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ECON-UA 12 ? Intermediate Macroeconomics

Professor: Jaroslav Borovi?cka

TA: Jonathan Payne

Fall 2016 Problem Set 2

Due on October 7, in your recitation

Please read the instructions on problem sets in the syllabus carefully. 1 True or false? For each of the following statements, decide whether it is true or false, and explain your

answer in at most two sentences using the material you learned in the course.

No points will be given without an appropriate explanation!

Question 1.1 The arbitrage equation can only hold in an economy in which returns are

not taxed. Question 1.2 Imagine an economy with two investors. Investor 1 can only invest into a

savings account while investor two can only invest into capital, and the two investors cannot

trade with each other. Then the arbitrage equation equalizing the returns on the savings

account and the return on capital still holds. Question 1.3 The user cost of capital also reflects changes in the price of capital. Question 1.4 The price-dividend ratio is high when the dividend growth rate is high

because prices and dividends have to grow at the same rate. Question 1.5 When we look at empirical evidence, we can observe that poor countries

catch up but only when we focus on countries that follow sensible economic policies. Question 1.6 A plausible model of capital depreciation for a factory is one where one

machine depreciates (is destroyed) every year, regardless how big the factory is. Question 1.7 There are two steady states in the Solow growth model but one of them is

more of a mathematical curiosity than of any economic relevance.

1 Question 1.8 When the productivity parameter A increases in the Solow growth model,

the increase in output in the new steady state (relative to the old steady state) will be

larger than the increase in output caused by the same increase in A in the static model of

production from Chapter 4. Question 1.9 The Solow growth model documents that capital accumulation alone can

generate long-run growth. Question 1.10 In the article ?The Myth of Asia?s Miracle?, Paul Krugman argues that

growth in some East Asian economies bears remarkable similarities to that of the communist

Soviet Union, and that this fast growth must necessarily slow down. 2 The Gordon growth formula Recall the Gordon growth formula for the price-dividend ratio ps /d.

Question 2.1 Show by taking the appropriate derivatives that the price dividend ratio

is a decreasing function of the interest rate R and an increasing function of the dividend

growth rate gd . Question 2.2 Recall the graph for the price-earnings ratio from lecture slides (the Robert

Shiller data) and assume that earnings are equal to dividends. Further assume that the real

interest rate is constant at 7%.

If the Gordon growth formulas is correct, what would the dividend growth rate have to

be to explain the highest peak in the P/E ratio data that occurred around 2000?

Similarly, what would the dividend growth rate have to be to explain the deepest through

the the P/E data that occurred in early 1920s? Question 2.3 Are the numbers you obtained plausible? What else could justify the

fluctuations in the P/E ratio? 3 Comparative statics in the Solow growth model In this exercise, we want to assess the contribution of capital accumulation to our understanding of differences in output per capita across countries. We will distinguish between 2 the results obtained from the static model of production and from the Solow growth model.

Question 3.1 Start with the Cobb Douglas production function

Y = AK ? L1?? . (1) We want to compute what is the change in output after a parameter (like the TFP A)

changes. We will proceed exactly in the same way as when we computed annual growth

rates. Denote Yold the amount of capital before the change, and Ynew the amount of capital

after the change in a parameter. Then we can define

Ynew

= 1 + gY

Yold

where gY is the growth rate of output from before until after the change. Substitute in

for Ynew /Yold from equation (1) and do the same for variables A, K, L. By using the

logarithmic approximation, derive the same formula for gY as the one that we were using

to compute annual growth rates (gY will be a function of gA , gK , gL and the parameter ?). Question 3.2 We will now do the same with the steady state relationship in the Solow

growth model. Take the formula for the steady-state capital-output ratio K ? /Y ? (it is a

function of s and ?), and substitute in for Y ? from the Cobb-Douglas production function.

Express K ? as a function of A, s, ?, L and ? only.

Once you are done, take again the production function and substitute in for K ? using

the formula that you just obtained to get an expression for Y ? as a function of A, s, ?, L

and ? only.

Hint: What you should ultimately get is

s ?

1

1??

L

(2)

Y ? = A 1??

?

but make sure you provide all the intermediate steps in your answer. Question 3.3 Take the expression for output as a function of parameters. We now want

to express the cumulative growth rate of output between the old and the new steady state

when one of the parameters ?, s, A, L, changes (treat ? as constant). In order to do so,

define again the cumulative growth rate of output from the old to the new steady state as

?

Ynew

? = 1 + gY

Yold and correspondingly the growth rates of ?, s, A, L; for instance

snew

= 1 + gS .

sold

Remember that gY is not an annual growth rate of output ? it is the total percentage

change in output between the old and the new steady state. Similarly, gS is the percentage

change in the saving rate parameter.

3 Now find the growth rate of output gY as a function of gA , gS , g? and gL .

?

.

Hint: The coefficient on gS that you should obtain is 1??

Important: It is important to understand what growth rate are we computing here.

This is not an annual growth rate, it is the cumulative growth rate that expresses the

percentage change in the steady state level of output after one of the variables of interest

(?, s, A, L) changes by a given percentage. It may therefore be more appropriate to call gY a

?percentage change? rather than the growth rate but given the similarities with our previous

computations, we will stick to the term growth rate, with the implicit understanding of

what is going on.

