Heijdra Foundations of Modern Macroeconomics (Oxford, 2002)

The Foundation of Modern Macroeconomics

Table 13.4. Menu costs and the markup

in Table 13.4 show that menu costs amounting to no more than 0.20% of revenue will make non-adjustment of prices an equilibrium in the sense that TINA > FIFA . The entry labelled "welfare gain" measures the gain in welfare (expressed in terms of an output share) which results from the monetary shock when there is no adjustment in prices:

where Uc cya (1 - a) 1 - a is the marginal utility of income, Vo is initial welfare, and VNA is welfare following the shock but in the absence of price adjustment. So, if = 1.1, ay = 0.1, and a = 106 , the monetary shock gives rise to a huge 29.1% rise in welfare. Finally, the entry labelled "ratio" is the ratio of the welfare gain and the macroeconomic menu costs. For the particular case considered here, the ratio

is 146.12, so that a small menu cost gives rise to very large welfare effects.

In Table 13.4 we hold the elasticity of marginal cost constant (at ay = 0.1) and consider various combinations of the markup (p) and the substitution elasticity of labour supply (a). Just like Blanchard and Kiyotaki (1987, p. 658) we find a number of key features in these simulations. First, the welfare measure does not vary a lot

with the different parameter combinations. Second, for a given value of a, the markup does not affect menu costs and the ratio very much. Third, for a given value of it, menu costs are strongly dependent on the value of the labour elasticity.

Table 13.5. I

AM = 0.05 = 1.25

a = 0.2 a = 0.5 a = 1 a = 2.5 a = 5 a = 106

a = 0.2 a = 0.5 a = 1 a = 2.5 a = 5 a = 106

e, for example, the em

._e. p = 1.25. If labour su revenue suffice to make r ratio drops very rap.,

example, if a = 1 then on venue) can stop the fir _

labour supply is not vt drives up wages (and ti-

ly that price adjusi,,,, In Table 13.5 we hold I

tr lbinations of the ela , 3). Essentially the same

the welfare gain is rather - )t affect menu costs an exerts a major effect on ---

Lialuation

simulation results gull. into trouble because no-

welfare ratio

29.11.61

29.44.22

I

30.62.65

30.86.76

30.813.12

30.829.12

30.948.68

30.9144.95

) more than 0.20% of revenue the sense that TINA > IIFA . The 'are (expressed in terms of an k when there is no adjustment

(13.82)

'-nme, 170 is initial welfare, and e of price adjustment. So, if Eck gives rise to a huge 29.1% ratio of the welfare gain and case considered here, the ratio

large welfare effects.

, t constant (at ay = 0.1) and id the substitution elasticity of "'87, p. 658) we find a number [re measure does not vary a lot d, for a given value of a, the

ry much. Third, for a given to value of the labour elasticity.

Chapter 13: New Keynesian Economics

Table 13.5. Menu costs and the elasticity of marginal cost

Take, for example, the empirically reasonable case for which the net markup is 25%, i.e. ,u = 1.25. If labour supply is infinitely elastic (a -÷ oo), menu costs of 0.2% of revenue suffice to make non-adjustment of prices optimal and the ratio is 145.73. This ratio drops very rapidly for lower, empirically more reasonable, values of example, if a = 1 then only unreasonably high menu costs (amounting to 3.51% of revenue) can stop the firm from finding price adjustment advantageous. Intuitively, if labour supply is not very elastic, the output expansion under non-adjustment drives up wages (and thus production costs) very rapidly and thus makes it more likely that price adjustment is profitable.

In Table 13.5 we hold the markup constant (at = 1.25) and consider various combinations of the elasticity of marginal cost (ay) and the labour supply elasticity Essentially the same picture emerges from this table as from the previous one: the welfare gain is rather insensitive to (a, ay )-combinations, the value of ay does not affect menu costs and the ratio very much, and the labour supply elasticity

exerts a major effect on menu costs and the ratio.

