Imperfect Competition and Nominal Rigidities in General Equilibrium: Implications for Macroeconomic Analysis and Policy. Florin O. Bilbiie HEC Paris Business School
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Contents 1 Imperfect competition with flexible prices: business cycles and monetary policy 1.1 The Economy . . . . . . . . . . . . . . . . . . 1.1.1 Households . . . . . . . . . . . . . . . 1.1.2 Firms . . . . . . . . . . . . . . . . . . 1.1.3 Price setting under flexible prices . . . 1.1.4 Flexible Price Equilibrium and market 1.2 Welfare and optimal policy . . . . . . . . . . 1.3 Steady state . . . . . . . . . . . . . . . . . . . 1.4 Loglinearized equilibrium solution . . . . . . 1.5 Determinacy of inflation and price level . . .
implications for . . . . . . . . . . . . . . . . . . . . clearing . . . . . . . . . . . . . . . . . . . .
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2 Sticky prices and the real effects of monetary policy 17 2.1 Why would prices not be flexible? A simple idea . . . . . . . . . 18 2.2 The Calvo model of price-setting . . . . . . . . . . . . . . . . . . 18 2.2.1 Equilibrium: differences with the flexible-price case . . . . 20 2.3 Loglinearized equilibrium . . . . . . . . . . . . . . . . . . . . . . 21 2.3.1 Inflation and relative price dynamics . . . . . . . . . . . . 21 2.3.2 The New Keynesian Phillips Curve and the aggregate supply side . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3.3 The IS curve or aggregate demand side . . . . . . . . . . 24 2.3.4 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . 24 2.3.5 Monetary Policy and Asset Prices . . . . . . . . . . . . . 25
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CONTENTS
CONTENTS
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In this part of the course, we seek to build a model that can be used to analyze monetary policy issues from a positive and normative perspective. On the positive side, we saw that models with flexible prices give little (if any) role to monetary policy in influencing the real allocation (consumption, output, etc.). A model with real rigidities (e.g. a model in which the real wage fails to adjust to clear the labor market) does not have very different implications along this dimension: imagine that money supply increases; nothing prevents the nominal wage and the price level to adjust proportionally (leaving the real wage unchanged), leading to unchanged real money balances. It is then apparent that nominal rigidities (a failure of prices to adjust instantaneously) are needed in order to build a model that gives a meaningful role to monetary policy. The transmission mechanism of monetary policy would then work through changing the incentives of private agents by changing the real interest rate, which is instead achieved by setting the nominal interest rate. On the normative side, we can address issues of optimal monetary policy in a microfounded model, i.e. one that models the optimizing behavior of agents explicitly. We will be able to study the optimal path of the interest rate set by the central bank in order to maximize utility of the representative agent. Since we used the wording ’price setting’, we have already understood that one assumption we need in order to introduce nominal rigidity is market power, or monopolistic competition (recall that in a perfectly competitive world, A Walrasian auctioneer quotes prices that clear the markets, and all agents take prices as given).
2
CONTENTS
Chapter 1
Imperfect competition with flexible prices: implications for business cycles and monetary policy We will work with the cashless nominal model introduced in Chapter 2, but will assume that consumption of the household is made by an aggregate of individual differentiated goods, each good being produced by a monopolistically competitive firm. Goods are imperfect substitutes, and we assume that the elasticity of substitution between any two goods is symmetric and equal to θ. Finally, we assume that there is an infinity of such goods, but the mass of the set of goods does not vary: in other words, no new products are being introduced and no products become obsolete. Some of us (and increasingly many) tend to think that marrying monopolistic competition with a fixed set of varieties is very unappealing, and there is some recent research along these lines1 . But for our purposes, we shall consider that the set of varieties is fixed and we normalize its mass to 1.
1 See
e.g. Bilbiie, Ghironi and Melitz (2007).
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4CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU
1.1 1.1.1
The Economy Households
The aggregator for consumption will be of the CES (constant elasticity of substitution) form first introduced by Dixit and Stiglitz2 : Ct =
1
(Ct (ω))
θ−1 θ
dω
0
θ θ−1
, θ > 1.
Refresher: Recall that elasticity of substitution is measured by:
elasticity of substitution = −
d ln(C (ω) /C (ω )) d ln(C (ω) /C (ω )) = − d ln (M RS (C (ω) , C (ω ))) d ln (Cω /Cω )
where Cω is the derivative of the consumption aggregator with respect to con1 1 θ−1 θ−1 1 3 sumption of good ω , (C (ω)) θ dω (C (ω))− θ . Substituting: 0
elasticity of substitution = −
d ln(C (ω) /C (ω )) 1
d ln (C (ω) /C (ω ))− θ
=θ
The household will as usually maximize the present discounted value of future utility, where we assume that utility within the period takes the form u (Ct ) − v (Lt ) , as in Chapter 2. The problem can be decomposed in two steps. In the first, static allocation problem, the agent decides how to allocate total expenditure for the period t (denoted by Xt = Pt Ct ) among the differentiated goods Ct (ω); the second step is just the problem we solved before: the intertemporal optimization problem (note that this is where the amount of expenditure in period t is being decided). The Static Allocation Problem The static allocation problem is hence: max
Ct (ω)
1
(Ct (ω))
0
θ−1 θ
dω
θ θ−1
subject to
2 In
fact, this is largely a misnomer, since Dixit and Stiglitz consider preferences specified θ Nt θ−1 θ−1 over a time-varying set of goods, Ct = (Ct (ω)) θ dω , where Nt is the number of 0
goods consumed and produced every period which is determined in equilibrium by a free entry condition equating the value of introducing a new good with its cost. See Bilbiie Ghironi and Melitz (2007) if you are interested in these issues. 3 A discplaimer for those mathematically inclined: Mathematically, this is plainly wrong. We are ’differentiating’ over a continuous set with respect to a function, and dC (ω) has in fact zero measure. Try to think of the integral as a sum (and of the goods set as a discrete rather than continuous one), and think of this derivative as the limit of the derivative of the sum when we go to a continuous set (the space between goods becomes arbitrarily smaller).
