Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

ome effect in labour supply of this specification is that e elasticity of labour supply ►upply is highly elastic and increase in the real wage. Estic and a large change in ['Dour supply.

aces a demand for its prodmment (see (13.16)). Since 'acing firm j can be written

(13.59)

(13.60)

onsumption to real money

"ly more general descripthe following production

(13.61)

al physical product of the firm is U-shaped (see II) and (13.10) coincide.

11

(13.a62)

rder condition:

)1

(13.63)

Land Y,(P,,P, Y) and profit

Chapter 13: New Keynesian Economics

where MCi is marginal costs of firm j and where we have substituted the price elasticity of demand (9) in going from the second to the third line. An active firm is one which produces a positive amount of goods (Yi(.) > 0) and sets price according to:

P; = pmci pi = wN 41-y)/y - 1 > 1. (13.64)

This pricing rule generalizes the one derived in section 1 of this chapter (i.e. equation (13.13)) by allowing for an upward-sloping marginal cost curve (if y < 1). Apart from this generalization, another important thing to note is that in section 1 it was assumed that the firm sets its output level in a profit-maximizing fashion taking other producers' output levels as given (the Cournot assumption). In contrast, in this section the firm sets its price in an optimal (profit-maximizing) fashion, taking other producers' prices as given (the Bertrand assumption). In the absence of menu costs the two assumptions yield the same pricing rule. As is shown below, however, this equivalence does not necessarily hold in the presence of menu costs.

We now have all the ingredients of the model, though still abstracting from menu costs. The main equations have been collected in Table 13.3. Equation (T3.2) expresses consumption (and equilibrium real money balances) as a function of factors influencing real wealth. It is obtained by using (13.54)—(13.55), (13.58), imposing money market equilibrium (13.48), and substituting the government budget constraint (we again abstract from civil servants and set LG = 0 in (T2.4)). Equation (T3.3) is the expression for aggregate profit income. It is obtained by substituting the optimal price (13.64) into the definition of profit income (13.62) and simplifying by using the definition of Y in (13.17). Finally,

Table 13.3. A simplified Blanchard-Kiyotaki model (no menu costs)

P = w(if a -> 00)

I

Notes: w ydaa (1 - a)1-1-1 > 0 and ,u, 01(9 - 1).

383

The Foundation of Modern Macroeconomics

(T3.4) is the price-setting rule in the symmetric equilibrium, and (T3.5) is the labour supply function.

Before turning to the implications of menu costs, we first study the properties of the model under perfect price flexibility. By studying the flex-price version of the models first, it is easier to understand the implications of menu costs later on. It is clear that money is neutral: multiplying and Mo by > 0 does not change anything real and just changes all nominal variables (such as nominal wealth, I, and nominal profit, 11) equi-proportionally. On the monetary side of things the models of Tables 13.2 and 13.3 are thus similar in that they both exhibit monetary neutrality when prices and wages are flexible. There is an important difference between the two models, however, in the area of fiscal policy. Indeed, because there is no income effect in labour supply (see (T3.5)), fiscal policy is completely ineffective in the model of this section. Using the expressions in Table 13.3 it is easy to show that a tax-financed increase in public consumption leads to one-for-one crowding out of private consumption, no effect on output, real profits, employment, and real wages, and an increase in the price level. In that sense the model used here is even more classical than the one used in the previous section. In the next subsection we study if and to what extent the notion of menu costs can give this hyper-classical model a more Keynesian flavour.

The basic menu - cost insight

Sometimes the answer to an apparently simple question can be quite surprising. A beautiful example of this phenomenon is provided by Akerlof and Yellen's (1985a) question whether "small deviations from rationality make significant differences to equilibria". Alternatively, the question could be rephrased in terms of transactions costs: can small costs of changing one's actions have large effects on the economic equilibrium and social welfare? Nine out of ten people would probably answer this question with an unequivocal "no". The thought experiment would probably lead them to reject the notion that a small "impulse" can produce a "large effect". In terms of Matsuyama's (1995) terminology, most people are unfamiliar with the notion of macroeconomic complementarities and cumulative processes. It turns out, however, that the answer to Akerlof and Yellen's question can be quite a bit more complex.

In the context of our model, the task at hand is to investigate whether, following a shock in aggregate demand, price stickiness can (a) be privately efficient and (b) exist in general equilibrium, whilst (c) the effect on social welfare can be large. If both parts (a) and (b) are demonstrated, Akerlof and Yellen's question is answered in the affirmative. Part (a) can be easily demonstrated to hold in our model and relies on a simple application of the envelope theorem. The proof of part (b) is more complex as it relies on the general equilibrium implications of price stickiness. Once (a) and

(b) have been demonstrated, part (c) follows readily.

