Exercise problems for Advanced Macroeconomics, Schemes and Mind Maps of Advanced Macroeconomics

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Exercise problems for

Advanced Macroeconomics

Third edition

Christian Groth

September 8, 2016

Department of Economics University of Copenhagen

Contents

Preface iii

Remarks on notation iv

1 Refresher on technology and firms 1

2 Public debt and fiscal sustainability 9

3 More about budget deficits and public debt 13

4 Overlapping generations in discrete and continuous time 21

5 More applications of the OLG model. Long-run aspects of fiscal policy 29

6 The q-theory of investment 43

7 Uncertainty, expectations, and speculative bubbles 55

8 Money and prices 65

9 Short-run IS-LM dynamics in closed and open economies 69

10 Financial intermediation, business cycles 89

Appendix A. Solutions to linear differential equations 101

ii

Remarks on notation

For historical reasons, in some of the exercises the “level of technology” (assumed measurable along a single dimension) is denoted , in others . Whether we write ln  or log  the natural logarithm is understood. In discrete-time models the time argument of a variable,  appears always as a subscript, that is, as  In continuous-time models, the time argument of a variable may appear as a subscript rather than in the more common form () (this is to save notation).

iv

Chapter 1

Refresher on technology and

firms

I.1 Short questions (answering requires only a few well chosen sentences and possibly a simple illustration)

a) Consider an economy where all firms’ technology is described by the same neoclassical production function,  =  ( )  = 1 2    with decreasing returns to scale everywhere (standard notation). Sup- pose there is “free entry and exit” and perfect competition in all mar- kets. Then a paradoxical situation arises in that no equilibrium with a finite number of firms (plants) would exist. Explain.

b) As an alternative to decreasing returns to scale at all output levels, introductory economics textbooks typically assume that the long-run average cost curve of the firm is decreasing at small levels of production and constant or increasing at larger levels of production. Express what this assumption means in terms of local “returns to scale”.

c) Give some arguments for the presumption that the average cost curve is downward-sloping at small output levels.

d) In many macro models the technology is assumed to have constant returns to scale (CRS) with respect to capital and labor taken together. What does this mean in formal terms?

e) Often the replication argument is put forward as a reason to expect that CRS should hold in the real world. What is the replication argument? Do you find the replication argument to be a convincing argument for the assumption of CRS with respect to capital and labor? Why or why not?

d) If we want to extend the domain of definition of the production function to include ( ) = (0 0) how can this be done while maintaining continuity of the function?

I.4 Write down a CRS two-factor production function with Harrod- neutral technological progress look. Why is the assumption of Harrod- neutrality so popular in macroeconomics?

I.5 Refresher on stocks versus flows. Two basic elements in long-run models are often presented in the following way. The aggregate production function is described by  =  (  ) (*)

where  is output (aggregate value added),  capital input,  labor input, and  the “level of technology”. The time index  may refer to period , that is, the time interval [  + 1)  or to a point in time (the beginning of period ), depending on the context. And accumulation of the stock of capital in the economy is described by

+1 −  =  −  (**)

where  is an (exogenous and constant) rate of (physical) depreciation of capital, 0 ≤  ≤ 1. Evolution in employment (assuming full employment) is described by +1 −  =    − 1  (***)

In continuous time models the corresponding equations are: (*) combined with

^ ˙() ≡ ()

^ ˙() ≡ ()

= ()  “free”.

a) At the theoretical level, what denominations (dimensions) should be attached to output, capital input, and labor input in a production function?

b) What is the denomination (dimension) attached to  in the accumu- lation equation (**)?

c) Might there be a consistency problem in the notation used in () vis- à-vis () and in () vis-à-vis (***)? Explain.

CHAPTER 1. REFRESHER ON TECHNOLOGY

AND FIRMS

d) Suggest an interpretation that ensures that there is no consistency problem.

e) Suppose there are two countries. They have the same technology, the same capital stock, the same number of employed workers, and the same number of man-hours per worker per year. Country  does not use shift work, but country  uses shift work, that is, two work teams of the same size and the same number of hours per day. Elaborate the formula (*) so that it can be applied to both countries.

f) Suppose  is a neoclassical production function with CRS w.r.t.  and . Compare the output levels in the two countries. Comment.

g) In continuous time we write aggregate (real) gross saving as () ≡  () − () What is the denomination of ()?

h) In continuous time, does the expression () + () make sense? Why or why not?

i) In discrete time, how can the expression  +  be meaningfully inter- preted?