Evaluation

The simulation results graphically illustrate that the standard menu-cost model runs into trouble because non-adjustment of prices after a monetary shock is only an

395

The Foundation of Modern Macroeconomics

equilibrium if labour supply is highly elastic (Blanchard and Kiyotaki, 1987, p. 663). For an empirically reasonable value of the labour supply elasticity, there are very strong incentives to adjust prices and nominal frictions produce only small nonneutralities. 15 Ball and Romer (1990) argue that the menu-cost argument can be rescued if the economy has both real and nominal rigidities. By real rigidity they mean the phenomenon that "real wages or prices are unresponsive to changes in economic activity" (Ball and Romer, 1990, p. 183). Nominal rigidity can either take the form of small menu costs or departures from full rationality (as in Akerlof and Yellen, 1985a, 1985b). Taken in isolation, real rigidity does not imply price inflexibility. But in combination with nominal rigidity, a high degree of real rigidity translates into substantial effects of monetary shocks. In the model considered in the previous subsection, a high labour supply elasticity leads to substantial real rigidity. Indeed, for a oo , the real wage is constant (see equation (T3.5)) and thus completely insensitive to economic activity. Ball and Romer (1990) discuss a number of alternative models leading to real rigidities, such as the efficiency-wage model of the labour market and the imperfect-information customer-market model of the goods market.

Rotemberg (1987, pp. 80-81) has identified a number of problematic aspects of the menu-cost insight. First, the menu-cost equilibrium may not be unique. In the context of our model, his argument runs as follows. Recall that ZMIN represents the minimumC20amount of menu costs for which it is profitable for an individual firm j not to adjust its price given that all other firms also keep their prices unchanged! But if one firm changes its price when Z = ZMIN, it generally becomes profitable for all other firms to change their prices also, so we have two equilibria: the firms either all adjust their prices or they all keep them unchanged. Let us now define Z'A',„„, as the minimum amount of menu costs for which an individual firm j keeps its price unchanged even if all other firms would change their prices. Clearly, ZMIN exceeds ZmIN.

Furthermore, if Z > Z„, the menu cost equilibrium is unique. For the intermediate case, however, with Z E (ZMIN, 4/N ) there are three equilibria: one with no firm adjusting, another with all firms adjusting, and an intermediate case in which a fraction 0 of the firms adjusts (0 < 0 < 1). Rotemberg (1987, p. 90) argues that the multiplicity of equilibria is a weakness for any economic model. Essentially, with multiple equilibria it is impossible to predict the economy's reaction to particular policy shocks.

A second problem with the menu-cost insight is that it could equally well be applied to quantities instead of prices. Indeed, if there are costs of adjusting quantities (e.g.acbecause capital has to be installed in advance of the price-setting decision, as in Shapiro,ap1989, pp. 350-351) it may well be optimal for the firm to adjust its price and leave output unchanged (Rotemberg, 1987, p. 77).

15 As we show in Chapter 15, the competitive real business cycle (RBC) model runs into the same problem because it can only generate large output movements following real shocks if the (intertemporal) labour supply elasticity is very large.

finally, as is argued by r

. portant practical

anat it does not general - 1 to two approaches N.

.J.3.2 Quadratic pric,

gr an influential artik....

mic model of price

,aadratic (just as in

Above). Intuitively, his rr conceptual) steps. 11

_.,fisting of the soluti, . :s. Normalizing the firm j is denoted 1 ti ;adratic approximation d incorporates adju ,

:lie firm can then be

.there (1 + p) -1 is the log P7 T . In the prey

actual prices, (1)1,,

:n in the absence of

,e "deviation costs" fight-hand side of (1.J.: a "suboptimal" level, i. on the right-hand siL:L

.t are due to price adj ak.,ustment costs. ' I

The first-order conc.. (13.83) and settii

" See Danziger (199,

COStS.

and Kiyotaki, 1987, p. 663). ply elasticity, there are very produce only small non- nenu-cost argument can be stifles. By real rigidity they unresponsive to changes in Nominal rigidity can either 'till rationality (as in Akerlof gidity does not imply price a high degree of real rigidity In the model considered in tv leads to substantial real it ( see equation (T3.5)) and and Romer (1990) discuss a such as the efficiency-wage ion customer-market model