Intermezzo

The envelope theor, theory. Broadly sped. function due to a cl not the decision van_ In more colloquial ten the top (Rotember,,. 1! Consider the formal that is the obit (vector of) exogenou

an optimum of f (x,

af(x,z) _ 0 ax — •

But (a) can itself be it choice for the deci

By plugging x* back nu value function:

V (z) Max f (x,

It is useful to note that functions throughou ity function (13.38) is maximum attainable price index (the paran the true price index t of a minimum valuL Using the optimal v

objective function totally differentiatin z I

dV (z) pf(x,/ dz ax

The second term on the function of the cha, term on the right-hand is induced by the

the optimum the obi

Mum. 1111111111111110

Ilium, and (T3.5) is the labour

re first study the properties of the flex-price version of the s of menu costs later on. It is Nfo by > 0 does not change :h as nominal wealth, I, and tary side of things the models h exhibit monetary neutrality difference between the two

,because there is no income mpletely ineffective in the

13.3 it is easy to show that a one-for-one crowding out of employment, and real wages, nodel used here is even more next subsection we study lye this hyper-classical model

I

()n can be quite surprising. ry Akerlof and Yellen's (1985a) nake significant differences to sed in terms of transactions large effects on the economic e would probably answer this eriment would probably lead produce a "large effect". In -, le are unfamiliar with the umulative processes. It turns 's question can be quite a bit

..-stigate whether, following a vately efficient and (b) exist welfare can be large. If both 's question is answered in the Id in our model and relies on )f of part (b) is more complex rice stickiness. Once (a) and

Chapter 13: New Keynesian Economics

ipmep

Intermezzo

The envelope theorem The envelope theorem is extremely useful in economic eory. Broadly speaking the theorem says that the change in the objective function due to a change in an exogenous parameter is the same whether or not the decision variable is adjusted as a result of the change in the parameter. In more colloquial terms, the theorem says that objective functions are flat at

the top (Roternberg, 1987, p. 76).

Consider the formal demonstration by Varian (1992, pp. 490-491). Suppose that f(x, z) is the objective function, x is the decision variable, and z is the (vector of) exogenous variables and parameters. The first-order condition for an optimum of f (x, z) by choice of x is:

But (a) can itself be interpreted as an implicit function relating the optimal choice for the decision variable (x*) to the particular values of z, say x* = x* (z). By plugging x* back into the objective function we obtain the so-called optimal value function:

It is useful to note that we have in fact encountered many such optimal value functions throughout the book. For example, in this chapter the indirect utility function (13.38) is an example of a maximum value function: it expresses maximum attainable utility (the objective) in terms of full income and a true price index (the parameters that are exogenous to the household). Similarly, the true price index for the composite differentiated good (13.9) is an example of a minimum value function.

Using the optimal value function (b) we can determine by how much the objective function changes if (an element of) z changes by a small amount. By

The second term on the right-hand side of (c) is the direct effect on the objective function of the change in z keeping the decision variable unchanged. The first term on the right-hand side is the indirect effect on the objective function that is induced by the change in x* itself. The point to note, however, is that in the optimum the objective function is flat (i.e. (a) shows that 0 for

385

The Foundation of Modern Macroeconomics

x = x*) so that the indirect effect is zero. Hence, equation (c) reduced to:

d17(z) a f (x* (z), z) a v (z) (d) dz az az

This is the simplest statement of the envelope theorem. The total and partial derivatives are the same, i.e. at the margin the change in the objective function is the same whether or not the decision variable is changed.

We close with an anecdote from times past. As is argued by Silberberg (1987), the discovery of the envelope theorem is due in part to a dispute between the famous economist Jacob Viner and his draftsman Dr Y. K. Wong. Viner was working on his famous paper about the relationship between short-run (ACSR) and long-run average cost (ACLR) curves (see Viner, 1931). He instructed Dr Wong to draw ACLR in such a way that it was never above any portion of any ACSR curve and that it would pass through the minimum points of all ACSR curves. Dr Wong, being a mathematician, refused to do so and pointed out to Viner that his instructions were actually inconsistent. Unfortunately, Viner, not being a mathematician, could not understand Dr Wong's point and ended up drawing ACLR through all the minima of the ACSR curves (see his chart IV and footnote 16). Samuelson (1947), being both an economist and mathematician, ultimately solved the puzzle by pointing out that ACLR is the envelope of all ACSR curves. Wong was right after all! If this anecdote has any lesson at all, it must be that economists should also be reasonably good mathematicians to avoid falling into puzzles that cannot be solved by graphical means alone.

