Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

Epilogue

er the past twenty five s in the mid-1970s, the level and Turnovsky books contain extensive \ tensions. In contrast, Al model is still treated •ely from the advanced

eals that the once domi-

'he economy, has fallen )nal expectations revo-

"- e IS-LM model. In part levelop dynamic models Ind give more attention books therefore contain !five agents (households,

-ational expectations).2

:in disarray, with pro- +_ieve the position once

..:roeconomics is some-

g ring, we find the "new ze market clearing. In with models incorporious degrees of market

,v (2000a) and Blanchard

: (2001), Turnovsky (1997,

rro (1997) and Auerbach and

Epilogue

Even though this picture of macroeconomics as a boxing match may bear some similarity with the reality of the late 1970s through to the early 1990s, we agree with Goodfriend and King (1997) who argue that the subject has been moving towards a new synthesis over the last decade or so. Goodfriend and King (1997, p. 255) identify the key aspects of, what they call, the New Neoclassical Synthesis (NNS):

•The NNS takes from the new classicals the notion that macroeconomic models should be explicitly dynamic, and should incorporate rational expectations and intertemporal optimization.

•The NNS takes from the new Keynesians the assumptions of imperfect competition and costly price adjustment.

•The NNS takes from the RBC approach the insistence on quantitative models of economic fluctuations.

Just like the old neoclassical synthesis (see Chapter 1), the NNS thus contains elements from new classicals and new Keynesians alike. Instead of as a boxing match, it appears as if the current intellectual debate in macroeconomics is more aptly characterized as a tango. In one sense this is true. For example, there is not much debate about the usefulness of the rational expectations hypothesis any more. Similarly, economists of all signatures routinely construct models based on dynamically optimizing behaviour of the economic agents. In another sense the metaphor of macroeconomics as a tango is somewhat flawed.

In his comment on Goodfriend and King (1997), Blanchard (1997) agrees that macroeconomists need to use three ingredients to study economic fluctuations, namely intertemporal optimization, nominal rigidities, and imperfect competition. He argues that putting together these ingredients and understanding their interaction constitutes the core business of all macroeconomists. 3 He goes on to argue that what distinguishes us macroeconomists is the relative weight one places on the different ingredients and the short-cuts one is willing to make.

Blanchard suggests that a useful way to think about these ingredients (and their interaction) is by means of a triangle, like in Figure E.1. In the top corner one could place the approach of Prescott (1986) with its emphasis on dynamically optimizing behaviour. In the bottom left corner one could place the approaches of Fischer (1977) and Taylor (1980) which emphasize nominal wage or price stickiness. Finally, in the bottom right corner one could think of Blanchard and Kiyotaki (1987) and Akerlof and Yellen (1985a) who focus on imperfections in the goods and labour markets.

Blanchard argues that most economists are located somewhere in the triangle, but he finds it an overstatement to infer from this phenomenon that a synthesis has

3 Blanchard (1997, p. 290) argues that this is exactly what the "old" neoclassical synthesis was all about: "In rather schizophrenic fashion, intertemporal optimization was at the core of the formalization of consumption and investment, imperfect competition the underlying rationale for markup pricing, and nominal rigidities used as a general justification for the Phillips curve. Since then, we have tried to improve on the shortcomings. But the goal is the same".

653

The Foundation of Modern Macroeconomics

Intertemporal

optimization

Figure E.1. Aspects of macro models

been achieved. The triangle is not a small one and diverse views can be accommodated within its boundaries. In particular, economists with classical leanings would probably locate themselves in the top half of the triangle whereas their Keynesinspired colleagues would be more comfortably located in the bottom half of the triangle. To the extent that there is a synthesis, it mainly refers to methods and not to particular applications of these methods.

