Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

p 11111111t

of profits, using the (time- ; uation (4.4) is altered to:

Chapter 4: Anticipation Effects and Eco'nomic Policy

indexes for now, we obtain:

[- • -] = K f(r + 8)X FK pi (1 - si)(13 + X [I - 6K] + r XK

=+ (1 — si)(DKK +

. and R(t) is a discounting , t rates up to t:

In the final step we have used the linear homogeneity of (1). (i.e. (13. (1)Ii FKK), equation (b), and the following result:

(c)

-11 term e-R(t) from the ►. and (4.10) to the case d a time-varying rate of

.'orange multiplier (X(0)), ale of the firm (V(0)), we

T." t)] e-R(t) (d)

V(0) in\

(0)

(0)K(0) q" ;1(0)) 4(0) = '1(0).

The nominal stockmarket value of the firm is P(0) V(0) and the nominal replacement value of its capital stock is Pi (0)K(0). As a result, Tobin's average q is P(0) V(0)/(PI (0)K(0)), which equals V(0)/(p/ (0)K(0)).

4.1.2 Fiscal policy: Investment stimulation

f (d) can be expanded by ion (c). Ignoring time

The model can now be used to investigate the immediate, transitional, and longrun effects of governmental efforts to stimulate investment. Omitting the (now almost superfluous) time index, the model consists of equations (4.3), (4.6), (4.8),

The Foundation of Modern Macroeconomics

and (4.11):

Despite its simplicity, the model allows several economically interesting variations to be considered within the same framework. Clearly, in view of (4.15), some assumption must be made about the real wage rate w. At least three types of labour market assumptions can be distinguished: (i) the model is interpreted at firm level and the real wage is assumed to be exogenously given (and constant); the model is interpreted at the level of the aggregate economy and (ii) full employment of labour is postulated or (iii) a macroeconomic labour supply equation is added to it (e.g. equation (1.11) with Pe = P). We consider these three cases in turn.

The effects of the investment subsidy under constant real wages

If the real wage rate is constant, the assumption of perfect competition in the goods market (and the implied homogeneity of the production function) renders the model very simple indeed. Of course, aside from the microeconomic interpretation given above, this case is also relevant for an entire economy with rigid real wages. Since the production function is homogeneous of degree one (constant returns to scale), the marginal products of labour and capital are homogeneous of degree zero (see the Intermezzo). This implies that FN(N, K) can be written as FN(1, KIN), which depends on the capital-labour ratio only. Equation (4.15) says that w = FN(1, KIN), which uniquely determines the K IN ratio for the firm, which is constant over time because w is constant over time. This also implies that the marginal product of capital is constant, since FK(N, K) = FK(1, KIN) = FK, a constant.

By assuming a constant real wage, the labour demand equation can be ignored, and the model consists of equations (4.13)—(4.14). The qualitative content of the model can be summarized graphically by means of Figure 4.1. The K = 0 line represents all combinations of K and q such that the capital stock is in equilibrium. In view of (4.13), this implies that gross investment is exactly equal to replacement investment along the K = 0 line. Formally, we obtain from (4.13):

aK )1<=0 =

In words, a higher capital stock necessitates a higher level of steady-state gross investment. This is only forthcoming if q is also higher.

4.

home 4.1. it

I

ArUne down ana

=— I

a lo i■t ‘nts ott the

4111111is .‘ :carnal

=o -

vs 0 Wile re"."'

-IL 7

(4.13)

(4.14)

(4.15)

ally interesting variations ly, in view of (4.15), some ' • least three types of labour tl is interpreted at firm level 1 (and constant); the model

d (ii) full employment of 'ply equation is added to it

e cases in turn.

al wages

!ct competition in the goods - 4; on function) renders the croeconomic interpretation nomy with rigid real wages. one (constant returns to lomogeneous of degree zero -itten as FN (1, KIN), which 51 says that w = FN (1, KIN), which is constant over time at the marginal product of

:onstant.

d equation can be ignored, qualitative content of the !re 4.1. The K = 0 line repstock is in equilibrium. In actly equal to replacement

om (4.13):

(4.16)

(4.17)

level of steady-state gross

Chapter 4: Anticipation Effects and Economic Policy

•If

K*

Figure 4.1. Investment with constant real wages

Equation (4.16) also implies that an increase in the investment subsidy shifts the k = 0 line down and to the right:

The after-subsidy cost of investing falls and as a result firms are willing to invest the

same amount for a lower value of q.