It/

er of problematic aspects of i may not be unique. In the 11 that ZMIN represents the able for an individual firm j

'1 7eir prices unchanged! But if becomes profitable for all ■ equilibria: the firms either 1. Let us now define 41N as ividual firm j keeps its price Clearly, 4/N exceeds ZMIN. ique. For the intermediate Iiilibria: one with no firm - mediate case in which a 1987, p. 90) argues that the model. Essentially, with ►my's reaction to particular

it could equally well be ie costs of adjusting quanti- f the price-setting decision, for the firm to adjust its

'e (RBC) model runs into the following real shocks if the

Chapter 13: New Keynesian Economics

Finally, as is argued by Rotemberg (1987, pp. 85-91) and Blanchard (1990, p. 822) an important practical disadvantage of the menu-cost approach to price adjustment is that it does not generalize easily to a dynamic setting. 16 For that reason we now turn to two approaches which do not have this disadvantage.

13.3.2 Quadratic price adjustment costs

In an influential article, Rotemberg (1982) has formulated a rather attractive dynamic model of price adjustment in which adjustment costs are assumed to be quadratic (just as in the investment literature surveyed in Chapters 2 and 4 above). Intuitively, his model solves the problem of dynamic price adjustments in two (conceptual) steps. In the first step, a path of "equilibrium" prices is determined

consisting of the solution that firm j would choose if there were no costs of adjusting prices. Normalizing the current (planning) period by t = 0, this equilibrium path for firm j is denoted by the sequence {Pit }c)°, 0 . In the second step, Rotemberg takes a quadratic approximation of the firm's profit function around this equilibrium path

and incorporates adjustment costs. He shows that the dynamic objective function of the firm can then be written as follows:

pi, log Pr,. In the presence of price adjustment costs, the firm chooses a sequence of actual prices, {1)1,,r 0 , in order to minimize the costs of deviating from the optimum in the absence of price adjustment costs (c20). Equation (13.83) shows that these "deviation costs" are composed of two terms. The first quadratic term on the right-hand side of (13.83) represents the intratemporal cost of setting the price at a "suboptimal" level, i.e. at a level different from P. The second quadratic term on the right-hand side of (13.83) parameterizes the intertemporal costs to the firm that are due to price adjustment costs. The higher is c, the more severe are the price

adjustment costs.

The first-order condition for the optimal price in period r is readily obtained by

16 See Danziger (1999) for a recent example of a dynamic general equilibrium model with menu costs.

397

The Foundation of Modern Macroeconomics

After some straightforward manipulation we find that (13.84) can be simplified to:

Equation (13.85) is a second-order difference equation in pi,, with constant coefficients and a potentially time-varying forcing term p7 T . In order to solve this equation we need two boundary conditions. The first is an initial condition which results from the fact that when the firm decides on its price pi,,, the price it charged in the previous period (pi, r _i ) is predetermined. The second boundary condition is a terminal condition saying that the firm expects to charge a price close to /3; r in the distant future (see Rotemberg (1982, pp. 523-524) for details):

It is shown by Kennan (1979, p. 1443) and Rotemberg (1987, p. 92) that the

r=0

where 0 < < 1 and )1/4.2 > 1. 17 The economic intuition behind the pricing-setting rule (13.87) is as follows. In the presence of price adjustment costs, the firm finds it optimal to adjust its price gradually over time. As a result, the optimal price in any period is the weighted average of the last period's price p1,_ i and the longrun "target" price given in square brackets on the right-hand side of (13.87). This target price itself depends on the present and future equilibrium prices (pi r , for

= 0, 1, ). In the special case where the equilibrium price is (expected to be) constant indefinitely, we have /37, = p7 and it follows that the target price is equal to p7. In the general case, however, the firm knows that it chases a moving (rather than a stationary) target because it recognizes future variability in the equilibrium price (say due to anticipated policy shocks).