Threads

Blanchard's triangle is quite useful for showing how the topics in this book fit together. To prepare students for the explicitly dynamic framework adopted in the later chapters of the book we start in Chapter 2 by studying the intrinsic dynamics that exist in IS-LM type models. The choice of the IS-LM model as a vehicle of exposition is a natural one in view of the fact that this model is still used widely in the intermediate textbooks (see above). Chapter 2 also presents a first view of the dynamic optimization approach in the form of Tobin's q theory of investment which is based on the notion of adjustment costs of investment. This investment theory is extended in Chapter 4 and applied in Chapters 14, 16, and 17.

Chapter 3 follows logically from Chapter 2 and shows some of the key implications of the rational expectations hypothesis (REH) within the context of (loglinear) IS-LM models. This chapter and the next also alert students to the crucial difference that exists between backward-looking and forward-looking stability. Both types of stability are encountered in the later chapters of the book within the context of dynamically optimizing models.

In Chapter 5 the student has a first encounter with nominal rigidities and with micro-based macroeconomics. The IS-LM model is no longer used and the

I

behavioural equatio - Some of the insights Chapter 13, where w( goods market can, in 1 for price stickiness.

In Chapter 6 we p sumption theory. Th. notions such as the 1 of tax smoothing I\ .. a first encounter with The intertemporal ha book.

In Chapters 7-9 we market. We first sty .. features of the labour] of trade unions) and I impinge on this mare

Chapter 10 intro applies to economic f yet another implicati4 ible to the econom not arise in the pre-R form expectations nomic environment. mechanism can potel

In Chapter 11 we -

we extend the anal opened up for trade i

of monetary and ti), of labour market rigic main implications ing rational expect model.

In Chapter 12 we d intertemporal optimi the student to a sim we study the optima rule") within the c( , chapter forms the s found in Chapter 14In Chapter 13 we nomics. As was point bottom half of the I

k views can be accommoclassical leanings would gle whereas their Keynesin the bottom half of the Ts to methods and not

he topics in this book fit

"mework adopted in the ng the intrinsic dynam-

-LM model as a vehicle of ()del is still used widely so presents a first view of 's q theory of investment , tment. This investment

14, 16, and 17.

some of the key implicathe context of (loglinear)

•s to the crucial difference stability. Both types of

Kok within the context of

h nominal rigidities and s no longer used and the

Epilogue

behavioural equations of households and firms are based on maximizing behaviour. Some of the insights from the quantity rationing literature are further developed in Chapter 13, where we show how the assumption of imperfect competition in the goods market can, in combination with price adjustment costs, provide a rationale for price stickiness.

In Chapter 6 we present the student with a first view of intertemporal consumption theory. This chapter allows us to familiarize the student with important notions such as the Ricardian Equivalence Theorem and the neoclassical theory of tax smoothing within a relatively simple model. In addition, the student has a first encounter with the notion of intertemporal substitution of consumption. The intertemporal household model is further extended in the later chapters of the book.

In Chapters 7-9 we present the various theories concerning the aggregate labour market. We first show how the competitive model can be used to interpret various features of the labour market. Next we show how imperfect competition (in the form of trade unions) and other kinds of imperfections (such as informational frictions) impinge on this market.

Chapter 10 introduces the notion of dynamic inconsistency, especially as it applies to economic policy. This chapter is a logical sequel to Chapter 3, and shows yet another implication of the REH, namely that economic policy may not be credible to the economic agents affected by it. Note that the credibility problem did not arise in the pre-REH literature because there agents were typically assumed to form expectations adaptively, i.e. not taking into account the structure of the economic environment. In this chapter we also show how an intertemporal reputation mechanism can potentially resolve the problem of dynamic inconsistency.

In Chapter 11 we tie up some loose ends held over from the first ten chapters. First we extend the analysis from Chapter 1 by showing how the standard IS-LM can be opened up for trade in goods and financial assets. Next we study the transmission of monetary and fiscal policy in a two-country world experiencing various kinds of labour market rigidities (i.e. real or nominal wage rigidity). Finally, we show the main implications for prices, output, and exchange rate fluctuations of introducing rational expectations (or, rather, perfect foresight) in the open economy IS-LM model.