In what follows assume that the capital share is equal to 13 , as we have usually assumed.

Question 3.4 If the TFP parameter increases by 10%, how does total output change in

the model of production and in the Solow growth model? Why is there a difference? Question 3.5 If population increases by 3%, how does the total output in the economy

change in the model of production and in the Solow growth model? Why (or why not) is

there a difference? Question 3.6 So far, we computed changes in total output. We will now focus on the

growth rates of output per capita.

Compute the growth rate of output per capita, YL , in the model of production and in

the Solow growth model.

Hint: Notice that you do not need to derive everything from scratch, just slightly

reorganize the results for gY . Question 3.7 What happens to the level of output per capita when population increases

by 10% in the model of production? What happens to the steady state level of output per

capita when population increases by 10% in the Solow growth model?

Provide an economic explanation for the results. Why do you observe an negative sign

in your answer for the model of production? And why not in the Solow growth model? 4 Experiments in the Solow growth model In the questions in this section, we will use the Solow growth model, described by equations

Yt = AKt? L1??

?Kt+1 = sYt ? ?Kt

with a given initial value K0 .

Question 4.1 What are empirically plausible values for ?, s and ??

4 (3) Question 4.2 Sketch the Solow diagram (with the stock of capital on the horizontal axis),

including curves for output, depreciation and saving. Provide appropriate notation for

axes and the curves. Depict the steady state value for capital, K ? . Question 4.3 Into a separate graph, plot the net investment curve as a function of

the stock of capital. Clearly depict the steady state level of capital in this graph. 4.1 Experiment 1: Productivity increase Question 4.4 Use a new Solow diagram to capture the following experiment. The economy

starts in a steady state (with steady state level of capital K ? ). Then at time t0 the TFP

parameter A increases.

Plot the depreciation, investment, and output functions before and after the change

in A, clearly denoting each curve. Also clearly denote the old steady state K ? and the new

steady state K ?? . Question 4.5 In two separate graphs (underneath each other), plot the time trajectories

of the following quantities for the experiment described in the preceding question:

1. Capital;

2. Output;

In each of the graphs, clearly depict the time t0 , and the original and new steady states

of each of the quantities. Question 4.6 Set up the problem of a profit-maximizing firm. The firm uses production

function (3) to produce output, pays each worker a competitive wage wt and rents each

unit of capital at a competitive rental rate rt . Take the first-order condition with respect

to capital to derive the relationship between the marginal product of capital and the rental

rate. Question 4.7 What is the steady-state level of the rental rate r ? ? How does it depend on

the productivity parameter A?

Hint: The following expression for the steady-state level of capital will be useful:

? K =  sA

? 5  1

1?? L. All you have to do is to use this expression to substitute out capital in the first-order

condition for the capital choice. Question 4.8 Plot the time trajectory for the rental rate r. Clearly depict the time t0 ,

and the original and new steady state for the rental rate. Is the new steady-state rental

rate r ?? lower or higher than the original steady-state rental rate r ? ? Question 4.9 The expenditure approach to GDP calculates GDP by adding up components that are determined by how income is used (what is it spent on). On the other

hand, the income approach to GDP calculates GDP by adding up components that are

determined by how income is distributed (who earns it).

When we studied the model of production and the Solow growth model, we have seen

equations that capture each of these approaches. So for each of the two approaches, write

down the particular equation and write in one sentence what this equation represents.

Hint: One of these equations was related to the zero-profit result of a firm with Cobb?

Douglas technology, while the other was a resource constraint. 4.2 Experiment 2: Decline in saving rate Question 4.10 Assume that the economy starts in a steady state with capital K ? . Imagine

that at time t0 the saving rate in the economy declines to zero.

What will be the new steady state level of capital K ?? ? What will be the new steady

state level of wages w?? ? Provide an economic justification of your answers.

Hint: The wage is equal to the marginal product of labor, which you can compute from

the production function. 5 Reading: Nobel Prize in Economics Sciences 2013 This problem is based on the article ?Trendspotting in asset markets?.

Question 5.1 The efficient market hypothesis postulates that asset prices should reflect

all currently available information. As a result, it is near to impossible to predict in which

direction should the stock market move, say, in the next week. Why is that?

Hint: Imagine that we would know with a reasonable degree of certainty that the stock

market will crash next week. What should happen then? Question 5.2 Even though the stock market movements are near to impossible to predict

over short time intervals, there are empirical regularities in asset price data that allow us

to predict stock market movements over longer horizons, say over the business cycle. 6 What is the standard risk-based (rational) explanation for these movements? Provide a

verbal interpretation and illustrate this using the Gordon growth formula.

Hint: We argued that risk adjustments may change over the business cycle. Why? How

is this captured by the Gordon growth formula? What is the consequence of changes in

these risk adjustments for the behavior of the price-dividend ratio over the business cycle? Question 5.3 The behavioral finance literature would argue that while risk adjustments

may contribute somewhat to asset price fluctuations, there are other, more important factors

at play. Explain at least some of these factors that may generate movements in asset prices. References 7

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This question was answered on: Feb 21, 2020

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