For points off the K = 0 line, the dynamics of the capital stock is also provided by equation (4.16):

The graphical interpretation is as follows. In point A the capital stock is in equi-

librium. If K is slightly higher (say at A' to the right of point A), (4.19) predicts that depreciation exceeds gross investment so that the capital stock falls over time,

i.e. K < 0. This dynamic effect is indicated by a horizontal arrow towards the K = 0 line. Obviously, for points to the left of the K = 0 line, the arrows point the other way (see point A"). The basic insight is, of course, that the capital accumulation process is self-correcting, i.e. for a given value of q, K has an automatic tendency to

return to the K = 0 line.

The q = 0 line represents all points for which the firm's investment plans are in equilibrium. By differentiating (4.14) we obtain:

The Foundation of Modern Macroeconomics

where we have used the fact that the marginal product of capital is constant. From (4.20) it is clear that the q = 0 line is horizontal:

This is intuitive: since both the rate of interest and the marginal product of capital are constant (and hence independent of K) , q itself is also constant and independent of K in the steady state. If the (exogenous) rate of interest rises, future marginal products of capital are discounted more heavily, so that the steady-state value of q falls:

For points off the q = 0 line, the dynamic behaviour of q is also provided by (4.20):

a q

_

The graphical interpretation is as follows. At point B the value of q is consistent with an equilibrium investment plan. Now take a slightly higher value of q, say the one associated with point B', directly above point B. Clearly, in view of the fact that both r and FK are constant, this higher value of q can only satisfy the arbitrage equation (4.12) if a (shadow) capital gain is expected, i.e. if q > 0. The opposite holds at points below the q = 0 line (say point B', as is indicated with the arrows in Figure 4.1). Intuitively, therefore, the q-dynamics is inherently unstable. Slight moves away from the 0 line are not self-correcting but reinforcing.

By combining the information regarding the K-dynamics and q-dynamics, the forces operating on points in different regions of Figure 4.1 are obtained and summarized by the arrows. For example, at point B' there are automatic forces shifting the (q, K) combination in a north-easterly direction. In Figure 4.2, a number of representative trajectories have been drawn. Note especially what happens if a trajectory crosses through the K = 0 line. Take point A, for example. As it moves in a southeasterly direction, it gets closer and closer to the K = 0 line. As it reaches this line (at point A'), however, the value of q keeps falling and the level of gross investment becomes too low to sustain the given capital stock. As a result, the trajectory veers off in a south-westerly direction towards point A" (never to be heard of again).

From the different trajectories that have been drawn in Figure 4.2, it can be judged that the model appears to be very unstable: all trajectories seem to lead away from the steady-state equilibrium point at Eo . There is, however, one path that does give rise0to stable adjustment, namely the q = 0 line itself. Consider, for example, point C. It lies on the 4 = 0 line (so there are no forces operating to change the value of q over time), but it lies to the left of the K = 0 line. But, the K-dynamics is stable, so the capital stock will automatically rise towards its level at point E o . A similar conclusion holds for point C'.

A

r e Ow stead!

a - Liie model

comb':

---inninecl at any 1-, -

"-*

:

.xf re-es - :1)1 • -,-s

_ ,.mot ec no-...y be Amor ultimately, ai

the acilustinenz sadd

L * *

tin the vital stoci

capital is constant. From

(4.21)

.;inal product of capital onstant and independent rises, future marginal the steady-state value of

(4.22)

I

also provided by (4.20):

(4.23)

value of q is consistent y higher value of q, say lrly, in view of the fact ly satisfy the arbitrage

. if q > 0. The opposite 1 !cated with the arrows ierently unstable. Slight reinforcing.

cs and q-dynamics, the l are obtained and sum- `omatic forces shifting ire 4.2, a number of repit happens if a trajectory As it moves in a south- e. As it reaches this line el of gross investment alit, the trajectory veers

he heard of again).

re 4.2, it can be judged ;cern to lead away from one path that does give ider, for example, point to change the value of

K-dynamics is stable, 1 at point Eo. A similar

Chapter 4: Anticipation Effects and Economic Policy

q

q*

K*

Figure 4.2. Derivation of the saddle path

In conclusion, for each given initial level of the capital stock,

And this is very fortunate indeed, because one would have an embarrassment of riches if this were not the case. Indeed, suppose that the model were globally stable, so that "all roads lead to Rome", i.e. combinations would eventually return to point Eo. That would lead to a very troublesome conclusion, namely that the shadow price of capital (q) is not

determined at any point in time!

