Notes by KS  /  last changed Mon Nov 26 17:43:10 EST 2012  /  Private, Not For Distribution

Black font is a paraphrase of Gali 08, blue is a direct quote, red is my comment



Gali 2008: "Monetary Policy, Inflation, and the Business Cycle"

Chapter 1 Introduction

1.1 Real Business Cycle (RBC) model

The RBC model "firmly established the use of DSGE models as a central tool" for macro. The DSGE framework developed in the 80's [Kydland & Prescott 82, Prescott 86].

Methodologically: "Behavioral equations describing aggregate variables were thus replaced by first-order conditions of intertemporal problems facing consumers and firms". "Rational expectations" replaces "ad hoc assumptions on the formation of expectations".

Conceptually:

• Business cycles are efficient and seek equilibrium in the presence of technology shocks, and hence, in opposition to Keynes, stabilization policies are not necessary and may even be counterproductive
• Technology shocks are the cause of business cycles
• The monetary factor is not modeled and it is not necessary to model it to explain business cycles

Cooley & Hansen 89 add a monetary sector to this RBC model, and this results in the classical monetary model, treated in chapter 2. It was influential in academic circles but not used by banks---because it finds that central banks don't have the "power to influence output and employment, at least in the short run" [references Friedman & Schwartz 1963 for empirical support]. Mentions the "Friedman Rule", which is to keep the nominal rate zero. Conclusion was that the classical model is incomplete.

1.2 The new Keynesian model

The New Keynesian (NK) model developed in the late 70's, parallel to the RBC theory, and then incorporated the DSGE framework.

In their simplest forms, both RBC and NK assume

• "An infinitely-lived representative household that seeks to maximize the utility from consumption and leisure, subject to an intertemporal budget constraint"
• "A large number of firms with access to an identical technology, subject to exogenous random shifts"

Although endogenous capital accumulation is a key element in RBC theory and not in the canonical versions of the NK theory, "it is easy to incorporate and is a common feature of medium-scale versions [of NK]."

The major and important differences between the resulting NK theory and the RBC model are

• Monopolistic competition. Prices "are set by private agents to maximize their objectives, as opposed to being determined by anonymous Walrasian auctioneer to clear all (competitive) markets at once." That is, the markets are more like posted-price than call-auction.
• Nominal rigidities. There are constraints on the frequency of price changes, or there may be costs associated with adjusting prices, and similar frictions apply to wages.
• Short-run non-neutrality of monetary policy. "As a consequence of the presence of nominal rigidities, changes in short term interest rates...are not matched by one-for-one changes in expected inflation, thus leading to variations in real interest rates. This leads to adjustments in output and employment, but in the long run the system reverts back to its natural equilibrium."

In this NK framework, response to shocks is generally inefficient. And, "...the non-neutrality of monetary policy...makes room for potentially welfare-enhancing interventions by the monetary authority in order to minimize the existing distortions." Also, the NK models are arguably immune from the Lucas Critique.

1.2.1 Evidence of nominal rigidities and policy non-neutralities

Two distinctive features and essential ingredients of the NK models require empirical evidence: nominal rigidities and the real effects of monetary policy.

1.2.1.1 Evidence of nominal rigidities

References empirical studies based on micro data of price and wage rigidities. The evidence points to price durations in the range 6-12 months.

1.2.1.2 Evidence of monetary policy non-neutralities

Gali says establishing this evidence is more difficult than establishing rigidities. I would add that it's more subtle, interesting, and problematic, given that it involves more complicated statistical inference.

This subject is more difficult. The crux of the matter is that in general, policy depends on measurable economic variables and vice-versa. The trick is to disentangle cause and effect "...by identifying changes in policy that are autonomous, i.e., that is, not the result of the central bank's response to movements in other variables."

Recent literature is cited, by authors such as Christiano, Eichenbaum, & Evans 99 [some graphs reproduced]; Sims 92; Gali 92; Bernanke & Mihov 98; and Uhlig 05; Peersman & Smets 03; and Romer & Romer 89.