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1.1. THE ECONOMY
1
Pt (ω) Ct (ω) dω = Xt .
0
where Pt (ω) is the individual price of variety ω and Xt is total nominal expenditure Xt = Pt Ct , where Pt is the welfare-based price index that we need to find. The Lagrangian associated to this problem is
1
(Ct (ω))
θ−1 θ
dω
0
θ θ−1
−
1
Pt (ω) Ct (ω) dω − Xt
0
. There is one optimality condition for each good, and the first order condition for a generic good is:
1
(Ct (ω))
dω
1 θ−1
(Ct (ω))− θ = Pt (ω)
(1.1)
Ct (ω) Ct
− 1θ
= Pt (ω)
(1.2)
θ−1 θ
0
Rewriting:
1
We need to solve for the Lagrange multiplier. To do so, we multiply the first order condition 1.1 by Ct (ω) :
1
(Ct (ω))
θ−1 θ
dω
0
1 θ−1
(Ct (ω))
θ−1 θ
= Pt (ω) Ct (ω)
and integrate over all gods to obtain:
0
1
(Ct (ω))
θ−1 θ
dω
1 θ−1
1
(Ct (ω))
θ−1 θ
dω
0
0
1
(Ct (ω))
1 = Pt (ω) Ct (ω) dω → 0
θ−1 θ
dω
θ θ−1
= Xt
Using the definitions of total consumption and total expenditure we have: Ct = Pt Ct → =
1 Pt
i.e. the Lagrange multiplier is equal to the inverse of the price level. Substituting this into 1.2, we finally obtain the demand function for a generic good: −θ Pt (ω) Ct (ω) = Ct . (1.3) Pt Importantly, we will call PtP(ω) the relative price of good ω in terms of the t numeraire and (minus) θ is the elasticity of demand with respect to this relative price.
6CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU The expression for the welfare-based price index is obtained by plugging the demand function 1.3 into the consumption aggregator:
Ct
=
1
0
Pt (ω) Pt
−θ
Ct
θ−1 θ
θ θ−1
dω
→
θ θ−1 1 1−θ θ−1 1 = Pt Pt (ω) dω →
0
Pt
=
1
1−θ
Pt (ω)
0
dω
1 1−θ
Note that you can obtain exactly the same results by solving the dual problem of minimizing nominal expenditure subject to obtaining a minimum level of consumption. The Dynamic Problem As in the cashless model of Chapter 2, the household will face the budget constraint: (1 + It )−1 Bt+1 + Vt Nt+1 + Pt Ct = Bt + Wt Lt + (Dt + Vt ) Nt , where Dt are nominal profits (dividends) on the shares the household holds in the good-producing firms, Vt is the price of a share and Nt the number of shares the household holds at the beginning of period t : this is a state variable. Bt are discount bonds that promise a unit of currency tomorrow and cost (1 + It )−1 today. Therefore, as we saw earlier in the course, the net return on bonds will be It . Note that this is the NOMINAL return on bonds, i.e. the return in currency units. The household makes a portfolio decision, maximizing the present discounted value of utility of consumption by choosing bond and share holdings. We will solve this problem in nominal terms, assuming that the household chooses the amounts of B and N to hold, not the quantities (case in which we would divide by the price level and we would solve a ’real’ problem). The two methods should give the same solution. ∞ t Bt Dt + Vt Wt Vt −1 Bt+1 max β U + N + L − (1 + I ) − N − v (L ) t t t t+1 t t=0 Bt+1 , Nt+1 Pt Pt Pt Pt Pt
The Euler equations for bonds and shares are: Bt+1 Nt+1 Lt
Pt : (1 + It ) UC (Ct ) = βEt UC (Ct+1 ) Pt+1 Vt Dt+1 + Vt+1 : UC (Ct ) = βEt UC (Ct+1 ) Pt Pt+1 Wt : UC (Ct ) = vL (Lt ) Pt −1
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1.1. THE ECONOMY
The second equation states that the value of a share today in utility terms is equal to the present discounted value of its payoff tomorrow. Forward iteration would lead to expressing the value of a share as a present discounted value of all future dividend streams (we saw this before). We have not talked about firms and about equilibrium yet, but I will anticipate in order to simplify our problem. We have normalized the number of goods consumed and produced to 1, so we expect that by market clearing (household will hold ’all’ shares) we can normalize the number of shares to 1 too, i.e. Nt+1 = Nt = 1. Under this normalization, the budget constraint becomes: −1
(1 + It )
Bt+1 + Pt Ct = Bt + Wt Lt + Dt .
The Euler equation for share will then merely define recursively the value of a share once all the other equilibrium values for consumption, prices and dividends have been obtained. For further use, note that the Fisher parity condition holds: 1 + Rt = (1 + It )
Pt 1 + It = , Pt+1 1 + Πt+1
whereΠt+1 is net inflation and Rt is as before the REAL net return on a REAL bond, one that gives you the right to units of consumption. You would get this by no-arbitrage if you assumed that there were both nominal and real bonds in this economy.