13.3.3 Staggered price contracts

In a number of papers, Calvo has proposed an alternative approach to modelling sluggish aggregate prices (see e.g. Calvo, 1982, 1983, 1987 and Calvo and Vegh, 1994). His basic idea, which derives from the early papers by Phelps (1978) and Taylor (1980), makes use of the notion that price contracts are staggered. Calvo (1987, p. 144) adopts the following price-setting technology. Each period of time "nature" draws a signal to the firm which may be a "green light" or a "red light" with probabilities n- and 1 - 7, respectively. These probabilities are the same for all

17 Readers of the Mathematical Appendix will recognize that X i and A2 are, respectively, the stable and unstable characteristic roots of the difference equation in (13.85).

firms in the economy. A fire _ optimally in that i

ght is received.

In order to solve the pric: can follow the same 4,

the pricing friction firm j wo

P!'. But with the pric:

, given in equation (13.6.: substituting the assurr

:..notion (13.83) we obtain:

\ 2 Q0 = (13j,0 - KO)

( 1 )2 [7r 2

+11+ p

▪ (1 - 702 (A,o

ene interpretation of this ex - m has a green light so it c. 3.88) gives the cost of dc 7 = 1) the firm may or may

-0 it will again be able to price /37 1 . If it gets a red lig..

)) and face the deviation r.vriod. In period t = 2 the the firm last received a greet Since the pattern should 1

to be set by the firm in the f rms involving pj,o:

Qo = (Po --P7,0) 2 + ( 11

-

where the remaining tern. in the discounting factor et - een light in any period, tt attached to future equilib:.. The firm chooses pi3 O in a

y 0C20/apho = 0 which ca:

(13.84) can be simplified to:

( 1 + p

cPk' (13.85)

1 in piir with constant coeffi- n order to solve this equation fi al condition which results the price it charged in )nd boundary condition is a e a price close top; in the

details):

(13.86)

[berg (1987, p. 92) that the be written as:

(13.87)

1 behind the pricing-setting `ment costs, the firm finds result, the optimal price in ''s price pi,_ i and the long- t-hand side of (13.87). This equilibrium prices (p7T , for m price is (expected to be) hat the target price is equal t it chases a moving (rather

ability in the equilibrium

firms in the economy. A firm which has just received a green light can change its price optimally in that period but must maintain that price until the next green

light is received.

In order to solve the pricing problem of a firm which has just received a green light we can follow the same approach as in the previous subsection. In the absence of the pricing friction firm j would always want to set its price equal to its equilibrium price 17. But with the pricing friction the firm aims to minimize the deviation cost, S2o, given in equation (13.83) but with c = 0 (there are no price adjustment costs). By substituting the assumptions about the pricing technology into the objective

function (13.83) we obtain:

The interpretation of this expression is as follows. In the current period (t = 0) the firm has a green light so it can set its price. The first term on the right-hand side of (13.88) gives the cost of deviating from p7, in the current period. In the next period (r = 1) the firm may or may not get a green light again. If it does (with probability 7) it will again be able to set its price in the light of the then relevant equilibrium

price /37 1 . If it gets a red light, however, it will have to keep its price unchanged (at No) and face the deviation costs associated with this choice made in the previous

period. In period r = 2 there are three different possibilities depending on when

the firm last received a green signal.

Since the pattern should be clear by now and we are only interested in the price to be set by the firm in the planning period, we can rewrite (13.88) by gathering all terms involving Pi,o:

ive approach to modelling ' 0 S7 and Calvo and Vegh, pers by Phelps (1978) and tracts are staggered. Calvo Dlogy. Each period of time ven light" or a "red light"

lities are the same for all

KI 12 are, respectively, the stable

where the remaining terms do not involve po. The pricing friction thus shows up in the discounting factor employed by the firm. The higher is the probability of a green light in any period, the less severe is the friction, and the lower is the weight

attached to future equilibrium prices.