In Chapter 12 we discuss a number of micro-founded models of money based on intertemporal optimization by households. This chapter also allows us to introduce the student to a simple model of money demand based on uncertainty. Finally, we study the optimal quantity of money (and Friedman's famous "full liquidity rule") within the context of a simple dynamic model. Together with Chapter 6, this chapter forms the stepping stone for the multi-period dynamic analyses which are found in Chapter 14-17.

In Chapter 13 we discuss some of the recent literature on new Keynesian economics. As was pointed out above, new Keynesians can be placed somewhere in the bottom half of the Blanchard triangle so it is not surprising that the chapter features

655

The Foundation of Modern Macroeconomics

an extensive discussion of imperfect competition and price stickiness. We show how the assumption of monopolistic competition in the goods market can provide the micro-foundations for the multiplier. We also demonstrate with the aid of an explicitly forward-looking theory of firm behaviour how price adjustment costs can provide the microeconomic foundations behind short-term price stickiness.

In Chapter 14 we present a brief overview of the main theories of economic growth that have been developed over the last forty-five years. We start with the classic analysis of Solow and Swan which is based on a Keynesian savings function. Next we replace the ad hoc savings function by explicitly modelling the consumptionsavings decisions of an infinitely lived representative household. (This chapter thus completes our discussion of the forward-looking theory of consumption commenced in Chapter 6.) The growth properties of the model are not much affected by this switch to a micro-based savings theory. Once the theory with forward-looking consumers has been developed, it is relatively straightforward to show how endogenous human capital accumulation or the accumulation of patents (and new product varieties) can give rise to so-called endogenous growth.

In Chapter 15 we give a brief overview of the recent RBC literature. Especially the early proponents to this approach are firmly located in the top half of the Blanchard triangle. More recently RBC practitioners have started to explore the interior parts of the triangle. We start this chapter by extending the dynamic consumption theory (of the previous chapter) to include a joint decision regarding the consumption of goods and leisure. In the extended model the household substitutes leisure across time, i.e. there is an intertemporal substitution effect in the supply of labour. We demonstrate the macroeconomic implications of this intertemporal labour supply effect both for deterministic shocks in government consumption (fiscal policy) and for stochastic technology shocks.

In Chapters 16-17 we study the macroeconomic and welfare-theoretic implications of abandoning the representative-agent framework on the household side. In both chapters we assume that individual households have finite lives and that they are not linked with each other via operative bequests. The resulting overlapping-generations structure implies another kind of imperfection, namely the incompleteness of markets. At any point in time, only the currently living generations are active in the market place but their economic behaviour affects the conditions facing future generations. This property of overlapping-generations models implies that virtually all policy measures affect both efficiency and the intergenerational distribution of resources.

Views

To the astute observer it is clear from our choice of topics where we ourselves are located within the Blanchard triangle—somewhere in the middle. We conclude this

book with some final of and what we expect z We find it a bad idea unavailable or incom p aesthetically pleasing. of the world we ham' There is, for example. are rigid (see e.g. Blinc determinant of con money illusion (Shafir the ad hoc IS-LM-AS IT seems to us that a goo( Another reason for 1 sible eventually to co models. In this conte: from its ashes) of the suspect that large au the microeconometric in this direction by I). in computing hardwa cal content of comr heterogeneity, mark,

In our view, good m following some simpl advice. Macroeconom merely on one of sess, namely a firm g are needed are a goo( tions, and of history. mark in applied macr in this context are J( Fischer, Rudiger Don good macroeconorn

The Economist, the B feature prominently macroeconomic art.,

ice stickiness. We show o,', market can provide :ate with the aid of an 4,-e adjustment costs can n price stickiness.

wies of economic growth

'start with the classic

isavings function. Next

!i ng the consumptionxi • hold. (This chapter ry of consumption com- - - e not much affected by iv with forward-looking - to show how endoge- •tents (and new product

'erature. Especially the op half of the Blanchard ', lore the interior parts nic consumption theory in g the consumption of

, stitutes leisure across le supply of labour. We -mporal labour supply ion (fiscal policy) and

- e-theoretic implicaDn the household side. have finite lives and requests. The resulting f imperfection, namely the currently living omic behaviour affects - apping-generations h efficiency and the

re we ourselves are fiddle. We conclude this

Epilogue

book with some final observations on what in our view constitutes macroeconomics and what we expect to be major themes in the years to come.