1.1.2
Firms
We assume that there is an infinity of monopolistically competitive firms, each of which produces one of the differentiated consumption goods ω using labor as the only input, employing the production function: Yt (ω) = At F (Lt (ω)) ,
(1.4)
where At is aggregate technology as in the RBC model and F is an increasing and concave production function. Firms take wages as given, and all firms will face the same market nominal wage of Wt . The marginal cost of production is found by minimizing total variable cost: min Wt Lt (ω) s.t. Yt (ω) = At F (Lt (ω))
Lt (ω)
The Lagrangian of this problem is Wt Lt (ω) + Λt [Yt (ω) − At F (Lt (ω))] , where Λt (ω) is the shadow value of relaxing the production constraint, i.e. the marginal cost of producing one unit of the good. The first order condition is: Wt = Λt (ω) At F (Lt (ω)) so nominal marginal cost will be: Λt (ω) =
Wt , At F (Lt (ω))
(1.5)
8CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU where At F (Lt (ω)) is the marginal product of labor. Note that in the special case of constant returns to scale in production (F (Lt (ω)) = 1), marginal cost will be equal across firms: Λt (ω) = Λt = Wt /At .
1.1.3
Price setting under flexible prices
Monopolistic competition implies that firms have price-setting power. Firms will hence choose the price of their good Pt (ω) that maximizes profits Dt (ω) = Pt (ω) Yt (ω) − Wt Lt (ω), subject to the demand function. Firms understand the demand function of the consumer: −θ Pt (ω) Yt (ω) = Yt (1.6) Pt . Suppose that firms are free to reset prices every period. The problem will be: max Pt (ω) Yt (ω) − Wt Lt (ω) s.t. 1.4 and 1.6
Pt (ω)
We can for for Lt (ω) by inverting the production function 1.4, Lt (ω) =
substitute F −1 YtA(ω) and Y t (ω) from 1.6 to obtain: t max Pt (ω)
Pt (ω)
Pt (ω) Pt
−θ
Yt − Wt F
−1
Pt (ω) Pt
−θ
Yt At
The firm understands that its decisions have no effect on aggregate variables at the margin, so the first-order condition is: (1 − θ)
Pt (ω) Pt
−θ
Yt + θ
Pt (ω) =
Pt (ω) Pt
−θ−1
Yt Wt 1
= 0 → At Pt F F −1 Yt (ω) At
θ Wt → θ − 1 At F (.)
Pt (ω) = MΛt (ω) , M ≡
θ >1 θ−1
(1.7)
Therefore, under full price flexibility, price is equal to a constant markup over marginal cost. The relative price of each good will simply be a constant markup over real marginal cost: Pt (ω) Λt (ω) =M Pt Pt
1.1.4
Flexible Price Equilibrium and market clearing
It can be shown that the equilibrium will be unique, but this goes a little beyond our scope in these notes and we shall take it with a grain of salt. We conjecture
1.2. WELFARE AND OPTIMAL POLICY
9
that the unique equilibrium is symmetric, i.e. it involves all firms producing the same quantity of output Yt (ω) = Yt at the same marginal cost Λt (ω) = Λt (hence employing the same labor demand Lt (ω) = Lt ) and setting the same price Pt (ω) = Pt . This is an equilibrium since it satisfies all the equilibrium conditions above. Goods market clearing implies that Y t = Ct Equilibrium in asset markets will imply, as we already discussed, Nt+1 = Nt = 1 and Bt+1 = 0. We take a closer look at the equilibrium in order to understand that under flexible prices, the real allocation is independent of monetary policy (which we have not yet specified!).From the pricing condition 1.7 we have: 1=M
Wt 1 Pt At F (Lt )
Substituting for the real wage from the intratemporal ’labor supply’ condition of the household, we obtain 1 UC (Ct ) =M vL (Lt ) At F (Lt )
(1.8)
Substituting the goods
market clearing condition and the aggregate production −1 Yt function Lt = F At , we obtain: U (Y ) 1
C t = M
. Y Yt vL F −1 Att At F F −1 A t
This equation determines the level of output under flexible prices (which we shall call, following Friedman, the natural rate of output) as a function of exogenous shocks only (in our model, these are only shocks to technology, but we could have introduced shocks to government spending as in the previous chapter, or to preferences - see Woodford, Chapter 3). Importantly, the natural rate of output under flexible prices is determined independently of any monetary factors. This is a neutrality result that will become clearer when studying the loglinearized equilibrium.
1.2
Welfare and optimal policy
Welfare can be analyzed in this model in a standard way, i.e. by assuming that a planner maximizes the representative household’s utility subject to the resource constraints. As in the competitive equilibrium, this too can be split in two stages: a static allocation problem of choosing how much resources to allocate to each individual good and hence how much to consume of each good, given a total amount of consumption and labor, and a ’dynamic’, or aggregate
10CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU one pertaining to the choice between consumption and labor. You will be asked in the homework to solve this problem, but I give you the optimality conditions below. The static problem The optimality conditions you will find are: Ct (ω) = Ct and Lt (ω) = Lt for ∀ω, i.e. the same quantity of labor is allocated to each product and the same quantity of each product is being consumed; this is due to the symmetry of the utility and production functions. The ’dynamic’ or ’aggregate’ problem. Once individual allocations are determined, the last optimality condition dictating the choice between consumption and labor will be: vL (Lt ) = At F (Lt ) , (1.9) uC (Ct ) i.e. as usually, the marginal rate of substitution between consumption and labor has to be equal to the marginal product of labor, i.e. the rate at which labor can be transformed into the consumption good. It is immediately apparent by direct comparison of 1.9 with its competitive equilibrium counterpart 1.8 that the two equilibria differ insofar as M > 1, i.e. there are positive markups for the prices of goods (note the analogy with the distortionary tax inefficiency in the previous chapter). The presence of a markup in the pricing of goods (and not in the pricing of leisure!) introduces a wedge between the marginal rate of substitution and the marginal rate of transformation. The leisure ’good’ is relatively too cheap, so too much of this good is being purchased, too little labor is being supplied and hence too little output is being produced. VERY importantly, note that this distortion is NOT associated with the presence of monopolistic competition (and markups) in goods markets! This is something that misleads incredibly many (and pretty smart) people, do not be among them. The proof of this claim is very simple. Suppose that labor supply is inelastic, then condition 1.8 simply disappears, and optimality implies that out of a fixed labor endowment, a symmetric quantity is allocated to the production of each good. Hence, the symmetric competitive equilibrium is efficient! In the elastic-labor case, the distortion that occurs is related to the absence of a markup in the pricing of leisure. You can see this by noting that efficiency in the elastic-labor case can be restored by subsidizing labor at a rate 1 + τ L that is equal to the inverse of the price markup and financing this by lump-sum taxes. Effectively, this policy induces a ’markup’ for the price of leisure (real wage) by making labor relatively more attractive. The competitive-equilibrium optimality condition becomes: UC (Ct ) M 1 = , vL (Lt ) 1 + τ L At F (Lt )
11
1.3. STEADY STATE
which is equivalent to the planner solution iff 1 + τ L = M.Effectively, this induces a ’markup’ for leisure, making it as expensive as the consumption good - and hence restoring efficiency, consistently with our intuition that efficiency requires that markups be aligned.