The firm chooses No in order to minimize S20. The first-order condition is given by as- 0 /api3O = 0 which can be written as:

The Foundation of Modern Macroeconomics

Since the infinite sum on the left-hand side of (13.90) converges to (1 + p) I Or p) we can rewrite (13.90) as follows:

where pg denotes the common "new" price set in period 0 by all firms facing a green light in that period. Note that we have assumed that all firms are identical so that the firm index no longer features in (13.91). The firms facing a red light in the planning period (r = 0) keep their prices as set in some past period using a rule like (13.91), i.e.:

for s = 1, 2, • • • oo. Since ir(1 — 71-) 0 is the fraction of firms which last adjusted prices s periods before the planning period, we can define the aggregate price level in the planning period as follows:

The actual aggregate price level in the planning period (po) is thus the weighted average of the aggregate price in the previous period (p_i) and the newly set price (pg). By substituting (13.91) in (13.93) we get the following expression for p o :

As is pointed out by Rotemberg (1987, p. 93), the pricing rule that results from the Calvo friction (given in (13.94)) is indistinguishable from the aggregate version of the pricing rule under adjustment costs (given in (13.87)). The nice thing about both pricing rules is that they can be readily estimated using time series data for actual economies. Rotemberg (1987, p. 93) for example, cites evidence that 8% of all prices are adjusted every quarter in the US, implying a mean time between price adjustments of about three years. 18

18 The expected time of price fixity (ETPF) is:

ETPF = 7r x 1 + n- (1 — 7r) x 2 + • • • + 741. — 7r)n-1 n +

00

= E (1— 70(1 + .5) = 1/7r. s=o

See King and Wolman (1996, p. 10).

13.4 Punchlines

We started this chapter 17, :..onopolistic competiti, market there are many sm, v and thus possess a sm ptimally exploit its ma: :. The model provides mic _.n the number of firms i sumption boosts output, t households poorer which 1

t. us to increase labour rtially mitigates the fall

_.. profits prompts entry d

even (the Chamberlinian

-.en the increase in the n wnsumer wage. The mui„

critically on the labour sui

Under monopolistic c,

taplier and the welfare of competition. Under mots market and the econoni :

,vernment spending out

.:are-enhancing, dire, ..

Next we introduce mor

.ty from real money b

,:.i.scussed in detail in Chaff. - oney is held by econor t invalidate the classical

simply inflates all nominal hanged.

. coney ceases to be a niL c competition is essent -is (and not some arki..

as the economy. We stud es as an equilibria.

taLLAence of small costs ai 'at at the top, it may be (

le wake of an expansi, et.: - :)ut. Provided labour s

_ - ee of real

iddiAlze the fixity of both

converges to (1 + p) (71- + p)

(13.91)

period 0 by all firms facing a that all firms are identical so firms facing a red light in the to past period using a rule like

ms which last adjusted prices re aggregate price level in the

(13.93)

xi (Po) is thus the weighted p_ ) and the newly set price wing expression for p o :

(13.94)

rule that results from the Dm the aggregate version of ")). The nice thing about d using time series data for e, cites evidence that 8% of a mean time between price

Chapter 13: New Keynesian Economics

13.4 Punchlines

We started this chapter by constructing a small general equilibrium model with monopolistic competition in the goods market. On the supply side of the goods market there are many small firms who each produce slightly unique product variety and thus possess a small amount of market power. Each firm sets its price to optimally exploit its market power.

The model provides microeconomic foundations for the multiplier. In the short run the number of firms is fixed and a tax-financed increase in government consumption boosts output, though by less than one-for-one. The tax increase makes households poorer which prompts them to decrease consumption and leisure (and thus to increase labour supply). The increase in output raises profit income which partially mitigates the fall in consumption. In the long run the short-run increase in profits prompts entry of new firms which continues until all firms exactly break even (the Chamberlinian tangency solution). If households like product diversity then the increase in the number of product varieties causes an increase in the real consumer wage. The multiplier is not very Keynesian as the output expansion relies critically on the labour supply response (a new classical feature).

Under monopolistic competition, there exists an intimate link between the multiplier and the welfare effect of public spending which is absent under perfect competition. Under monopolistic competition there is a distortion in the goods market and the economy is "too small" from a societal point of view. By raising government spending output rises and that in itself constitutes a move in the right, welfare-enhancing, direction.