I.6 The Solow growth model can be set up in the following way (dis- crete time version). A closed economy is considered. There is an aggregate production function,  =  ( ) (1)

where  is a neoclassical production function with CRS,  is output,  is capital input,  is the technology level, and  is the labor input. So   is effective labor input. It is assumed that

 =  0 (1 + ) where  ≥ 0 , (2)  =  0 (1 + ) where  ≥ 0. (3)

Aggregate gross saving is assumed proportional to gross aggregate income which, in a closed economy, equals real GDP,  :

 =  0    1  (4)

Capital accumulation is described by

+1 =  +  −  where 0   ≤ 1  (5)

The symbols   and  represent parameters and the initial values  0   0  and  0  are given (exogenous) positive numbers.

CHAPTER 1. REFRESHER ON TECHNOLOGY

AND FIRMS

g) As Solow once said (in a private correspondence with Amartya Sen^1 ): “The idea [of the model] is to trace full employment paths, no more.” What market form is theoretically capable of generating permanent full employment?

h) Even if we recognize that the Solow model only attempts to trace hypo- thetical time paths with full employment (or rather employment corre- sponding to the “natural” or “structural” rate of unemployment), the model has at least one important limitation. What is in your opinion that limitation?

1. We consider the same economy as that described by (1) - (5) in Problem I.6.

a) Find the long-run growth rate of output per unit of labor,  ≡  .

b) Suppose the economy is in steady state up to and including period  − 1  Then, at time  (the beginning of period ) an upward shift in the saving rate occurs. Illustrate by a transition diagram the evolution of the economy from period  onward

c) Draw the time profile of ln  in the ( ln ) plane.

d) How, if at all, is the level of  affected by the shift in ?

e) How, if at all, is the growth rate of  affected by the shift in ? Here you may have to distinguish between temporary and permanent effects.

f) Explain by words the economic mechanisms behind your results in d) and e).

g) As Solow once said (in a private correspondence with Amartya Sen^2 ): “The idea [of the model] is to trace full employment paths, no more.” What market form is theoretically capable of generating permanent full employment?

h) Even if we recognize that the Solow model only attempts to trace hypo- thetical time paths with full employment (or rather employment corre- sponding to the “natural” or “structural” rate of unemployment), the (^1) Growth Economics. Selected Readings, edited by Amartya Sen, Penguin Books, Mid-

dlesex, 1970, p. 24. (^2) Growth Economics. Selected Readings, edited by Amartya Sen, Penguin Books, Mid-

dlesex, 1970, p. 24.

model has at least one important limitation. What is in your opinion that limitation?

I.8 A more flexible specification of the technology than the Cobb-Douglas function. Consider the CES production function^3

 = 

^ + (1 − )^

where   and  are parameters satisfying   0 , 0    1  and   1   6 = 0

a) Does the production function imply CRS? Why or why not?

b) Show that (*) implies

 

μ  

and

μ  

c) Express the marginal rate of substitution of capital for labor in terms of  ≡ 

d) In case of an affirmative answer to a), derive the intensive form of the production function.

e) Is the production function neoclassical? Hint: a convenient approach is to focus on   expressed in terms of  and consider the cases   0 and 0    1  separately; next use a certain symmetry visible in (*); finally use your answer to a).

f) Draw a graph of  ≡  as a function of  for the cases   0 and 0    1  respectively. Comment and compare with a Cobb-Douglas function on intensive form,  = .^4

g) Write down a CES production function with Harrod-neutral technical progress.

I.9 A potential source of permanent productivity growth (this exercise presupposes that f) of Problem I.8 has been solved). Consider a Solow-type growth model, cf. Problem I.6. Suppose the production function is a CES function as in (*) of Problem I.8. Let  ∈ (0 1) ^1 ^   ( + ) and ignore technical progress.