1.3
Steady state
In the steady state of this economy (in which we assume that technology is constant and normalize it to 1), the equation resulting by combining the intratemporal optimality condition with the pricing condition (the combination of which delivers 1.8) and the production function pins down steady-state hours4 : UC (F (L)) 1 =M vL (L) F (L)
(1.10)
Once we solved for hours we can solve for all other variables in steady state as we did before. I will emphasize here one difference with respect to the models we have seen up to now, which comes from the fact that there is monopolistic competition, and hence we have positive profits. From the pricing condition 1 t 1 = MW Pt At F (Lt ) , we have that the real wage in steady state is: 1 W = F (L) , P M so the share of labor income in total output is: WL 1 F (L) L ζ F (L) L = = < 1, where ζ ≡ . PY M F (L) M F (L) From the definition of profits, we obtain that the share of profits in steady-state output is: D WL ζ =1− =1− > 0. PY PY M Finally, note that as before the Fisherian parity condition implies that 1+R =
1+I = β −1 1+Π
In much of the remainder we will assume that steady-state inflation is zero, Π = 0. Finally, the share Euler equation implies that the net nominal return on shares in steady state is also: D =1+I V or that the price/earnings ratio is equal to the price of a discount bond. 4 Recall that hours are constant for a separable utility function if and only if u (C) = ln C so that income and substitution effects cancel out. Here we consider the more general case and we abstract from growth considerations (e.g. asuming that technology does not grow).
12CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU
1.4
Loglinearized equilibrium solution
We will denote variables obtained in this flexible-price equilibrium with a star. In the loglinearized equilibrium, the Euler equation for bonds is c∗t = Et c∗t+1 −
1 [it − Et π t+1 ] , γ
where γ = −uCC C/uC > 0 is the coefficient of relative risk aversion (in the RBC model we had implicitly assumed this was 1 since we assumed log utility). The intratemporal optimality condition gives: ϕlt∗ = wt∗ − γc∗t,
(1.11)
where as throughout the notes ϕ = vLL L/vL .Goods market clearing equates consumption with output: c∗t = yt∗ . The loglinearized production function is: yt∗ = at + ζlt∗ , where ζ ≡ FL L/F is the elasticity of F with respect to hours, which is equal to 1 under constant returns to scale (linear production). Real marginal cost is given loglinearizing the
by
real (deflated by Pt ) version of 1.5, denoting λt = Λt Λ W t ln Pt / P and wt = ln W : Pt / P λ∗t = wt∗ − at + ηlt∗ ,
where η ≡ −FLL L/FL > 0 is the elasticity of the marginal product of labor with respect to labor. This will also be the elasticity of labor demand. Since in the symmetric flexible-price equilibrium we have a constant markup rule, real marginal cost will be constant λ∗t = 0, implying (we can think of this as a labor demand): wt∗ = at − ηlt∗ (1.12) Note that under constant returns, we will merely have η = 0, and the usual expression for a horizontal labor demand wt∗ = at . Combining labor demand 1.12 and labor supply 1.11, we obtain: (ϕ + η) lt∗ = at − γc∗t, and further substituting the production function and the goods market equilibrium: ϕ+η ∗ (yt − at ) = at − γyt,∗ ζ so the natural rate of output is: yt∗
=
1+ γ+
ϕ+η ζ ϕ+η ζ
at .
13
1.5. DETERMINACY OF INFLATION AND PRICE LEVEL We find the other flexible-price variables easily, for example labor supply: lt∗ =
1−γ at , ζγ + ϕ + η
Note that for log utility, γ = 1, we would simply have yt∗ = at and lt∗ = 0 - and by now you should be able to say why this is the case. The bottomline is that monetary policy plays no role whatsoever in determining the dynamics of real variables under flexible prices. The real return on a hypothetical real asset under flexible prices will be: 1+ rt∗ = γ Et c∗t+1 − c∗t = 1+
ϕ+η ζ ϕ+η γζ
[Et at+1 − at ]
This is called the Wicksellian interest rate (it should be noted that Friedman developed the concept of a natural rate of output as that value of output that occurs under Wicksell’s natural rate of interest!). Finally, note one often-forgotten implication of a monopolistic competition model, namely its asset pricing implications. Once we have solved for all variables we can solve for asset prices by using the loglinearized version of the Euler equation for shares, something we will turn to when discussing asset pricing implications.
1.5
Determinacy of inflation and price level
Even though nominal variables, including those determined by monetary policy directly, do not affect the real allocation, it is instructive to study how the nominal variables are in fact determined in equilibrium. The Fisherian parity condition is: rt∗ = it − Et πt+1 , where rt∗ has been found above as a function of purely exogenous forces. Suppose that the central bank sets interest rates according to the Taylor rule introduced previously it = φπ π t + εt , this implies that inflation is governed by: φπ πt = Et πt+1 + rt∗ − εt .