1.3 Organization of book

• Chapt 2 Classical monetary model -- efficiency of equilibrium allocation and neutrality of monetary policy
• Chapt 3 NK model -- added product differentiation, monopolistic competition, staggered price setting -- labor markets still competitive -- equilibrium includes the NK Phillips curve, a dynamic IS equation, and a description of monetary policy -- sticky prices shown to make monetary policy non-neutral.
• Chapt 4 Normative discussion of monetary policy -- simple rules -- Taylor rule and a constant growth rule.
• Chapt 5 Addresses the criticism of optimal (NK) policy in Chapt 4 that there is no tradeoff between inflation stabilization and output gap stabilization
• Chapt 6 Extends NK by introducing imperfect competition and staggered nominal wage setting in labor markets
• Chapt 7 Develops open economy version of NK
• Chapt 8 Reviews general lessons, describes briefly extensions

Chapter 2 A classical monetary model (CMM)

This model is characterized by perfect competition and full flexible (not sticky) prices. Money plays a very limited role---in §§1-4 it is just the unit of account, and in §5 it generates utility to households.

2.1 Households

The representative household seeks to maximize the utility

(1)      E₀ Σ β^t U(C_t, N_t)

where C is consumption and N is labor (number of hours, but can be considered continuous), and the sum is from 0 to ∞ unless otherwise mentioned. The utility function U is assumed increasing in C and nondecreasing in N. The second derivatives of U (marginal utilities) are assumed nonincreasing in C and N. Below, U_t denotes U(C_t, N_t).

E_t is the expectation operator at time t. It seems that the system will be driven exogenously with random variables, in particular through the technology coefficient that determines production. The notation E_t[x] indicates the expectation of x, given that all random variables for all periods up to and including t have already been chosen. Thus, variables at time t are not random and are constant with respect to E_t. That is, E_t[x_t] = x_t. This becomes important in what follows.

The household's maximization of U is subject to the flow budget constraint

(2)      P_tC_t + Q_tB_t    ≤    B_t−1 + W_tN_t − T_t

where P is the price of CG, W is the nominal wage. B is the quantity of one-period nominally riskless bonds, purchased in period t and maturing in period t+1. Each bond (a discount bond) costs Q_t at time t and pays a unit of money at maturity. T is lump-sum additions of subtractions to income, so it can include taxes and dividends. The household knows P, Q, and W at time t when deciding on C, N at time t.

I think the sign in the T_t term should be "+", as it is in chapter 3, see below. That is, the convention is that T_t is an addition to income.

How is B determined? If the flow constraint were an equality constraint, it would be determined from the other variables as the one-period, discounted bonds bought with unspent money (positive B) or as loans (negative B). What is the interest rate on such loans? Can we assume that the loans are one-period and discounted at the same effective rate as the one-period discounted bonds that would be bought?

Notice that the income (RHS) at time t is determined by the choice of labor N at time t, along with the choice of consumption. That is, there is no delay between the choice of labor and the resultant household income. The bond maturity of one time step ties everything to the one time scale, with no allowance (in this model) for smoothing or delays.

To me it seems we should take the budget constraint (2) as an equality, which I will do in these notes unless I say otherwise.

There is also a "solvency constraint", that the limit as t goes to infinity of the expected value of B is nonnegative. This prevents the accumulation of debt.

2.1.1 Optimal C and N

First, at a fixed time t, vary C and N while respecting the budget constraint, and consider everything else in the utility function fixed. Stationarity and the budget constraint demand that

     U_c,t dC + U_n,t dN = 0

where the second subscripts on U_t denote partial derivatives, and

     P_tdC = W_tdN

This leads to Gali's first necessary condition

(4)      −U_n,t/U_c,t = W_t/P_t

Notice that if U is separable in C and N, U_c is a function of C alone, U_n is a function of N alone, and (4) allows us, in principle, to find N(C) (at time t for every t). For the simple power law utility coming soon, the calculation is simple. It is enough then to look for C. (An example with nonseparable utility is given in §2.5.2.)

An intertemporal condition is next. Suppose we consider reallocating consumption between periods t and t+1, keeping everything else fixed. To balance an increase in consumption at time t+1 we must consume less at time t, investing the savings in one-period bonds at the discount rate Q, so there is the condition

     P_t+1dC_t+1 = −P_tdC_t/Q_t

The RHS is the number of bonds we can buy at time t with P_tdC_t units of money, and each is worth one unit of money at time t+1.

Note: This is similar to the budget constraint above but is intertemporal. Therefore a decrease in consumption at time t in general affects P_t+1. If this is taken into account it leads to the condition

     P^*_t+1dC_t+1 = −P_tdC_t/Q_t

where P^*_t+1 is the price of CG at time t+1 when consumption at time t is perturbed by dC_t. But this agrees with P_t+1dC_t+1 to first order in dC_t.