(^3) CES stands for Constant Elasticity of Substitution. (^4) This function can in fact be shown to be the limiting case of the CES function (in

intensive form) for  → 0 

Chapter 2

Public debt and fiscal

sustainability

B o rrow ed fro m J e p p e D ru ed a h l:

Figure 2.1: Some background material for this section of the exercises.

II.1 Consider the government budget in a small open economy (SOE) fully integrated in the world market for goods and financial capital. Time

CHAPTER 2. PUBLIC DEBT AND FISCAL

SUSTAINABILITY

is discrete, the period length is one year, and there is no uncertainty. Let  and  be non-negative constants and let

 =  0 (1 + )(1 + )^ = real GDP,  = real government spending on goods and services,  = real net tax revenue ( = gross tax revenue − transfer payments),  = real public debt at the start of period   = real interest rate in the SOE = world market real interest rate.

We assume that any government budget deficit is exclusively financed by issuing debt (and any budget surplus by redeeming debt).

a) Interpret  and 

b) Suppose the current inflation rate in the SOE equals  Given this inflation rate and given , what is the level of the nominal interest rate, ? You should provide the exact formula, not an approximation. Let  = 0 03 per year and  = 0 02  What is  exactly? Instead, let  = 0 04 per year and  = 0 15 (as in many countries in the aftermath of the second oil crisis 1979-80). What is  exactly? Compare with the result you get from the standard approximative formula.

c) Returning to variables in real terms, write down the real budget deficit and an equation showing how +1 is determined

From now on assume that  =  a constant. Consider a scenario with  0  0  1 +   (1 + )(1 + ) and  =   a positive constant less than one.

d) What does government solvency mean and what does fiscal sustainabil- ity mean?

e) Find the maximum constant  which is consistent with fiscal sus- tainability (as evaluated on the basis of the expected evolution of the debt-GDP ratio). Hint: the difference equation +1 =  +  where  and  are constants,  6 = 1 has the solution  = ( 0 − ∗)^ + ∗ where ∗^ = (1 − )

II.2 Consider a small open economy (SOE) facing a constant real interest rate   0  given from the world market for financial capital. We ignore business cycle fluctuations and assume that real GDP,  grows at a constant exogenous rate   0. We assume   

CHAPTER 2. PUBLIC DEBT AND FISCAL

SUSTAINABILITY

b) Let  ≡ (− 1 ) Derive the law of motion (difference equation) for  assuming the deficit ceiling is always binding. Hint: GBD =  + ( − )

Suppose  is such that 0  1 + (1 − )  (1 + )(1 +  )

c) For an arbitrary  0  0  find the time path of  Briefly comment. Hint: the difference equation +1 =  +  where  and  are constants,  6 = 1 has the solution  = ( 0 − ∗)^ + ∗ where ∗^ = (1 − )

d) How does a rise in  affect the long-run debt-income ratio? Comment.

e) Let the steady-state value of  be denoted ∗^ and assume  0  ∗ Illustrate the time path of  in the ( ) plane. Comment.

f) How does ∗^ depend on ? Comment.

g) How does ∗^ depend on ? Comment.

h) What could the motivation for having   1 be? Comment.

II.4 Short questions

a) What is meant by the No-Ponzi-Game condition of the government?

b) The No-Ponzi-Game condition of the government and the intertemporal budget constraint of the government are closely related. In what sense?

c) “A given fiscal policy is sustainable if and only if it maintains com- pliance with the intertemporal budget constraint of the government.” True or false? Briefly discuss.

d) In the absence of uncertainty and credit frictions, if     a govern- ment can run a permanent debt rollover without experiencing solvency difficulties (standard notation). Briefly explain.

e) How is the inequality in d) modified in the presence of uncertainty and credit frictions?

II.5 The Ricardian equivalence issue. What is meant by Ricardian equivalence? Under the assumption of rational expectations and at most a “weak” bequest motive, overlapping generations models refute Ricardian equivalence. How?