What happens to the optimal price of firm j if aggregate demand changes by a small amount? The answer is provided by the envelope theorem (see the Intermezzo). In particular, (13.59) and (13.64) together yield an expression for the optimal price in terms of the parameters that are exogenous to firm j, i.e. P7 =

PAP, Y, WN):

By substituting 17 (.) into (13.62) we obtain the maximum profit function, rh (P, Y, WN ), of firm j:

By differentiating this expression with respect to aggregate demand we obtain the result that it doesn't really matter to the profit of firm j whether or not it changes

= [Vi() dY

a n,(1

[ aP, I

= 0 At

:ere MC(.) is short-hal 4... e timum. Hence, to a IA: a change in aggregate de :imally following the The envelope result ca

, ted by Akerlof and

k . el are put on the hoi demand is Y0 and the o

T optimal price-pruiit

.usv consider what happ C - eris paribus the nor. (L,,iisumption good (P). - profit function shifts u:

.-. expansion leads to an increase in the optima

profit hill (point B

But this is not the end -Aperiences a boost in

We hold constant the pi, -ndhex, P, constant. In doing ,--

-a is allowed because theftatc, , carries a small weight in t

Formally, (13.62) implies - have positive profits (as c.: -

" > MCi . Furthermore. (1 aY,/aY. Combining th:

...Ise it raises their pr. *, 14 In contrast, if the m.

A. This strong result I,

..-.-aiand elasticity (0) and 0- "^?n proportional to the give kw firm j to change its price

The Foundation of Modern Macroeconomics

(Y (Y1)

Figure 13.3. Menu costs

accordingly. But this means that it needs to employ more workers. Since all firms are in exactly the same position as firm j they will also want to employ more workers so that aggregate demand for labour will rise. This is where the labour market comes in. Clearly, if the labour supply elasticity is very large (a oo), firm j (and all other firms) can obtain the additional units of labour at the initial nominal wage rate (n). In that case the real wage is rigid (see (T3.5)) and thus, if the price index P does not change neither will the nominal wage rate WN . So all we need to show now is why the price index would be rigid.

Assuming for the time being that labour supply is infinitely elastic (a oo) it is possible to demonstrate the menu-cost insight graphically with the aid of

Figure 13.3. For given values of P and WN , the aggregate demand shock would increase the profits of firm j from 111(P, n) to ni(p, 171, n) if it adjusted its

price optimally (which is the move from A to B). If instead firm j keeps its price unchanged, the profit increase would be the vertical distance between points C and A and the envelope theorem suggests that the profit loss due to non-adjustment of the price is second order, i.e. the vertical distance DC in Figure 13.3 is very small. But that suggests that small menu costs can make non-adjustment of the price a profitable option for firm j. Indeed, provided the menu costs (Z) are larger than the vertical distance DC, keeping Pi unchanged is the optimal choice for firm j, i.e. Pi will be set equal to its old optimal level (P1 (P, Yo, n)) if the following condition is satisfied:

ni(PI(P, Yo, Y1, WOv) > r17(P, Y1, n) - Z, (13.68)

:_)re the left-hand sid tki price and faces the 1 as the net profit of firm th,urs the menu cost. S as firm j, they also do

- umption that P is coi iabour supply case (a — :nand shock has no c

.►e effects of fiscal computed as follows. -I

in (T3.2). Since PY — WNL) we car

Y=C+G,

mescal policy is highly e1

dy\mcE ( J(

UG) T

the superscript ,rnment consum: qtr ind and profit leer :)ecause of the h, - sot change either. Th; e old real wage rat

—• Lillie in the form one exactly coven

ax_ _.:nption is unc

Consumption as in L. )n (13.61) the e

(10/p)

r

we have

e set their prices

:ium. •fonetary policy, •

> 0) stimulate

r1,(Pp WO/)

P Yo, WOsi)

P.

re workers. Since all firms are nt to employ more workers so •'re the labour market comes - oo), firm j (and all other he initial nominal wage rate Ind thus, if the price index P IVN . So all we need to show

s infinitely elastic (a oo) graphically with the aid of - sate demand shock would P, Y1 , 147(;') if it adjusted its rr, stead firm j keeps its price ance between points C and oss due to non-adjustment of in Figure 13.3 is very small. on-adjustment of the price a i costs (Z) are larger than the imal choice for firm j, i.e. P1 if the following condition is

(13.68)

Chapter 13: New Keynesian Economics

where the left-hand side of (13.68) is the profit level of firm j when it charges the old price and faces the higher aggregate demand, Y 1 . The right-hand side of (13.68) is the net profit of firm j if it changes its price in the face of higher demand and incurs the menu cost. Since by assumption all firms are in exactly the same position as firm j, they also do not change their price if (13.68) holds and the maintained assumption that P is constant is thereby confirmed. Hence, for the infinitely elastic labour supply case (a oo) a menu-cost equilibrium exists for which an aggregate demand shock has no effect on prices and the nominal (and real) wage rate.