The first-order stationarity condition is

(4-intertemporal)      U_c,tdC_t + βE_t[U_c,t+1dC_t+1] = 0

where the expectation is over the random variables at time t, that in general affect utility at time t+1. Combining this with the intertemporal budget constraint gives Gali's second condition

(5)      Q_t = βE_t[(U_c,t+1/U_c,t)(P_t/P_t+1)]

The following separable form for U, with the required nonincreasing marginal utility, is assumed in "much of what follows":

     U(C_t, N_t) = C_t^1−σ/(1−σ) − N_t^1+φ/(1+φ)

The optimality conditions (4) and (5) can be solved explicitly with this form, yielding Gali's (6), which is an explicit intratemporal relation between C and N; and (7), which involves an intertemporal expectation.

(6)      W_t/P_t = C_t^σN_t^φ

(7)      Q_t = βE_t[(C_t+1/C_t)^−σ(P_t/P_t+1)] = E_t[ β(C_t+1/C_t)^−σ(P_t/P_t+1)]

(β is a constant.) Using lower-case letters to denote natural logs of variables, yields the nice log-linear form for the optimality condition (6):

(8)      w − p = σc + φn

We'll use natural logs throughout. This can be interpreted as a competitive labor supply schedule, determining the "quantity of labor supplied as a function of the real wage, given the marginal utility of consumption (which under the assumptions is a function of consumption only)".

This can be written as

(8a)      n = (w − p)/φ − σc/φ

which means that labor increases with the difference between wage and CG price (called the "real wage" below); and decreases with consumption. Question: are these implications consistent with intuition?

The second optimality condition, Gali's (7), involves an intertemporal expectation, and requires some approximations to write in a simple form in terms of readily interpretable quantities. This is where things start to get interesting, because one approximation is around a "steady state with constant rates of inflation and consumption growth". Gali gives details of the expansion around the steady state in Appendix 2.1. I'll go slowly here.

First, the bond discount Q = 1/(1 + yield), so we can define the interest rate as −log Q, since

     i ≡ −log Q = log(1 + yield) ≈ yield

for small yield.

I'll call this Approximation 1, which might conceivably cause a problem if the interest rate is high.

Taking −log of (7) gives us

(7a)      i_t = − log { E_t[β(C_t+1/C_t)^−σ(P_t/P_t+1)] }

Now define

     ρ ≡ −log β

     Δc_t+1 = c_t+1 − c_t

and inflation (of log prices) as

     π_t+1 = p_t+1 − p_t

and rewrite (7a) as

(7b)      i_t = − log { E_t[ exp( −ρ − σΔc_t+1 − π_t+1) ] }

and further, raising e to this power,

(7c)      1 = exp( i_t )E_t[ exp( −ρ − σΔc_t+1 − π_t+1) ]

Gali starts Appendix 2.1 with

(44)      1 = E_t[ exp( i_t − ρ − σΔc_t+1 − π_t+1) ]

which is described as a rewrite of (7).

Slipping the interest factor inside the expectation is OK here because i_t is determined at time t, and is therefore a constant with respect to the expectation E_t, as noted above. Actually, i_t is not even a random variable if the interest rate is determined exogenously, but becomes a random variable when the central policy depends on random system variables. Were it not for this transparency the derivation would be vulnerable to the Lucas critique.

To see that the exponent in (44) is close to zero, assume a hypothetical perfect-foresight steady state with constant inflation π and constant growth γ, so that the interest rate can be written

     i = ρ + π + σγ

Substituting for ρ in (44), the exponent can be written

     (i_t − i) − σ(Δc_t+1 − γ) − (π_t+1 − π)

This is close to zero by the definition of what is meant by a steady state, with γ interpreted as the steady-state increase in (log) consumption. We are therefore justified in approximating the exponential in (44) with the first two terms in its Taylor series, exp(x) ≈ 1 + x. Using this and the fact that E_t[x_t] = x_t in (44) yields

(9)      c_t = E_t[c_t+1] − (1/σ)(i_t − E_t[π_t+1] − ρ)

This is how Gali writes it. I'll call this two-term Taylor series Approximation 2. In the general, the procedure of (1) taking logs and (2) using "a first-order Taylor series expansion about a point (usually a steady state)" is called "log linearization" (Eric Sims, Notes, Graduate Macro II, Spring 2010, Notre Dame). It's a standard tool in modern macro, especially in formalisms like the NKM.

In the framework so far the household has no explicit motive motive for holding money balances. In some cases Gali will "postulate a demand for real balances" given by

(10)      m_t − p_t = y_t − ηi_t

"A money demand equation similar to (10) can be derived under a variety of assumptions. For instance, in §2.5 it is derived as an optimality condition for the household when money balances yield utility."