We find it a bad idea to scrap models merely because their micro-foundations are unavailable or incomplete. Although we acknowledge that a micro-based model is aesthetically pleasing, a weakly founded model may better capture pertinent aspects of the world we happen to live in than a model with the wrong micro-foundations. There is, for example, ample evidence suggesting that nominal prices and/or wages are rigid (see e.g. Blinder, 1994 and Bewley, 1999), that current income is a strong determinant of consumption (Mankiw, 2000b), and that agents do "suffer" from money illusion (Shafir et al., 1997). Furthermore, as is demonstrated by Gall (1992), the ad hoc IS-LM-AS model does quite a decent job at matching postwar US data. It seems to us that a good macro model should pay attention to this kind of evidence.

Another reason for tolerating weakly founded models is that it may well be possible eventually to come up with credible microeconomic foundations for such models. In this context one can think of the remarkable comeback (like Phoenix from its ashes) of the IS-LM model initiated by McCallum and Nelson (1999). We suspect that large advances will be made in the coming years on the link between the microeconometric evidence and their macroeconomic implications (cf. the plea in this direction by Browning et al., 1999). With the ongoing technological advance in computing hardware and software, it will be feasible to strengthen the empirical content of computable equilibrium models by paying more attention to agent heterogeneity, market imperfections, and nominal rigidities.

In our view, good macroeconomics is not designed from the armchair and blindly following some simple methodological prescriptions will not lead to good policy advice. Macroeconomics is not only a science but also an art form. This book focuses merely on one of the necessary skills that any good macroeconomist should possess, namely a firm grasp of the technical tools of the trade. The other skills that are needed are a good understanding of the empirical evidence, of actual institutions, and of history. Some of the best macro theoreticians have also made their mark in applied macroeconomics and in policy circles. Names that spring to mind in this context are John Maynard Keynes (him again), Robert Mundell, Stanley Fischer, Rudiger Dornbusch, Larry Summers, Michael Bruno, and Joe Stiglitz. A good macroeconomist should not be blind to the real world and periodicals such as

The Economist, the Brookings Papers on Economic Activity, and Economic Policy should feature prominently in the macro curriculum. By reading the works of the great macroeconomic artists, the tools discussed in this book should come to life.

657

Mathematical Appendix

A.1 Introduction

In this mathematical appendix we give a brief overview of the main techniques that are used in this book. In order to preserve space, for most cases we simply state the results and refer the interested reader to various sources—of differing levels of sophistication—where the mathematical background for these results is explained in more detail. The transform methods used in sections A.6.1 and A. 7.2 are explained in more detail because they are somewhat unfamiliar to most economists. Klein (1998) and Pemberton and Rau (2001) are both good single sources for the mathematical techniques employed in this book, both in terms of coverage and the level of sophistication. SydsEeter et al. (2000) is a very convenient reference book describing most of the tricks used by economists.

A.2 Matrix Algebra

A.2.1 General

A matrix is a rectangular array of numbers aij where i = 1, 2, . . . , m is the row index

and j = 1, 2, , n is the column index. A matrix of dimension m by n thus has m rows and n columns:

If m = n = 1 then A is a scalar, if m = 1 and n > 1 it is row vector, and if n = 1 and m > 1 it is a column vector. If m = n then the matrix A is square and we call the diagonal containing the elements all, a22, • • • , a nn the principal diagonal. There are a number of special matrices. The zero matrix contains only elements equal to zero