Chapter 3

More about budget deficits and

public debt

III.1 Consider a small open economy facing an exogenous constant real interest rate  Suppose that at time 0 government debt is  0  0  GDP is denoted  and grows at the constant rate    Assume government spending,  satisfies  =  and that net tax revenue,  satisfies  =   where  and  are positive constants and  = 0 1  2 ....

a) What is the minimum size of the primary budget surplus as a share of GDP required for satisfying the government’s intertemporal budget constraint as seen from time 0 (the beginning of period 0)? Derive your result by two different methods, that is, by using first the debt arithmetic method focusing on the dynamics of the debt-income ratio and next the method based on the intertemporal government budget constraint.

b) What key condition in the setup is it that ensures that both methods are appropriate and give the same result?

III.2 A budget deficit rule. Let time be continuous and suppose that money financing of budget deficits never occurs. Consider a budget deficit rule saying that the nominal budget deficit must never be above  · 100 per cent of nominal GDP,     0  that is, the requirement is

^ ˙ ≤   (*)

where ˙ ≡  (given  = () is nominal government debt) and  =  () is a price index, whereas  =  () is real GDP.

where  0 ,  0 , and  are given (until further notice,  is constant). Thus, the pension payments are, along with interest payments on government debt, the only government expenses. The government always preserves solvency in the sense that sooner or later tax revenue is adjusted to satisfy the intertemporal government budget constraint (more about this below).

The representative young individual: An individual belonging to generation  chooses saving,  and bequest, +1, to each of the descendants so as to maximize

X^ ∞

=

s.t.

 1  +  =  −  +   2 +1 + (1 + )+1 = (1 + ) +  +1 ≥ 0 

and taking into account the optimal responses of the descendants. Here 1 + ¯ ≡ (1 + )(1 + ) where    ≥ 0 (both  and  constant). Also   − 1 is constant. The period utility function  satisfies the No Fast assumption and ^0  0  ^00  0. Negative bequests are forbidden by law.

a) How comes that the preferences of the single parent can be expressed as in (*)?

b) Derive the first-order conditions for the decision problem, taking into account that two cases are possible, namely that the constraint +1 ≥ 0 is binding and that it is not binding. Interpret the first-order condi- tions.

Suppose it so happens that  =  and that, at least for a while, circum- stances are such that the agents are at an interior solution (i.e., +1  0) We define a steady state of this economy as a path along which  1  and  2  do not change over time.

c) Is the economy in a steady state? Why or why not? Hint: combine the first-order conditions and use that  = 

The link between the intertemporal budget constraint of the government and that of the dynasty:

CHAPTER 3. MORE ABOUT BUDGET DEFICITS

AND PUBLIC DEBT)

As seen from the beginning of period  the intertemporal government budget constraint is:

X^ ∞

=

+− 1 (1 + )−−^1 =

X^ ∞

=

^ ˜+(1 + )−−^1 −  ⇒ (i)

X^ ∞

=

(1 + )+^

X^ ∞

=

(1 + )+^

•  ⇒ (ii)

X^ ∞

=

=  (iii)

d) Briefly explain in economic terms what each row here expresses.

e) The intertemporal budget constraint of the representative dynasty is

X^ ∞

=

(1 + )+^

[ 2 + + (1 + ) 1 +] =  + 

where  is aggregate financial wealth in the economy and  is aggre- gate human wealth (after taxes):

X^ ∞

=

(1 + )+^

Briefly explain in economic terms these two equations.

f) Suppose that in period  + 1  is increased (a little) to a higher con- stant level, before the bequest +1 is decided. Is the consumption path ( 2 +  1 +)∞ =1 affected? Why or why not?

g) Given  suppose that for some periods there is a (small) tax cut so that ˜+  +− 1 + +, that is, a budget deficit is run. Is the consumption path ( 2 +  1 +)∞ =0 affected? Why or why not?

Implications of    : Now suppose instead that    (but still   ) and that the economy is, at least initially, in steady state.

h) Will the bequest motive be operative? Why or why not?

i) Suppose  is increased (a little) to a higher level without  being immediately adjusted correspondingly. Is resource allocation affected? Why or why not?