The effects of fiscal and monetary policy in a menu-cost equilibrium can be computed as follows. The model consists of equations (T3.1) and the second expression in (T3.2). Since aggregate profit income equals revenue minus the wage bill (fl PY — WNL) we can write the system as:

Fiscal policy is highly effective in the menu-cost equilibrium:

where the superscript "MCE" stands for menu-cost equilibrium. The increase in government consumption raises aggregate demand and thus each individual firm's demand and profit level. Due to the menu costs all firms keep their price unchanged and because of the horizontal labour supply curve (a oo) the nominal wage does not change either. The firms can hire all the additional units of labour they need at the old real wage rate. The representative household receives the additional firm revenue in the form of additional wage payments and profit income. The additional income exactly covers the higher taxes levied by the government so that private consumption is unchanged and the output effect is simply the effect due to public consumption as in the original Haavelmo (1945) story. In view of the production function (13.61) the employment expansion can be written as:

where we have used symmetry (Li = L/N for j = 1, . . . , N) plus the fact that firms have set their prices as a markup over marginal cost in the initial (pre-shock)

equilibrium.

Monetary policy, consisting of a helicopter drop of nominal money balances (dM0 > 0) stimulates output, employment, and consumption, and the existence

389

The Foundation of Modern Macroeconomics

of menu costs thus destroys monetary neutrality:

The increase in money balances leads to an increase in consumption spending and further multiplier effects via the expanded income of the representative household, i.e. after n rounds of the multiplier process spending has increased by P dY = P dC = an] dMo and the demand for money has increased by dM = (1- a)[1 + a + a2 + • • • + an] dMo. Since the marginal propensity to consume is less than unity,

the multiplier process converges to the expressions in (13.73).

In summary, we have succeeded in demonstrating that with a very high labour supply elasticity so that the labour supply curve is horizontal), small menu costs can lead to nominal price and wage inflexibility, which in turn drastically alters the qualitative properties of the model. Indeed, as was shown in the previous subsection, the flex-price version of the model possesses extremely classical properties in that money is neutral and fiscal policy only affects the price level. In contrast, in a menu cost equilibrium, both fiscal and monetary policy affect output and employment thus giving the model a much more Keynesian flavour. Below we demonstrate that both the nominal rigidity (price stickiness due to menu costs in price adjustment) and the real rigidity (constant real wage due to a horizontal labour supply curve) are of crucial importance in this result. Before doing so, however, we must demonstrate part (c) of our menu-cost investigation by demonstrating that there are first-order welfare effects associated with the aggregate demand effects we found above (see page 384 above).

As before, we use the indirect utility function to compute the welfare effects of aggregate demand shocks in a menu-cost equilibrium. By using (13.69)-(13.70) in

—* co imposed) we find a number of alternative expressions for

In going from the first to the second expression we have used the definition for aggregate profit income (I1 PY - WNL) and in going from the second to the third expression we have used the labour supply equation (T3.5). Fiscal policy clearly has

first-order welfare effects

dV ) MCE = a" (1 --- d dG T

YL

=— — - p. (

nere the second equali ment consumption r

-ynes' story in section

more hours of work. Sip no surplus from suppl\ !iie household for ha .% Ai,

Hence, only the additi vernment spending ), terms of MCPF as:

0 < MCPF17'fcE - —1

Mere we have used

e(1 - a) 1-". The exist:. sot obliterate the social N.. onetary policy also

indeed, using the final e

;le term outside the b

*marginal utility of nomi side of (13.77) there Li

effect and the pr,

:dity effect exists be( ,uboptimal if real 1111 1111 Chapter 12, the inefi

arce (fiat money)

mon, ceteris paribus cons irk .ire gain because it lc ;...t economy closer to

the nature of competitic

-v

- YL L

(13.74)

c used the definition for

.nm the second to the third Fiscal policy clearly has

The term outside the brackets on the right-hand side of (13.77) represents the marginal utility of nominal income. Inside the square brackets on the right-hand side of (13.77) there are two effects which may be labelled, respectively, the uidity effect and the profit effect. As is pointed out by Blanchard and Kiyotaki, the liquidity effect exists because even the competitive equilibrium (for which 1/0 = 0) is suboptimal if real money enters utility (1987, p. 654 n. 13). As is explained in Chapter 12, the inefficiency results from the fact that people economize on a resource (fiat money) which is not scarce from a societal point of view. For that reason, ceteris paribus consumption, an increase in real money balances constitutes a welfare gain because it lowers the marginal utility of real money balances and brings the economy closer to Friedman's satiation point. This effect operates regardless of the nature of competition in the goods market.