= 0 for i = 1, 2, .... I n matrix with ones on t

A.2.2 Addition, subt

Two matrices A and B c A has elements [ad and

by adding correspond

I

B = C, withc

for i = 1, 2, . , m and j

A – B = D, with d

for i = 1, 2, , m and j Matrices can be mu: • by that scalar, i.e. B k rules and properties foil

kA = Ak k(A + B)=kA (k + 1)A = kA -

(k1)A =

( – 1)A = –A A+(– 1)B=A-1

Two matrices can be matrix product AB is d (matrix A) is the same matrix). If A is m by r a

for i = 1, 2, , m and Even if BA is defined i (yielding AB) does not operation yielding BA (A, B, and C are confoi,

A(B+C)=AB+.: (A + B)C = AC 4- .2 A(BC) = (AMC

k(AB)=A(kB)

A 0=0A =(

A / = / A = -

al Appendix

w of the main techniques ost cases we simply state es—of differing levels of r these results is explained 1 and A. 7.2 are explained o most economists. Klein foe sources for the mathe- ,:overage and the level of reference book describing

2, .. . , m is the row index nsion m by n thus has m

(A.1)

row vector, and if n = 1 k square and we call the

, .11 diagonal. There are a elements equal to zero

Mathematical Appendix

(aii = 0 for i = 1, 2, . . . , m and j = 1, 2, . . . , n). The identity matrix, I, is a square n by n matrix with ones on the principal diagonal and zeros elsewhere.

A.2.2 Addition, subtraction, multiplication

Two matrices A and B can be added if and only if they have the same dimension. If A has elements [aii] and B has elements [bii] then the matrix A + B is obtained by adding corresponding elements:

for i = 1, 2, . . . , m and j = 1, 2, . , n. Subtracting matrices works the same way:

for i = 1, 2, . . . , m and j = 1, 2, . . . , n.

Matrices can be multiplied by a scalar, k, by multiplying all elements of the matrix by that scalar, i.e. B kA then bij kau for i = 1, 2, . , m and j = 1, 2, . . . , n. Some rules and properties follow immediately (k and 1 are both scalars):

kA=Ak

k(A + B)=kA + kB

(k + 1)A = kA + lA (A.4) (kl)A = k(1A)

( – 1)A = –A A + (-1)B=A –B.

Two matrices can be multiplied if they are conformable for that operation. The matrix product AB is defined if the column dimension of the matrix on the left (matrix A) is the same as the row dimension of the matrix on the right (the B matrix). If A is m by r and B is r by n then by this rule AB is defined as follows:

for i = 1, 2, . , m and j = 1, 2, . , n. Unless m = n the product BA is not defined. Even if BA is defined it is not equal to AB in general. So premultiplying B by A (yielding AB) does not give the same matrix in general as premultiplying A by B (an operation yielding BA). Some properties of matrix multiplication are the following (A, B, and C are conformable matrices, 0 is the zero matrix, and k is a scalar):

A(B + C) = AB + AC

(A + B)C =AC + AB

A(BC) = (AB)C (A.6) k(AB) =A(kB)

A0=0A = 0

AI=IA=A

659

Mathematical Appendix

A.2.3 Transposition

The transpose of matrix A is denoted by AT (or sometimes by A'). It is obtained by interchanging the rows and columns of matrix A. Hence, if A is m by n and B AT then B is n by m and bij aii. Some properties of transposes are:

A.2.4 Square matrices

In this subsection we gather the key results pertaining to square matrices (for which the row and column dimensions are the same). The trace of the n by n matrix A, denoted by tr(A), is the sum of the elements on its principal diagonal:

tr(kA) = ktr(A) tr (AB) = tr (BA)

The determinant of a square matrix A, denoted by IAI (sometimes by det(A)) is a unique scalar associated with that matrix. For a two-by-two matrix the determinant is:

For a three-by-three matrix the determinant can be computed as follows:

=an [a22a33 a23a32J a12 [a2la33 — a23a31] + a13 [a21a32 — a22a31]

=alla22a33 — ana23a32 — ai2a2033 + ai2a23a31 + anan a32 a13a22a31.

