graph bellow we can note that consumption today and tomorrow will remain constant. The savings will decrease: s = y ↓ - c , but c remains constant => s↓

c2- On the graph bellow we can note that consumption today and tomorrow will remain constant. The savings will increase: s = y↑ - c , but c remains constant => s↑

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

Question 6 Given information: y  200 y'  150 t  40 t'  50 r  0.05

a)

If the consumer could borrow and lend at the real interest rate, r 0.05, then the

consumer’s lifetime budget constraint would be given by: c

c y  t   yt  . (1  r ) (1  r )

Plugging in the numbers from this problem, we obtain: c  0.95c'  255.2.

In the figure below, the initial budget constraint is given by BE 1D. The budget constraint has a kink at the initial endowment point E1 = (160,100), because the consumer cannot borrow, and therefore cannot consume more than 160 in the first period. Because the consumer has perfect-complements preferences, the indifference curves are kinked at c  c. 11

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b)

Intermediate Macroeconomics

Econ 3102-002

With perfect-complements complements prefere preferences, nces, the consumer picks point A in figure on the

previous page. Plugging in c  cc into the budget constraint and solving, we find that c = c = 130.7 and so s  y  t  c  160  130.7  29.3. In this case, the fact that the consumer cannot borrow does not matterr for the consumer’s choice, as the consumer decides to be a lender. c)

When t = 20 and t'  71, the consumer’s lifetime wealth remains unchanged at 255.2.

However, the budget constraint shifts to BE 2F, figure on the previous page, with the new endowment point at E2 = (180,79). This change does not matter for the consumer’s choice, again because he or she chooses to be a lender. Consumption is still 130.7, but now savings is s  y  t  c  180  130.7  49.3.

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d)

Intermediate Macroeconomics

Econ 3102-002

Now first-period income falls to 100. Wealth is now equal to w = 155.2. In the figure

above, the budget constraint for the consumer is AE 1D, so when the consumer chooses the point on his or her budget constraint that is on the highest indifference curve, any point on the line segment BE1 will do. Suppose that the consumer chooses the endowment point E 1, where c = 60 and c  100. This implies that s = 0, and the consumer is credit-constrained in that he or she would like to borrow, but cannot. Now with the tax change, the budget constraint shifts to AE 2G, with the endowment point E2 = (80,79). Thus the consumer can choose c = c on the new budget constraint, and solving for consumption in each period using the budget constraint c  0.95c'  155.2,

we get c = c = 79.5, and s = 0.5. Here, notice that first-period consumption increased by almost the same amount as the tax cut, although lifetime wealth remains unchanged at 155.2. Effectively, the budget constraint for the consumer is relaxed. Therefore, for tax cuts that leave lifetime wealth unchanged, lenders will not change their current consumption, but creditconstrained borrowers will increase current consumption.

Question 7 The consumer faces a borrowing constraint that places a ceiling on the level of current consumption. The consumer may consume more than the current endowment, y  t, but less than the amount of the lifetime endowment, we. The consumer’s budget line is as in the first figure below. The budget line becomes vertical at c = x. An example of such a budget line is depicted in the two panels of the figure as ABD. As one possibility, the constraint is nonbinding as in the figure below. The consumer chooses point H. A change in the level of x has no effect on such a consumer. Alternatively, the consumer depicted in the second figure below originally chooses the corner solution, point B. The consumer achieves the level of utility corresponding to indifference curve, I1. An increase in x produces the new budget line, ACJ. This consumer now chooses point G. She increases current consumption and decreases both current saving and future consumption.

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Fall 2016

Intermediate Macroeconomics

Econ 3102-002

This consumer is able to improve her level of utility to that corresponding to indifference curve, I2.

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