391

The Foundation of Modern Macroeconomics

In contrast to the liquidity effect, the profit effect in (13.77) is only operative under monopolistic competition (i.e. if 1/0 is finite). This works via the profit income of households. An increase in the money stock boosts output and profit income and this causes an additional welfare gain to the representative household over and above the liquidity effect. Since both effects work in the same direction, the total welfare effect of an increase in nominal money balances in a menu-cost equilibrium is unambiguously positive and first order.

Some simulations

In the previous subsection it was demonstrated (for the case with a horizontal labour supply curve, i.e. a -÷ oo) , that with small menu costs both monetary and fiscal policy can have first-order effects on welfare. We have thus confirmed the basic menu-cost insight of Akerlof and Yellen (1985a, 1985b). In Tables 13.4 and 13.5 we present some numerical simulations with a more general version of the menu-cost model. In particular, we investigate the robustness of the menu-cost insight with respect to changes in key parameters such as the labour supply elasticity (a), the markup (,u), and the elasticity of the marginal cost function (ay (1 — y)/y).

In order to perform the simulations, numerical values must be chosen for all the parameters that appear in Table 13.3. The following so-called calibration approach is adopted. We set up the model such that the parameters of special interest (a, ay, and ,u) can be varied freely. We adopt a number of quantities/shares that are held constant (at economically reasonable values) throughout the simulations. In particular, the number of firms is No = 1000 (large), the steady-state revenue share of overhead

labour cost is coF WN NF/PY = 0.05, the output share of government consumption is COG G/Y = 0.1, and the velocity of money vm Mo/(PY)o = 6. We assume that the initial money supply is MO = 1 and that initial output and employment are normalized at unity, Yo = Lo = 1. For a given configuration of (a, ay, p,) it is possible to compute the initial steady state for the endogenous variables (Y, C, P, WN /P, L, II /P) by using the calibration parameters (a, yL, F, k) appropriately, i.e. in such a way that the steady state is consistent with the share and parameter information we have

imposed above.

Since this way of calibrating a theoretical model may not be familiar to all readers, we show in detail how we can retrieve the remaining variables and parameters. We denote the initial steady-state value with a subscript "0". It follows from (T3.1) that

Co = (1 - coG)Y0 = 0.9 and Go = coG = 0.1. By rewriting the money velocity defini-

tion we find Po = Mo/(vmY0) = 1/6. From (T3.2) we derive we C/Y = avm /(1 - a) which can be solved for a = cocl(wc+vivr) 0.13. By defining the pure profit share as

1- EL [rII(PY)] 0 , it follows from (T3.3) and the definition of coF that EL = y I A+ (oF (where y -=-E 1/(1 + ay)). By definition EL -,- [WNLI (PY)] 0 from which we derive an expression for the initial real wage (WN /P) 0 = EL. We can make this expression for the real wage consistent with (T3.5) by setting yL = (WN / P) 0 a' (1. - a) 1-a. In view of the definition of WF we find that F = (Y/N)o/ (WN/11 0 = (NO EL )-1 . The value for

retrieved from (T3.4):

•e an example, for the L. Approach yields the follow

In order to numerically Kiyotaki (1987, p. 65 she form of a 5% increase i

7e scenarios. In the fii.. price of their product i

ntrast, in the no-adjust►e

Jt to meet the aggre,..

assuming that the menu me employed to change pi

(T3.1)-(T3.2)

nFA = (P' Y PY -

1

re the superscript "FA *.em can be solved nu-

1VN.

In contrast, in the no-i P = P0) and the system .

.:action under no adjustrn n NA = po y wN [k y

Is system of equations c Y, L, C, and WN .

In the final step, we corn est value of menu co , -in, i.e. for which TI FA ju

.rnber of indicators for di four different values 1

*Ames for the labour stir -- entry labelled "menu o n-adjustment is an equil

menu costs = 100 x

(

where (WN )NA and YNA ar( xice is not adjusted. So,