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Edouard Schaal Spring 2016

Econ-UA 12: Intermediate Macroeconomics

Problem Set 5

This problem set is due Wednesday, April 13th in class at the end of the lecture. You are allowedto discuss the problems with your classmates. It is, however, in your best interest to try to solvethese problems by yourself first.

Part A. ABC Exercises

Chapter 4. Analytical problem 1.  Use the saving-investment diagram to analyze the eff ects of the following on national saving, investment, and the real interest rate. Explain your reasoning.

1. Consumers become more future-oriented and thus decide to save more.

2. The government announces a large, one-time bonus payment to veterans returning from a war.

The bonus will be financed by additional taxes levied on the general population over the nextfive years.

3. The government introduces an investment tax credit (off set by other types of taxes, so totaltax collections remain unchanged).

4. A large number of accessible oil deposits are discovered, which increases the expected futuremarginal product of oil rigs and pipelines. It also causes an increase in expected future income.

Chapter 7. Numerical problem 5.  Consider an economy with a constant nominal money supply,a constant level of real output   Y  = 100, and a constant real interest rate   r  = 0.10. Suppose thatthe income elasticity of money demand is 0.5 and the interest elasticity of money demand is -0.1.[Hint:  please read section 7.5, pp.268-269 in ABC].

1. By what percentage does the equilibrium price level diff 

er from its initial value if outputincreases to  Y  = 106 (and  r  remains at 0.10)?

2. By what percentage does the equilibrium price level diff er from its initial value if the realinterest rate increases to  r  = 0.11 (and  Y  remains at 100)?

3. Suppose that the real interest rate increases to  r  = 0.11. What would real output have to befor the equilibrium price level to remain at its initial value?

Part B. Corporate Taxation

We study the eff ect of corporate taxation on investment. Let us consider the basic model of invest-ment that we studied in class. Firms choose their investment and their future level of capital stockto maximize their profits. We now assume that a government imposes a tax on firms: they have topay a fraction  t  of their total output in taxes. Their future real profits can now be written as:

πf  = (1− t)Af 

 K f 

0.5 N f 

0.5− ucf .K f −wf .N f .

Assume that the price of capital is constant over time  pK  = pf K  = 1.  The interest rate,  r , equals 5%

and the depreciation rate,  d, equals 15%. In addition, assume that future total factor productivity,Af , equals 1 and the initial capital stock,  K , equals 10.

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1. Assume for now that future labor,   N f ,   is fixed, so that we can ignore the terms related tolabor. Discuss the eff ect of tax  t  on the desired future capital stock on a graph with  K f  on thex-axis. Explain the economic intuition behind this result. [Hint : The graph should be similarto what we studied in class: you will plot the net sales of the firm for two cases, one in whicht = 0 and one in which  t > 0, as well as the total cost of using capital  ucf .K f ]

2. We will now verify this result analytically. Derive the first-order condition that characterizesthe optimal choice of future capital  K f . [Do not solve yet ]

3. For   t   = 0 and   N f  = 2, compute numerically the optimal future capital stock and level of investment.

4. For  t = 10% and  N f  = 2, compute numerically the optimal future capital stock and level of investment.

5. Are your answers to questions (3) and (4) consistent with the result obtained in question (1)?

6. We have assumed so far that  N f  remains fixed. This will not be true in equilibrium. Usingwhat you have learned about labor markets, what would be the eff ect of the corporate tax onN f ? How would this aff ect your answer to question (4)? [No derivation necessary, just explain with words what would happen ]

Part C. Cagan’s Model of Seignorage

Consider an economy in which the money demand function is given by

M 

P   = L(Y, r + πe) =  Y e−a(r+π

e)

where   Y   is real income,   r   is the real interest rate,   πe is the expected rate of inflation and   a >  0a constant parameter that describes how quickly the money demand reacts to interest rates. Wefocus on the case where prices move much faster than   Y   and   r, so we will treat the latter two asconstant. Our objective in this exercise is to understand why some countries can experience high

rates of inflation and why governments choose policies that lead to such situations.There are diff erent reasons why inflation may sometimes be desirable from the point of view of 

a government. The explanation we consider here is that government use the “inflation tax” to raiserevenue. Printing money is indeed a rather quick and easy way for the government to raise revenuewithout increasing taxes. We call seignorage the revenue generated by money creation. We canwrite the real revenue from seignorage as

S  = ∆M 

P 

where  ∆M  is the increase in the money supply.

1. Assume the money supply increases at a constant rate   g   =   ∆M M 

  . For simplicity, let us alsoassume that inflation expectation are fixed at   πe = 0 for now. What would be the rate of 

inflation in this economy? How does it relate to the growth rate of money supply  g ?

Now, the assumption that expectations of future inflation are fixed may be fine in the short run, butcertainly not in the long run. At some point, people should come to realize that there is inflationand should adjust their expectations.

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2. Assume that people have adaptive expectations and that their expectations converge to theactual rate of inflation in the long run. Assume, in addition, that the actual long run rate of inflation   π  is constant:

πe−→ π.

What is the only long run rate of inflation consistent with these beliefs? How does it relate tothe growth rate in the money supply  g ?

3. How does your answer in (2) diff er from (1)? Do your answers seem consistent? What needsto adjust between question (1) and (2) to ensure that the equilibrium is reached in the moneymarket? [Hint : the inflation rate is only determined in the long run; in the short run, pricesare free to adjust in a specific way to be consistent with both questions.]

4. The government chooses the growth rate of the money supply to maximize seignorage revenue.Write the real seignorage revenue as

S  = ∆M 

P   =

 ∆M 

M  | {z } =g

×

M 

P   .

 |{z} =Y e−a(r+π)

Substitute in the above expression the value you found for the long run inflation rate asa function of   g. Derive the first order condition with respect to   g. What is the level of money creation that maximizes real seignorage revenue for the government? What is thecorresponding optimal inflation rate? Is it positive?

The temptation of printing money to generate seignorage revenue is great in countries with fiscalproblems. This explains why we often observe high rates of inflation in countries with large budgetdeficits. This, of course, can be quite dangerous, since governments do not necessary internalize thenegative impact that inflation could have on the economy – the most dramatic consequence beinga hyperinflation. To prevent such problems, most industrialized economies have made their centralbanks independent from the government.

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