The particular type of stability that is exemplified by the model is called

there is exactly one stable adjustment path (called the saddle path) that re-establishes equilibrium after a shock. Technically speaking, the requirement that the economy be on the saddle path has more justification than just convenience: ultimately, an exploding solution is seen by agents not to be in their own best interests, so that they have good reason to restrict attention to the saddle-path solution. The remainder of this chapter will be used to demonstrate the remarkable predictive content of models incorporating saddle-point stability.

Consider the case of an

sidy. This means that at some time tA the government announces that s, will be increased "as of today". In other words, the policy change is implemented imme-

diately. For future reference, the implementation date is denoted by t1 . Hence, an unanticipated shock is a shock for which announcement and implementation dates coincide, i.e. tA = tI. The effects of the policy measure can be derived graphically with the aid of Figure 4.3. We have already derived that an increase in s1 shifts the

K = 0 line to the right, so that the ultimate equilibrium will be at point E 1 . How does the adjustment occur? Very simple. Since E o is on the q = 0 line (which is also the saddle path for this model), the higher subsidy gives rise to higher gross investment and the adjustment path is along the saddle path from E0 to E 1 . Note that the capital stock adjusts smoothly, due to the fact that adjustment costs make

89

The Foundation of Modern Macroeconomics

K0 K1

Figure 4.3. An unanticipated permanent increase in the investment subsidy

very uneven investment plans very expensive. The adjustment over time has also been illustrated in Figure 4.3.

As a second "finger exercise" with the model, consider an unanticipated permanent increase in the exogenous rate of interest r as illustrated in Figure 4.4. Equation (4.22) shows that this shifts the q = 0 line down due to the heavier discounting of future marginal products of capital. What does the adjustment path look like now? Clearly, the new equilibrium is at point E1 and the only path to this point is the saddle path going through it. Since K is fixed in the short run, the only stable adjustment path is the one with a "financial correction" at the time of the occurrence of the shock (in tA ): q jumps down from point E0 to point A directly below it. The intuition behind this financial correction is aided by solving the differential equation for q given in (4.14):

Hence, as was already hinted at above, q represents the discounted value of present and future marginal products of capital, so that an increase in r (either now or in

N.

F1160

44 AA 4 Aed...•, 4 mostaiate fmancial

_.s t u.. anal example c

ar --it

of .a x>.cdulec sik Isom to occur at sou et --tee : L.

- .Appals to the value u becx af.;,./a11:« -

Millabie. But by ho-.

::311.

arm at the time the

maul h .eshn,

= 0)1

—4 = 0

K

time

^Pnt

ustment over time has also

er an unanticipated permaated in Figure 4.4. Equation he heavier discounting of ament path look like now? path to this point is the run, the only stable adjust- e time of the occurrence of rectly below it. The intu- g the differential equation

(4.24)

I

ifcounted value of present se in r (either now or in

Chapter 4: Anticipation Effects and Economic Policy

go

A

4

Ko

Ko

K

K,

A :

ri

A

TO

Figure 4.4. An unanticipated permanent increase in the rate of interest

the future) immediately leads to a revaluation of this stream of returns. After the immediate financial correction, the adjustment proceeds smoothly along the saddle path towards the ultimate steady-state equilibrium point E1 .

As a final example of how the model works, consider the case where the firm hears at time tA that interest rates will rise permanently at some future date tI. This is an example of a so-called anticipated shock. Formally, an unanticipated shock is one that is known to occur at some later date. Obviously, the only real news reaches the agent at time tA . Everything that happens after that time is known to the agent. What happens to the value of q can already be gleaned from (4.24). Discounting of future marginal products becomes heavier (than before the shock) after the rate of interest has actually risen, i.e. for t > t1 . Hence, q must fall at the time the news becomes available. But by how much? This is best illustrated with the aid of Figure 4.5.

Consider the following intuitive solution principle: a discrete adjustment in q must occur at the time the news becomes available (i.e. at tA), and there cannot be a further discrete adjustment in q after tA . Intuitively, an anticipated jump in q would imply an infinite (shadow) capital gain or loss (since there would be a finite change in q in an infinitesimal amount of time). Hence, the solution principle amounts to

91

The Foundation of Modern Macroeconomics

k=o

(4=0)0

A

(4=0)1

K1 K0

K0

K E1

K1

q,I

q,I

B

time

Figure 4.5. An anticipated permanent increase in the rate of interest

requiring that all jumps occur when something truly unexpected occurs (which is at time tA). Obviously, at tA there is an infinite capital loss, but it is unanticipated.