(1.13)
Definition 1 The Taylor Principle We say that monetary policy is active (i.e. it satisfies the Taylor principle) if φπ > 1 and passive (Taylor principle is violated) if the opposite holds: φπ < 1. We will distinguish two cases according to this distinction.
14CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU Determinacy under the Taylor Principle In this case equation 1.13 can be solved forward for inflation: ∞ −j−1 ∗ πt = Et j=0 (φπ ) rt+j − εt+j
This is intuitive: present and future discretionary monetary easing (i.e. negative shocks εt+j < 0) or decreases in the ’natural’ rate of interest induce higher inflation today; the more aggressive monetary policy is, the shorter-lived these effects will be. But monetary policy’s role is very limited in that it merely pins down a rate of inflation which has no effect on the real allocation. Indeterminacy under a passive monetary policy (Taylor principle violated) What happens when the Taylor principle is violated, φπ < 1? Equation 1.13 cannot be solved forward, even though πt is a forward-looking variable. I will outline a general method of dealing with indeterminate equilibrium; this method allows us to see exactly why this case is labelled ’equilibrium indeterminacy’, and what can we say about the possible equilibria. I am doing this precisely in the spirit of ’learning techniques by means of examples’ that hopefully characterizes this course. Recall from the revision problems provided that solving rational expectation models implies solving for an expectation function, as well as the variables themselves. A natural way to solve our problem is then to introduce a new variable for this expectation: E / t ≡ Et π t+1 Define the expectation (forecast) error as: η t ≡ πt − E / t−1 = πt − Et−1 πt , indicating how far off the prediction using yesterday’s information set is from the actual, realized value. Using these definitions, we can rewrite our equation as: E / t = φπ E / t−1 + φπ η t − rt∗ + εt (1.14) We can try to solve equation (1.14) backwards (use repeated substitution or lag operators L, or whatever else) to get: E /t =
∞ 1 ∗ (φπ ηt − rt∗ + εt ) = φjπ φπ ηt−j − rt−j + εt−j . 1 − φπ L j=0
(1.15)
But we haven’t really solved for anything this way: expectations E / t are a function of past and present expectation errors ηt−j . The problem is that when φπ < 1 and πt is not a predetermined variable, we have no restrictions on either expectations or expectation errors that we can use so solve our equation.
1.5. DETERMINACY OF INFLATION AND PRICE LEVEL
15
This case is labelled ’equilibrium indeterminacy’ and this label is natural: the ’solution’ (1.15) expresses an endogenous variable, E / t as a function of another endogenous variable, i.e. expectation errors. Under equilibrium indeterminacy, the solution we found for the expectation function (1.15) immediately implies a solution for the actual variable (using the definition of an expectation error): ∞ ∗ πt+1 = ηt+1 + φjπ φπ ηt−j − rt−j + εt−j . (1.16) j=0
This shows that the solution for our variable πt+1 depends on fundamental shocks as well as ’expectation errors’, where the latter have not been pinned down. How many equilibria there are? An infinity, indexed by the expectation errors. Because these expectation errors are not determined, we say that ’sunspots’, shocks that are completely extrinsic to the model, have real effects under indeterminacy. Since there is nothing to pin down expectation errors ηt , let’s assume that it takes the arbitrary (but linear, since the model is linear) form: φπ ηt = ψ (rt∗ − εt ) + st
(1.17)
i.e. that expectation errors are an arbitrary combination of fundamental uncertainty (rt∗ and εt ) and purely non-fundamental uncertainty: sunspots st . Notably, ψ is an arbitrary constant. Picking one particular equilibrium path among the infinite possibilities boils down to: (i) specifying the stochastic properties of st and (ii) picking a value for ψ. This emphasizes two things: i. Indeterminacy introduces a role for non-fundamental, sunspot shocks in influencing equilibrium dynamics. ii. Indeterminacy affects the propagation of fundamental shocks in an arbitrary way dictated by the value of ψ (even when sunspot shocks are absent, st = 0). What we should remember from this discussion is that under passive monetary policy, the level of inflation (and implicitly, of nominal interest rates) is indeterminate. However, this indeterminacy has no implications for the real economy (output, consumption, wages, etc.) precisely due to the neutrality result we just proved. This leads to a natural name for this type of indeterminacy, namely ’nominal indeterminacy’. When prices are sticky, however, we will see that nominal indeterminacy will imply real indeterminacy, because inflation and monetary policy will affect the real allocation. Finally, we should also retain from this discussion that what rules out the effects of sunspots under an active monetary policy is a threat by the monetary authority to put the economy on an explosive path. It is this threat that induces agents choose the fundamental equilibrium as the only equilibrium, rather than equilibria in which sunspots matter. This emphasizes the role of forward-looking expectations in shaping outcomes in the New Keynesian model, differently from older Keynesian models.