This is an equation in log variables, what Gali calls log-linear form. m_t is described in §2.5 as representing the (log of) "holdings of money in period t". Therefore m_t − p_t = log(M_t/P_t) is thought of as the postulated real demand for money, being normalized by price. y is production, defined below, which is equal to c in equilibrium, and η ≥ 0 "denotes the interest semi-elasticity of money demand". This can be interpreted as meaning that the demand for money balances (which yield no interest) varies inversely with the interest rate. This is because keeping a cash balance costs the interest that would be lost by investing in bonds. This demand function for real real balances is derived in §2.5.1 assuming a separable household utility function that includes dependence on m_t.

2.2 Firms

A representative firm is assumed with the production function (at time t)

(11)      Y_t = A_tN_t^{1 − α}

Here the absence of capital as a factor of production is a significant departure from any of our models, and it will be interesting to see exactly how capital has been or could be incorporated in this and the following models in the general DSGE framework. This extension is discussed in §8.1 ("Extensions"). This is also the point at which Gali introduces an explicit random variable, A, which is assumed to "evolve exogenously".

"Each period the firm maximizes profit", with the price of CG and wage considered given:

(12)      P_tY_t − W_tN_t

And here is another departure, the firm's maximization of profit---in contrast with trying to maintain an optimal split between labor and capital with a budget established by income. Question: are these two principles of action in any way equivalent?

It's simple enough to substitute (11) in (12) and set the derivative with respect to N to zero. Gali interprets the result as the condition when the marginal cost of hiring labor to produce a unit of CG, W/Y_N, is equal to P.

(13)      W_t/P_t = A(1 − α)A_tN_t^{− α}

"In log-linear terms"

(14)      w_t − p_t = a_t − αn_t + log(1 − α)

This can be interpreted as a labor demand schedule.

This is a companion to the labor supply schedule (8). Gali calls the LHS, w − p, the "real wage".

2.3 Equilibrium

"The baseline model abstracts away from aggregate demand components like investment, government purchases, or net exports. Accordingly the goods market clearing condition is given by"

(15)      y_t = c_t

Combining the (intratemporal) household and firm optimality conditions (8) and (14) with the equilibrium condition (5) and the log-linear production function

(16)      y_t = a_t + (1 − α)n_t

yields the equilibrium levels of employment and output:

(17)      n_t^equil = ψ_naa_t + ϑ_n
(18)      y_t^equil = ψ_yaa_t + ϑ_y

where the constants ψ and ϑ are simple functions of α, σ, and φ. There is no general assumption of stationarity, so the expectations can depend on t. It turns out that in equilibrium log output is an increasing function of the technology constant because ψ_ya > 0.

I checked this algebra.

Note that the equilibrium presently computed does not depend on the investment discount Q (or i ≡ −log Q), or household utility discount β (or ρ ≡ −log β).

We can now use the intertemporal condition (9) to derive the "implied real interest rate", defined by

(21, rearranged and used here as a definition, not the "Fisherian equation")      r_t ≡ i_t − E_t[π_t+1]

This makes sense because i_t is the interest rate stemming from the bond discount rate, and π_t+1 is the increase in CG price from period t to period t+1, a measure of inflation. This equation defines the real interest rate in this development and is called the Fisher equation; it's actually an approximation valid for small rates, see the wikipedia entry for the Fisher equation. The exact equation is actually (1 + r_t) = (1 + i_t)/(1 + E_t[π_t+1]), because the value one period in the future is increased at the nominal interest rate but discounted by inflation.

At equilibrium with y_t = c_t, (9) becomes

(19)      r_t^equil = ρ + σE_t[Δy_t+1] = ρ + σψ_yaE_t[Δa_t+1]

where the latter equation uses the equilibrium value (18) for y.

The real wage ω_t ≡ w_t − p_t is, using (14),

(20)      ω_t^equil = ψ_ωa_t + ϑ_ω

where the constants ψ and ϑ are simple functions of α, σ, and φ.

This is an important conclusion from the RBC model: ...the equilibrium dynamics [sic] of employment, output, and the real interest rate are determined independently of the monetary policy. In other words, monetary policy is neutral with respect to those variables. ...output and employment fluctuate in response to variations in technology, which is assumed to be the only real driving force. In particular, output always rises in the face of productivity increase...[t]he same is true for the real wage. On the other hand, the sign of the employment is ambiguous, depending on whether σ (which measures the strength of the wealth effect of labor supply) is larger or smaller than 1."