(A.11)

We have computed IAI by going along the first row and seeking two-by-two determinants associated with each element on that first row. For element an we find the

associated two-by-two d is located. The resultir., In a similar fashion, the umn 2 from the original

row 1 and column 3 1— ment aij by IM111 we can

A cofactor is a minor v lows: if the sum of the positive and the cofa then the cofactor is rh. that the determinant o IAI ICiil + a12 ICI_ puted Al1by going alor- by going along any of for j = 1, 2, 3). It is not di same value for 1A1.

The procedure we The Laplace expansion c

I

fl

The determinant has a r

III = 1

101=0

IAI =IAT I

IAI = ( — 1)"

IABI = IBA)

I

•If any row (colum (columns) of A then

•If B results from A b

•If B results from A 131

•The addition (subti, IAI unchanged.

•The addition (sub, column leaves Al1 u

A'). It is obtained by : A is m by n and B AT 'c are:

(A.7)

-e matrices (for which of the n by n matrix A,

al di -tonal:

(A.8)

Mathematical Appendix

associated two-by-two determinant by deleting the row and column in which an is located. The resulting two-by-two determinant is called the minor of element an . In a similar fashion, the minor of element a12 is found by deleting row 1 and column 2 from the original determinant, and the minor of a13 is obtained by deleting

row 1 and column 3 from the original determinant. Denoting the minor of element ao by IMI1 we can define the cofactor of that element by 1Ciil = ( — 1) i±i IM111.

A cofactor is a minor with a sign in front of it. The sign is determined as follows: if the sum of the row and column indices (i + j) is even, then the sign is positive and the cofactor is equal to the minor. Conversely, if i + j is uneven, then the cofactor is minus the minor. Using these definitions we can now see that the determinant of the three-by-three matrix in (A.11) can be written as:

by going along any of the columns of the original determinant (IAI =

for j = 1, 2, 3). It is not difficult to verify that in each case we would have found the same value for 1A1.

The procedure we have just followed to compute IAI is called a The Laplace expansion of an n by n matrix is given by:

IAI

(A.9)

(cnrnetimes by det(A)) o-by-two matrix the

A.10)

as follows:

a a21 a22

13 a31 a32

332 — a22a31]

a32 — a13a22a31.

(A.11)

king two-by-two deter- ' ment al l we find the

The determinant has a number of useful properties (k is scalar):

111=1

10 1= 0 1A1= IAT I

1A1 =(— 1 )n = -n IkA

IABI = IBAI

•If any row (column) is a non-trivial linear combination of all the other rows (columns) of A then IAI = 0.

•If B results from A by interchanging two rows (or columns) then 1B1 = — 1A1.

• If B results from A by multiplying one row (or one column) by k then IBI = k lAl.

•The addition (subtraction) of a multiple of any row to (from) another row leaves 1Alunchanged.

•The addition (subtraction) of a multiple of any column to (from) another column leaves 1A1 unchanged.

661

Mathematical Appendix

The adjoint matrix of matrix A is denoted by adj A. It is defined as the transposed matrix of cofactors:

If IAI 0 0 then the matrix A is non-singular and possesses a unique inverse, denoted by A-1 :

If the matrix A has an inverse it follows that A -1A = AA -1 = I.

Intermezzo

Matrix inversion. For example, let A be:

A1 2

3 4

then we find by applying the rules that IA I = 4 - 6 -2 (non-singular matrix) so that the inverse matrix exists and is equal to:

To check that we have not made any mistakes we compute AA -1 and A-1 A (both should equal the identity matrix).

Assuming that the ind non-singular), we find th

I-1 =1

(A-1 ) -1 =A

(AT)-1 = (A -1 )T

(AB)- 1 = B-1 A - 1

IA-1 1 =

I

A.2.5 Cramer's Rule

Suppose we have a lined:

I

anxi + a12x2 + " - azixi + a22x2 + - - -4

anixi + an2x2 + - -

where aij are the coet endogenous variables. V equation as:

Ax = b ,

where A is an n by n mat

all ai2 -

a22

A =-

_ an i ant

Provided the coefficient the matrix equation is:

x = A- 1 b.

Instead of inverting the variables by means of c.

IA/

fox- rj = 1,

IAI