16CHAPTER 1. IMPERFECT COMPETITION WITH FLEXIBLE PRICES: IMPLICATIONS FOR BU
Chapter 2
Sticky prices and the real effects of monetary policy It is clear that in order to have real effect of monetary policy, we need to introduce some form of imperfect price adjustment. We will not have much to say in this course about why exactly do prices not adjust; we will simply assume that they do not, and study the implications of this assumption at the aggregate level. An enormous literature exists that tries to justify nominal rigidities, and I recommend those interested to start by reading the relevant chapter in Walsh’s textbook. For our purposes, I will just introduce the simple idea that the cost of not changing the price when aggregate conditions change can be justified by the presence of small costs of changing prices, i.e. ’menu costs’. I will then cover a simple model of staggered price setting originally used by Calvo; this model has become the workhorse of monetary economics in academia and policymaking institutions. The framework is often labelled ’New Keynesian’, but also ’Neo Wicksellian’, ’New neoclassical synthesis’, Neomonetarist, and so on. This heterogeneity in labels reflects its heterodoxy. In a nutshell, what generates neutrality of monetary policy under flexible prices is the irrelevance of monetary factors (i.e. of nominal variables such as inflation and nominal interest rates) for aggregate supply and aggregate demand. Sticky prices break up this neutrality by making aggregate supply dependent upon inflation. By market clearing, inflation will hence influence aggregate demand too, so there is a role for monetary policy to influence aggregate demand (by changing setting nominal interest rates, which instead affect the Euler equation for consumption). In a sense, neither the aggregate supply block nor the aggregate demand block will be independent of monetary policy. 17
18CHAPTER 2. STICKY PRICES AND THE REAL EFFECTS OF MONETARY POLICY
2.1
Why would prices not be flexible? A simple idea
To justify nominal rigidities, consider one of our monopolistically competitive firms producing ω, whose profit-maximizing price depends on the demand it faces. Suppose it starts from a given price P¯ (ω) and next period the demand it faces increases, making the new optimal (i.e., profit-maximizing) price be P ∗ (ω)- now the question is whether it would be optimal or not for the firm to adjust its price. If it does not adjust (it stays at P¯ rather than move to P ∗ ), we can calculate its profit loss by taking the Taylor expansion of the difference of profits under the two scenarios: 1 2 D P¯ (ω) −D (P ∗ (ω)) = D (P ∗ (ω)) P¯ (ω) − P ∗ (ω) + D (P ∗ (ω)) P¯ (ω) − P ∗ (ω) 2 (2.1) Note that by the Envelope Theorem D (P ∗ (ω)) = 0 (P ∗ (ω) is the profitmaximizing price) so that 2 1 D (P ∗ (ω)) − D P¯ (ω) = − D (P ∗ (ω)) P ∗ (ω) − P¯ (ω) 2
(2.2)
and the profit loss is second order. Hence, small menu costs of price adjustment are enough to prevent the firm from adjusting its price. Note that: 1. This does not hold under perfect competition: if prices do not adjust when demand increases, there would be excess demand, and the firm could increase the price and sell the same output as before, making a first order profit gain. But with imperfect competition, price exceeds marginal cost; hence firms are better off if they can sell more at the prevailing price. Because as we have seen the gain from adjusting is second order, a small menu cost can be enough to prevent adjustment, and at the same time to make firms produce more. 2. Although the loss from not adjusting under imperfect competition is second order, the macroeconomic and welfare effects can be of first order1 .
2.2
The Calvo model of price-setting
In order to introduce staggered price-setting, we follow Calvo (1983) and Yun (1996) and assume that rather than being perfectly able to readjust the price every period, firms face a particular type of friction. Namely, the opportunity to readjust the price will follow a Poisson process: with probability α, firms will not be able to readjust the price and will continue charging the same price; with probability 1 − α, firms will be able to reoptimize. Since the law of large numbers holds, α will also be the fraction of goods prices that remain unchanged 1 This could happen e.g. for ’aggregate demand externalities’ with imperfect competition, as in Blanchard and Kiyotaki (1987).
2.2. THE CALVO MODEL OF PRICE-SETTING
19
each period, while new prices are chosen for the other 1 − α of the goods. For simplicity, the probability that any given price will be adjusted in any given period is assumed to be independent of (i) the time elapsed since the price was set, and (ii) the current price. These assumptions are clearly unrealistic, but they make the problem very tractable since they greatly reduce the size of the state space required to characterize the dynamics. Note that the average −1 duration of a price will be (1 − α) . Since every firm that chooses a new price for its good in period t faces exactly the same decision problem, the optimal price Pto (ω) is the same for each of them, and so Pto (ω) = Pto in equilibrium (all prices that are chosen in period t have the common value Pto ) . The remaining fraction α of prices charged in period t are simply a subset of the prices charged in period t−1, with each price appearing in the period t distribution of unchanged prices with the same relative frequency as in the period t − 1 price distribution. (For this last argument it is crucial that each price has an equal probability of being adjusted in a given period.) Then the welfare-based price index in period t is given by: 1−θ Pt1−θ = (1 − α) (Pto )1−θ + αPt−1
(2.3)
Therefore, in order to determine the dynamics of this aggregate price index we only need to know its initial value and the optimal price Pto chosen by those producers who can reset prices in a given period t, price that is common to all producers. Note that the index 2.3 can equivalently be written after dividing 1−θ through by Pt−1 as (Πt = Pt /Pt−1 is the gross inflation rate): o 1−θ Pt Πt1−θ = (1 − α) + α, (2.4) Pt−1 Inflation is generated by price setting in period t (pursued by firms who can adjust their prices) differently from the average price level in the previous period. We turn next to the determinants of this optimal price. A generic firm that changes its price in period t will choose the price that maximizes the value of the firm, more precisely the present discounted value of future profits, where profits are discounted at a ’special’ rate that needs to take into account that the chosen price is expected to be in place for more than one period. Specifically, the generic firm correctly anticipates that its price set today, at time t, will remain unchanged next period with probability α, and generally in period t + s with probability αs (the sum of these probabilities gives the average duration of a price, (1 − α)−1 ). Therefore, the firm solves (maximizes the discounted sum of future nominal profits, hence using the relevant stochastic discount factor Qt,t+s used as pricing kernel for nominal payoffs): ∞ s o −1 Yt,t+s max Et (α Qt,t+s Pt Yt,t+s − Wt F Pto At+s s=0
subject to the demand equation at t+s, conditional upon price set s periods in advance:
20CHAPTER 2. STICKY PRICES AND THE REAL EFFECTS OF MONETARY POLICY
Yt,t+s =
Pto Pt+s
−θ
Yt+s .
(2.5)
We solve the problem by substituting the demand into the objective function and taking the derivative with respect to Pto to obtain: Et
∞ (αs Qt,t+s Yt,t+s Pto − MΛot,t+s = 0.