The emphasis is Gali's. On the other hand, the equilibrium values of nominal variables, like inflation, nominal interest rate, and nominal prices, are not determined uniquely by real forces---for that we need to specify the monetary policy. Adding a utility for holding money balances implies that the money supply is also indeterminate. Examples of monetary policy rules come next.

Note on the case of zero expected inflation in the CMM: zero inflation implies that the expected nominal interest rate i_t^equil is equal to the expected real interest rate r_t^equil [eq. (21)]. Furthermore, the latter is determined by the real system parameters [eq. (19)], including the expected value of the first difference of the driving technology shocks. Thus, stipulating zero expected inflation determines the equilibrium nominal interest rate. In the case of zero secular growth, E_t[Δa_t+1] = 0, and (19) shows that then this equilibrium nominal interest rate is ρ.

2.4 Monetary Policy and Price Level Determination

2.4.1 An Exogeneous Path for the Nominal Interest Rate  → indeterminate equilibrium

In this case we choose i_t to be an exogenous stationary process with mean ρ, "which is consistent with a steady state with zero inflation and no secular growth".

Yes, my discussion immediately above shows that the equilibrium nominal interest rate with no inflation and no growth is in fact ρ, and that justifies this claim.

Constant interest is a special case. From the definition of real interest rate

     E_t[π_t+1] = i_t − r_t

and r_t is determined independently of the monetary policy. Thus, expected inflation is determined, but not the inflation itself. Any path that satisfies

     p_t+1 = p_t + i_t − r_t + ξ_t+1

will be consistent with the equilibrium, where ξ_t+1 is any zero-mean random variable, called sunspot shocks. In this system "nonfundamental factors may cause fluctuations in one or more variables" and an equilibrium with this property is called an indeterminate equilibrium. In particular, in the CMM an "exogenous nominal interest rate leads to price level indeterminacy".

2.4.2 A Simple Inflation-Based Interest Rate Rule  → sometimes determinate and sometimes indeterminate equilibrium

The central bank adjusts the nominal interest rate according to the rule

     i_t = ρ + φ_ππ_t

where φ_π ≥ 0. Substituting i_t = E_t[π_t+1] + r_t this can be written

(22)      φ_ππ_t = E_t[π_t+1] + (r_t − ρ)

Summary

Case 1: φ_π > 1. (22) has only one stationary solution, that is, "a solution that stays in the neighborhood of the steady state". An explicit solution for π_t is obtained, and price levels are fully determined as a function of the path of the real interest rate and hence, by (19) as a function of fundamentals. The larger φ_π the smaller the impact of the real shock on inflation.

Case 2: φ_π < 1. This case is similar to that in §2.4.1, with an exogenous nominal rate. The path will be consistent with sunspot shocks, and price levels and inflation are not determined uniquely. This is an example of

The Taylor principle and corresponding Taylor rule: The central bank must adjust nominal interest rates more than one-for-one in response to inflation to bring about a determinate equilibrium.

The wikipedia entry for Taylor rule is very interesting, especially the section "Empirical relevance". According to the wikipedia page, Taylor proposed the rule in 1993, and Dale W. Henderson and Warwick McKibbin did so simultaneously.

2.4.3 An Exogenous Path for the Money Supply

In §2.4.3 "the central bank sets an exogenous path for money supply {m_t}" using the postulated money demand (10), then determines the nominal interest rate. The results are

• The exogenous path for money supply determines the nominal price level and inflation uniquely
• In the case when money growth (Δm_t) follows a simple AR(1) process, "the price level should respond more than one for one to the increase in the money supply, a prediction that contrasts starkly with the sluggish response of price level observed in empirical estimates of the effects of monetary policy shocks as discussed in chapter 1."
• The nominal interest rate is predicted to go up with the money supply. "In other words, the model implies the absence of a liquidity effect, in contrast with the evidence discussed in chapter 1."

2.4.4 Optimal Monetary Policy

We note that in the baseline CMC monetary policy does not affect real variables, but does affect nominal variables like prices. But because the utility of the household is a function only of consumption and hours worked---two real variables---no monetary policy is better than any other (in terms of utility). This result is clearly "extreme and empirically unappealing", but can be "overcome" by adding monetary wealth (money balances) to the household utility, which is done in the next section. However, the overall assessment of the CMC "cannot be positive". It fails because

• It "cannot explain the observed real effects of monetary policy on real variables"
• Its "predictions regarding the response of the price level, the nominal rate, and the money supply to exogenous monetary policy shocks are also in conflict with the empirical evidence."