Next we introduce money into the model by assuming that households derive utility from real money balances. (This money-in-the-utility-function approach is discussed in detail in Chapter 12 and constitutes the simplest way to ensure that fiat money is held by economic agents.) Monopolistic competition in and of itself does not invalidate the classical dichotomy. Indeed, a helicopter drop of money balances simply inflates all nominal variables equi-proportionally and leaves all real variables unchanged.

Money ceases to be a mere veil if prices are sticky. Here the assumption of monopolistic competition is essential because it explicitly recognizes that it is the individual firms (and not some anonymous auctioneer) who are responsible for setting prices in the economy. We study three major approaches under which price stickiness emerges as an equilibrium phenomenon. The menu-cost approach postulates the existence of small costs associated with changing prices. Since profit functions are flat at the top, it may be optimal for an individual firm not to increase its price in the wake of an expansionary (monetary or fiscal) shock and instead to expand its output. Provided labour supply is sufficiently elastic (and there is thus a sufficient degree of real rigidity) small menu costs (a source of nominal rigidity) can rationalize the fixity of both wages and prices in general equilibrium. In the menu-cost

401

The Foundation of Modern Macroeconomics

equilibrium, both fiscal and monetary policy are highly effective and money is not neutral. The Achilles heel of the menu-cost model is that it hinges on a highly elastic labour supply equation, a feature which is not supported by the empirical evidence.

A more pragmatic approach to price stickiness assumes that there are convex costs associated with changing prices. In this approach, the individual firm tries to steer the actual sequence of its price as close as possible to its "ideal" price path which would be attained in the absence of adjustment costs. The presence of adjustment costs ensures that the firm sets its actual price as a weighted average of last period's price and some long-run target price which is explicitly forward looking. At a macroeconomic level, the adjustment cost approach thus provides a microeconomic foundation for the expectations-augmented Phillips curve of Friedman and Phelps.

In the third approach to aggregate price stickiness, the pricing friction is stochastic. Each period of time "nature" draws a signal to the firm which may be a "green light" or a "red light" with given probabilities. These probabilities are the same for all firms in the economy. A firm which has just received a green light can change its price optimally (without adjustment costs) in that period but must maintain that price until the next green light is received. Although this theory differs substantially from the adjustment-cost approach at the microeconomic level, the two approaches give rise to an observationally equivalent macroeconomic pricing equation.

Further Reading

Mankiw and Romer (1991) is a collection of key articles on new Keynesian economics. Also see Gordon (1990) and Benassi, Chirco, and Colombo (1994) for overviews of new Keynesian economics. On monopolistic competition as a foundation for the multiplier, see Ng (1982), Hart (1982), Solow (1986), Blanchard and Kiyotaki (1987), Dixon (1987), Mankiw (1988), and Startz (1989). Recent contributions include Molana and Moutos (1992), Dixon and Lawler (1996), Heijdra and Ligthart (1997), and Heijdra, Ligthart, and van der Ploeg (1998). On the welfare properties of the monopolitically competitive equilibrium, see Mankiw and Whinston (1986). Benassy (199 1 a,b, 1993b), Silvestre (1993), and Matsuyama (1995) give excellent surveys of the early literature.

On price adjustment costs, see Mankiw (1985), Poterba, Rotemberg, and Summers (1986), Parkin (1986), Dixon and Hansen (1999), and Danziger (1999). Levy et al. (1997) present empirical evidence on the size of menu costs in supermarket chains. For the envelope theorem, see Dixit (1990). On the new Keynesian Phillips curve, see Ball, Mankiw, and Romer (1988) and Roberts (1995). The Calvo approach to price stickiness is widely used in monetary economics. See, for example, King and Wolman (1996, 1999), Clarida, Gall, and Gertler (1999), Goodfriend and King (1997), Rotemberg and Woodford (1999), and Yun (1996).

Kiyotaki (1988) and Benassy (1993a) show that under monopolistic competition it may not be optimal for households to have rational expectations. There is a large literature on

multiple equilibria and coordi

.eifer (1986), Diamond an

41989a), and Benhabib and r,

presented by Cooper (1999).7

:nes (1937).