With the aid of this solution principle, the adjustment path can be deduced. We work backwards. At the time of the interest rate increase the (q, K) combination must be on the new saddle path, i.e. at point B. If it were to reach B too soon (t < ti ) or too late (t > tj ), equilibrium would never be re-established without further jumps in q that are prohibited. Between tA and tj the dynamic forces determining q and K are those associated with the old equilibrium E0 (see the arrows). Working backwards, there is exactly one trajectory starting at point A at tA that arrives at point B at the right time ti. Hence, the unique path that re-establishes equilibrium after the shock is the one comprised of a discrete adjustment at tA from E0 to A, followed by gradual adjustment from A to B in the period before the interest rate has risen, arrival at point B at ti , followed by further gradual adjustment in the capital stock from B to E1

In comparison with the case of an unanticipated rise in the interest rate, the paths of q and investment are more smooth in the anticipated case (compare Figures 4.4

and 4.5, lower pant

...ed shock elt future.

fr :S of their

low we ha - . facing a col >mic le‘- t

.:al

t furthermore tl

._:. By

=(q , — OK ,

=(r + —4, 1

4.6, ti., NJ( :ompbcated in

ger, thee A and C, t ie =

L.. uldwe

411a tf- - st policy mat bk._ _Jr

taw principle in

and to the let s.

Aarcpt of capital! si

*-n_ At t

(4 = 0) 0

E,

ri

time

I.

t increase

expected occurs (which is loss, but it is unanticipated. path can be deduced. We the (q, K) combination must -each B too soon (t < ti ) or J without further jumps in rces determining q and K are - - )ws). Working backwards, that arrives at point B at k hes equilibrium after the it tA from E0 to A, followed e the interest rate has risen, tment in the capital stock

0 -

n the interest rate, the paths (1 case (compare Figures 4.4

Chapter 4: Anticipation Effects and Economic Policy

and 4.5, lower panel). The reason is, of course, that the firm in the case of an anticipated shock has an opportunity to react to the worsened investment climate in the future.

The effects of the investment subsidy with full employment in the labour market

Up to now we have interpreted the model given in (4.13)-(4.14) as applying to a single firm facing a constant real wage. Suppose that we re-interpret the model at a macroeconomic level, i.e. I and K now represent economy-wide gross investment and the capital stock, respectively, and the interpretation of q is likewise altered. Assume furthermore that the economy is characterized by full employment in the labour market. By normalizing employment to unity (N = 1), the model consists of:

K = I (q, si ) – SK, (4.25) 4= (r + 8)q – FK(1, K), (4.26)

where it is clear that the major change caused by our re-interpretation is that the marginal product of capital is no longer constant. Intuitively, since the labour input is fully employed, the economy experiences diminishing returns to capital, since FKK < 0. This also causes the 4= 0 line to be affected:

Intuitively, steady-state q is downward sloping in K because the more capital is used, the lower is its marginal product. As a result, the discounted stream of marginal products (which is q) falls.

In Figure 4.6, the saddle path is derived graphically. The dynamic forces are much more complicated in this case. This is because the steady-state level of q and the q-dynamics itself are now both dependent on K. In addition to trajectories from points like A and C, there are now also trajectories from points like B and D that pass through the 4= 0 line. The major alteration compared to our earlier case is that the saddle path no longer coincides with the 4= 0 line.

As a first policy measure, consider an anticipated abolition of the investment subsidy, as was for example the case in the Netherlands in the late 1980s. Using the solution principle introduced above, the effects of this announced policy measure can be derived with the aid of Figure 4.7. The ultimate effect of the abolition of the subsidy is to increase the relative price of investment goods and to shift the K 0 line up and to the left. In the long run the economy ends up at point E 1 , with a lower capital stock and a higher value of q (due to the higher steady-state marginal product of capital). Since the discrete adjustment in q must occur at the time of the announcement tA, and the economy must be on the new saddle path at the time of implementation t1, the adjustment path must look like the one sketched in the diagram. At tA there is a financial correction that pushes the economy from

93

The Foundation of Modern Macroeconomics

K

Investment with full employment in the labour market

K1

K

B

Figure 4.7. An anticipated abolition of the investment subsidy

to A directly aim = ec( n-.

az • After that, thc le striking '30

) tiAlS

Ott subsidy while it

rq-,ure 4.7. The c - e 7v

Of p4.„..

ti;,

de so by creating in

_r

bee introduce a pe at a t.

oul as intertemporal s

:al ..

•our simple me

of course). Our het _ t

maiming must he

q

•su! • idy we

z all this to A at t.

TN - adual adi■, -

,..en in the Lou

Eo to A - on

k..Qmpar, ilme able of q fa

,• -

middy is one wa)