That leads to the introduction of "nominal frictions" and the Basic New Keynesian Model in chapter 3. The rest of chapter 2 is concerned mainly with the introduction of money balances in the utility.

2.5 Money in the Utility Function

Optimality conditions are derived for the utility function with cash balances; this amounts to adjusting the flow budget constraint to accommodate the change in the model [eq. (27)], and the derivation of one new condition [eq. (28)]. Two examples with more specific assumptions on the form of the utility function follow in the next two sections.

2.5.1 An Example with Separable Utility

Note that the Taylor series applied to log(1−exp(−i_t)) in (29) is about some value i that is not defined---in order to get a linear rather than logarithmic dependence on i_t. It is evidently an equilibrium value for the rate i_t, but i don't see why such an equilibrium value must exist. In any event, this example case leads to the "conventional linear demand for real balances":

(31)      m_t − p_t = c_t − ηi_t = y_t − ηi_t

and this is used later in the book without explicit citation.

The example ends with "As in the analysis of the cashless economy, the usefulness of (30), or (31), is confined to the determination of the equilibrium values for inflation and other nominal variables whenever the description of monetary policy involves the quantity of money in circulation. Otherwise, the only use of the money demand equation is to determine the quantity of money that the central bank will need to supply in order to support, in equilibrium, the nominal interest rate implied by the policy rule."

I interpret this as meaning that when the policy rule depends on the money supply, (31) can be used to determine the rate path, and hence a derivation like that in §2.4.2, Case 1 will determine the nominal rate and inflation.

2.5.2 An Example with Nonseparable Utility

About five pages of analysis, which I will skip for now. A main conclusion, on p. 30: "Condition (39) points to a key implication of nonseparability (ω ≠ 0): Equilibrium output is no longer invariant to monetary policy, at least to the extent that the latter implies variations in the nominal interest rate."

2.5.3 Optimal Monetary Policy in a Classical Economy with Money in the Utility Function

"The problem facing a hypothetical social planner seeking to maximize the utility of the representative household is presented and solved."

This leads to Friedman's rule, i_t = 0 for all t, mentioned in §1.1. This "policy implies an average (steady state) rate of inflation π = −ρ < 0, i.e., prices will decline on average at the rate of time preference."

There follows some discussion of implementing Freidman's rule, then Notes on the Literature, and some exercises.



Chapter 3 The Basic New Keynesian Model

This involves two changes in the CMM:

• Imperfect competition in the goods market is introduced, wherein there is a continuum of firms, each of which produces a differentiated good and sets prices
• "..some constraints are imposed on the price adjustment mechanism by assuming that only a fraction of firms can reset their prices in any given period"

"The resulting framework...has become in recent years the workhorse for the analysis of monetary policy, fluctuations, and welfare."

3.1 Households

The utility function is identical to that in the CMM model, (2.1), where consumption is now replaced by the aggregate consumption index

     C_t ≡ (∫ C_t(i)^(ε−1)/ε di)^ε/(ε−1)

where we assume a continuum of goods indexed by i ∈ [0,1], and the integral is over that continuum. Note the parsimonious notation, which can be confusing. The point, though, is to define variables like P_t and C_t so that when possible the equations become formally identical to those in chapter 2. The ε is not discussed, but this seems to be a standard way to model diminishing utility of commodities wrt to other commodities (with ε > 1); when ε = 2, for example, this is

     C_t ≡ (∫ C_t(i)^1/2 di)²

The household seeks to maximize this utility subject to the budget constraint corresponding to (2.2), except the cost-of-consumption is now appropriately averaged:

     ∫ P_t(i)C_t(i)di + Q_tB_t    ≤    B_t−1 + W_tN_t − T_t,      t = 0, 1, 2 ...,

where P(i) is the price of good i. We also assume the same solvency constraint as in chapter 2, which is that the limit of the expected bond holdings at infinity is nonnegative for every t.

Of course there's no need for a continuum of goods. It's not realistic and it just makes the optimization steps harder to justify rigorously. For example, we want to say that

     (d/dC_t(j)) ∫ P_t(i)C_t(i)di = P_t(j)

I think it would be better to make the integration a finite sum, and I'll think of it that way. In Gali 11, which extends the NKM so that it models unemployment, he introduces a two-dimensional continuum of members of a representative household. It's the convention in this field.