(2.6)
s=0
where Λt,t+s is the marginal cost of producing an extra unit of output at time t + s from the standpoint of time t (i.e. for a firm setting price at t). Note that in the flex-price limit α → 0, we have the previous result derived under flexible prices: Pto = MΛot,t+s = MΛt , price is a constant markup over marginal cost. This markup is hence the desired, frictionless markup of the firm.
2.2.1
Equilibrium: differences with the flexible-price case
In the Calvo model, all firms setting prices at time t will make identical choices, and all firms not setting prices but charging last period’s price will also make identical choices. However, these two classes of firms (which coexist at any given t) make very different choices; indeed, at every time t there is dispersion in relative prices. This affects the real allocation as follows. The goods market clearing condition stipulates that total consumption has to be equal to total production. However, to find an aggregate production function, we need to aggregate carefully. Combining demand for a generic good with the production function for the same good (i.e. imposing market clearing good-by-good), we have: −θ −θ Pt (ω) P (ω) C t t At F (Lt (ω)) = Ct → Lt (ω) = F −1 Pt Pt At Integrating over goods and imposing labor market and aggregate good market clearing: −θ Pt (ω) Yt −1 Lt = F dω Pt At For illustrative purposes, we concentrate on the case whereby F (L) = Lζ , so this becomes: ζ1 − ζθ Yt Pt (ω) Lt = dω At Pt We can hence write a modified aggregate production function of the form: Yt =
At Lζt , ∆t
21
2.3. LOGLINEARIZED EQUILIBRIUM where2 ∆t ≡
Pt (ω) Pt
− θζ
dω
ζ
.
is a measure of dispersion of relative prices. In the flexible price limit, there is no dispersion in relative prices, and ∆∗t = 1. In that case (and in that case ONLY) ζ is the aggregate production given by Yt∗ = At (L∗t ) . In general, relative price dispersion reduces the output that can be produced with a given level of labor, and hence will create a deviation from the optimum. In that sense, relative price dispersion is a measure of distortions in this economy in a sense that will be made explicit below. Note, however, that up to a first-order, linear approximation around a zeroinflation steady state, price dispersion will be zero, so the linearized production function in the sticky-price case will still be: yt = at + ζlt . A proof of this claim is in the Appendix.
2.3
Loglinearized equilibrium
We consider a zero-inflation steady state around which we will loglinearize our equilibrium conditions. It is easy to show that the steady-state of he stickyprice economy is then identical to that of the flexible-price economy studied in the previous chapter. Indeed, some of the equilibrium conditions which do not pertain to the pricing decision will also be unaffected. What is crucial different under sticky prices is the dynamics of the inflation rate (and of the optimally chose price), to which we now turn.
2.3.1
Inflation and relative price dynamics
Loglinearizing the price index around a steady-state with zero inflation, Π = 1 (in such a steady state Pto = Pt = Pt−1 ), we obtain a relationship between the inflation rate and the relative price of a producer setting its price in period t: απt = (1 − α) (pot − pt )
(2.7)
Loglinearization of 2.6 around a zero-inflation steady state delivers pot = (1 − αβ) Et 2 For
∞
s=0
(αβ)s λot,t+s + pt+s ,
a general F, price dispersion would be defined by: At Pt (ω) −θ Yt F −1 dω. ∆t ≡ Yt Pt At
(2.8)
22CHAPTER 2. STICKY PRICES AND THE REAL EFFECTS OF MONETARY POLICY where λot,t+s is the log-deviation of the REAL marginal cost Λot,t+s /Pt+s at time t + s of a firm setting its price at t. As 2.8 makes clear, the deviation of the optimal price from its (constant-markup) steady-state level is induced by future movements in nominal marginal cost. In that sense, the optimal price is a markup over present and future marginal costs, where the weights of future marginal cost are determined by the probability that the chosen price will still be in place in the future. We can obtain dynamics in terms of aggregate, economy-wide variables only once we eliminate the two firm-specific variables: pot (by combining 2.7 and 2.8), and the firm-specific real marginal cost λot,t+s . Average real marginal cost λt+s will in general be different from the firm-specific marginal cost λt,t+s . We define average marginal cost as Λt /Pt = Wt /Pt At F (Lt ) , which in loglinearized form is λt = wt − at + ηlt . Combining this with the loglinearized production function yt = at + ζlt : λt+s = wt+s − 1 + ηζ −1 at+s + ηζ −1 yt+s The firm-specific marginal cost is instead: o λot,t+s = wt+s − 1 + ηζ −1 at+s + ηζ −1 yt,t+s η o yt,t+s − yt+s = λt+s + ζ θη o = λt+s − (p − pt+s ) ζ t
(2.9)
Substituting in the loglinearized pricing equation 2.8: pot = (1 − αβ) Subtracting pt : pot − pt
∞ ∞ ζ Et (αβ)s λt+s + (1 − αβ) Et (αβ)s pt+s , ζ + θη s=0 s=0
∞ ∞ ζ (αβ)s λt+s + (αβ)s Et π t+s Et ζ + θη s=0 s=1 ζ = αβEt πt+1 + (1 − αβ) λt + αβ Et pot+1 − Et pt+1 ζ + θη
= (1 − αβ)
Using 2.7 in this equation to eliminate the firm-specific, optimal relative prices, we obtain what is called the New Keynesian Phillips Curve: πt where ψ
= βEt πt+1 + ψλt , (1 − αβ) (1 − α) ζ = α ζ + θη
(2.10)
This equation links current inflation with future inflation and movements in average real marginal cost; since β < 1 we can solve this forward to obtain current inflation as a present discounted value of future deviations of average real marginal cost from its steady-state value: ∞ s πt = ψ β Et λt+s . s=0
2.3. LOGLINEARIZED EQUILIBRIUM
23
This clearly emphasizes the forward-looking nature of price-setting and the forward -looking nature of inflation. Inflation in this model results from attempts of firms who can adjust prices to set prices that are above the average price level pot > pt (recall the relationship between pot − pt and π t ) when they expect real marginal cost to be on average above the steady-state, flexible-price level. Equivalently, we can define the average gross markup Mt as the inverse of average real marginal cost, which in loglinearized form yields (where we use ln Mt µt ): µt πt
= −λt , implying: = βEt π t+1 − ψµt .