The household now has an additional decision to make: it "must decide how to allocate its consumption expenditures among the different goods." This is done in Appendix 3.1 by maximizing the Lagrangian

     C_t − λ(∫ P_t(i)C_t(i)di − Z_t)

This is to maximize utility subject for a fixed expenditure level Z_t. Differentiating wrt C_t(i) gives us the first-order conditions

     C_t^−1/ε(i)C_t^1/ε = λP_t(i)

Thus, for every i and j

     λ = C_t^−1/ε(i)C_t^1/ε/P_t(i) = C_t^−1/ε(j)C_t^1/ε/P_t(j)      ∀ i, j

or, assuming C_t ≠ 0,

     λ = C_t^−1/ε(i)/P_t(i) = C_t^−1/ε(j)/P_t(j)      ∀ i, j

Writing C_t(i) in terms of C_t(j) and substituting in the constraint that expenditures ∫ P_t(i)C_t(i)di = Z_t,

     C_t(j) = (P_t(j)/P_t)^−ε(Z_t/P_t)      ∀ j

where P_t is the aggregate price index, defined by

(A)      P_t ≡ (∫ P_t(i)^1−εdi)^1/(1−ε)

Substituting this in the definition of the aggregate consumption index C_t yields

(B)      Z_t = ∫ P_t(i)C_t(i)di = P_tC_t

That is, the total expenditure for consumer goods, defined as ∫ P_t(i)C_t(i)di, is the product of the aggregate price index and the aggregate consumption index. Note that this result depends on the optimality condition maximizing utility. Combining (A), (B) and using the definition of Z_t gets to the main result of Appendix 3.1, which is the demand schedule, the consumption as a function of price:

(1)      C_t(i) = (P_t(i)/P_t)^−εC_t

I checked all the algebra in Appendix 3.1. Note that this is the equality condition in Holder's inequality. Write (B) as

(C)      ∫ P_t(i)C_t(i)di = (∫ P_t(i)^μdi)^1/μ (∫ C_t(i)^νdi)^1/ν

where μ = 1−ε, ν = (ε−1)/ε, and 1/μ+1/ν = 1. The N&S condition for equality is the linear dependence of P_t(i)^μ and C_t(i)^ν, which is just equation (1).

Now the budget constraint can be written in a way formally identical to (2.2).

For some reason the sign of the term T_i in the budget constraint is "−" in in chapter and "+" here in chapter 3. It represents "lump-sum additions or subtractions to period income", and as an "addition to income" on the RHS of (2.2) should have a "+" sign, so the "− is evidently a typo in chapter 2. It hasn't been used yet, so it hardly matters.

The derivation of the optimization conditions (2.4) and (2.5) is exactly the same, and, in fact, the rest of the derivation in chapter 2 now follows, up to the treatment of firms in §2.2, in the same way, given that the starting points are identical. In particular, the same separable utility function leads to the same conditions (2.8) and (2.9).

3.2 Firms

"Assume a continuum of firms indexed" in the same way as the goods (a finite set would be better here as well, in my opinion)

(5)      Y_t(i) = A_tN_t(i)^1−α

The level of technology A_t is "assumed to be common to all firms and to evolve exogenously over time". "All firms face an identical isoelastic demand schedule given by (1), and take the aggregate price level P_t and the aggregate consumption index C_t as given."

Now come staggered prices, which is due to Calvo (1983) and works like this: "Each firm may reset its price only with probability 1−θ in any given period, independent of the time elapsed since the last adjustment. Thus, each period a measure 1−θ of producers reset their prices, while a fraction θ keep their prices unchanged. As a result, the average duration of a price is given by (1−θ)⁻¹. In this context, θ becomes a natural index of price stickiness."

The convention here is that if a firm sets its price at period t, it gets its first opportunity to reset it at period t+1. Therefore the average life is 1⋅(1−θ)+2⋅θ⋅(1−θ)+3⋅θ²(1−θ)+... = (1−θ)⋅( 1+2⋅θ+3⋅θ²+...) = (1−θ)⁻¹.

Note that the goods and firms are "differentiated" (so that they can be treated differently), but not "heterogenous", since the technology constant is the same across all firms. It is pointed out in §3.2.1 that "...all firms will choose the same price because they face an identical problem." This refers to the new price P_t^* for those prices that are reset from P_t−1. As we go from step t to t+1, some prices are preserved and some are reset, so in general at any step there will be a distribution of prices.