Inflation results from attempts by firms who can set prices to set prices above the average price level in response to present and future downward of the markup with respect to its frictionless, desired level of µ = M−1 obtained under flexible prices.
2.3.2
The New Keynesian Phillips Curve and the aggregate supply side
To gain further insight into inflation dynamics, we try to express average marginal cost as a function of the level of economic activity (output). λt = wt − 1 + ηζ −1 at + ηζ −1 yt .
From the loglinearized intratemporal optimality condition and aggregate production function we have: wt
so:
= ϕlt + γct ϕ ϕ = + γ yt − at , ζ ζ
λt = γ + (η + ϕ) ζ −1 yt − 1 + (η + ϕ) ζ −1 at ,
(2.11)
Recall that in the flexible-price equilibrium, λ∗t = 0, so the marginal cost schedule is: 0 = γ + (η + ϕ) ζ −1 yt∗ − 1 + (η + ϕ) ζ −1 at , (2.12)
which is indeed the equation we used to obtain the natural level of output yt∗ . Subtracting 2.12 from 2.11, we obtain: = χ (yt − yt∗ ) , where η+ϕ χ = γ+ ζ
λt
(2.13)
This equation emphasizes that movements in average real marginal cost (or equivalently, in average markup) are related to the output gap, i.e. deviations
24CHAPTER 2. STICKY PRICES AND THE REAL EFFECTS OF MONETARY POLICY of output from its flexible-price, natural level. Substituting 2.13 in 2.10, and defining output gap xt ≡ yt − yt∗ we obtain: πt = βEt πt+1 + ψχxt .
(2.14)
It is this equation that is usually labelled the ’New Keynesian Phillips Curve’, or the aggregate supply relationship (since it is derived using only equilibrium conditions pertaining to the ’supply’, production side).
2.3.3
The IS curve or aggregate demand side
Recall that the Euler equation written in terms of output is: yt = Et yt+1 −
1 [it − Et πt+1 ] , γ
and its flex-price equivalent (which implicitly defines the natural, Wicksellian interest rate) is: 1 ∗ yt∗ = Et yt+1 − rt∗ γ Subtracting and using the definition of output gap: xt = Et xt+1 −
1 [it − Et πt+1 − rt∗ ] . γ
(2.15)
This equation relates deviations of aggregate demand from its potential xt to its future value and to deviations of the ex-ante real interest rate it − Et πt+1 from its natural level rt∗ , where γ1 is the elasticity of aggregate demand to interest rates.
2.3.4
Monetary policy
Lets us step back for a moment and note that we need to solve for three endogenous variables: output gap, inflation and nominal interest rates. We have two equations (the New Keynesian Phillips Curve 2.14 and the IS curve 2.15), so we need one more. We are hence in a position to close the model by specifying a monetary policy rule, i.e. an equation governing interest-rate setting. Recall that the instrument of monetary policy is the nominal interest rate it . The transmission mechanism of monetary policy goes through influencing the real interest rate it − Et π t+1 by setting the nominal interest rate, which instead influences aggregate demand today relative to tomorrow via the IS curve 2.15, which instead affects inflation via the New Keynesian Phillips Curve 2.14. There are two possibilities: one is to specify an interest rate rule, or a Taylor rule, as we have done before, and another one is to solve for an ’optimal’ path of it that maximizes the welfare of the representative consumer. For the moment we will concentrate on the former option.
2.3. LOGLINEARIZED EQUILIBRIUM
25
Consider a simple modified Taylor rule that specifies interest rates as a function of expected inflation (rather than realized inflation, as previously): it = φπ Et π t+1 + εt .
(2.16)
I take this example because it captures the idea that central banks respond to expected inflation from the viewpoint of time t, and hence have a larger information set than merely the current rate of inflation. This rule (indeed, a modified version of it studied below) has been shown to be a good description of monetary policy in developed countries. Output and inflation stabilization In this version of the model there is no tradeoff between output and inflation stabilization. To see this, note that the flexible-price allocation can be implemented as long as the central bank manipulates nominal interest in order to perfectly match movements in the Wicksellian interest rate: it − Et πt+1 = rt∗ . Substituting this into 2.14 and 2.15 we obtain xt = 0 and πt = 0. However, this is only one of many possible equilibria. Indeed, while matching movements in the natural rate of interest is a necessary condition for implementing price stability, this is not a sufficient condition. It turns out that the additional necessary condition is the Taylor principle, i.e. φπ > 1. This brings us (back) to the discussion of equilibrium determinacy. Real (in)determinacy and the Taylor Principle TO BE COMPLETED
2.3.5
Monetary Policy and Asset Prices
TO BE COMPLETED
26CHAPTER 2. STICKY PRICES AND THE REAL EFFECTS OF MONETARY POLICY
Bibliography [1] Bilbiie, F. O., Ghironi, F. and Melitz, M 2007, Monetary Policy and Business Cycles with Endogenous Entry and Product Variety, NBER Macroeconomics Annual, vol. 22 [2] Christiano, L., M. Eichenbaum and Evans (2005): ”Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy“, Journal of Political Economy, vol. 113, no. 1, 1-46 [3] Gali, J., 2007 ’Monetary Policy, Inflation, and the Business Cycle’ Chapter 3, Mimeo [4] Walsh, C. "Monetary Theory and Policy" [5] Woodford, M., (2003), Interest and prices: foundations of a theory of monetary policy, Princeton University Press, Princeton, NJ.
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