Interpretation of the continuum of labor    The consumption and price, C_t(i) and P_t(i), are clearly functions on [0,1], and there should be no problem defining the corresponding aggregate indices in the manner above. The labor, N_t(i), however, seems to present the problem that production needs to be defined in a way consistent with the notation and equations in the CMM (Chapter 2). The aggregate labor at time t is (as on p. 46: "Market clearing in the labor market requires")

(D)      N_t = ∫ N_t(i)di

This must be the N_t used in the optimality conditions that are formally identical to those in Chapter 2 (pp. 42-43). In particular, the cost of labor must be written in the form W_tN_t for the balance equation corresponding to (2.2), and the form of the household utility must use this N_t. Let's examine the production that follows from Gali's definitions. From (5), the production corresponding to firm i is

(E)      Y_t(i) = A_tN_t(i)^1−α

and therefore the aggregate output of goods is

(F)      Y_t ≡ (∫ Y_t(i)^(ε−1)/ε di)^ε/(ε−1) = (∫ [A_tN_t(i)^1−α]^(ε−1)/ε di)^ε/(ε−1) = A_t(∫ [N_t(i)^1−α]^(ε−1)/ε di)^ε/(ε−1)

using as the definition the same norm as consumption, as in §3.3, p. 45. This would seem to be required by the equilibrium condition. Take as an example the case α = 1/2 and ε = 2:

(G)      Y_t = A_t(∫ N_t(i)^1/4 di)²

Without worrying for now about the interpretation of N_t(i), I'd like to try to find a clear question or problem with this formulation. A simple observation is that there can be many different N_t(i) with the same aggregate labor N_t (D), but with different aggregate outputs (F or G).

This seems to be the same point mentioned in Christiano et al. 2010 [C10], p. 294: "A notable feature of the New Keynesian model is the absence of an aggregate production function. That is, given information about aggregate inputs and technology, it is not possible to say what aggregate output, Y_t, is. This is because Y_t also depends on how inputs are distributed among the various intermediate good producers."

It may be that the only "labor" that matters in the analysis in Chapter 3 (NKM) is N_t, since Gali bends over backwards to press the new analysis into the same mold as Chapter 2 (CMM). Thus, there seems to be no solution given for anything like the N_t(i), and that would explain the apparent vagueness of the continuum-of-firms model---it's a fiction designed to derive the price dynamics in the next section, which seems to be good enough for Gali's purposes. I'll check this and try to confirm with other sources.

3.2.1 Aggregate Price Dynamics

Begin with Appendix 3.2. Calculate the aggregate price from the definition, letting S = the set of firms not reoptimizing:

     P_t = [∫_S P_t−1(i)^1−εdi + (1−θ)(P*_t)^1−ε ]^1/(1−ε) = [θ(P_t−1)^1−ε + (1−θ)(P*_t)^1−ε ]^1/(1−ε)

The second equality,

     ∫_S P_t−1(i)^1−εdi = θ(P_t−1)^1−ε

follows from the fact that the distribution of prices among firms not adjusting prices in period t−1 is the same as the distribution of all prices in that period. Dividing both sides by P_t−1 and defining Π_t ≡ P_t/P_t−1,

(6 and 34)      Π_t^1−ε = θ + (1−θ)(P*_t/P_t−1)^1−ε

Notice that in a steady state with zero inflation (Π = 1), P*_t = P_t−1 = P_t, ∀ t. (p. 62)

"Furthermore, a log-linear approximation to the aggregate price index around that steady state yields" (p. 44)

(7)      π_t = (1−θ)(p*_t − p_t−1)

Call this Approximation 3, which, like Approximation 2, assumes we are near equilibrium. To derive eq. (7) from (6) take the log of (6)

(4)      (1−ε)π_t = log(θ + (1−θ)(P*_t/P_t−1)^1−ε) = log[1 + (1−θ)((P*_t/P_t−1)^1−ε − 1) ] ≈ (1−θ)[(P*_t/P_t−1)^1−ε − 1]

since the last log is of the form log(1+x) for small x. Divide by (1−ε):

(H)      π_t ≈ ((1−θ)/(1−ε))[(P*_t/P_t−1)^1−ε − 1]

Letting R = (P*_t/P_t−1) and r = log R, (H) is

(I)      π_t ≈ ((1−θ)/(1−ε))[R^1−ε − 1] = ((1−θ)/(1−ε))[e^r(1−ε) − 1] ≈ ((1−θ)/(1−ε))[r(1−ε)] = (1−θ)r = (1−θ)(p*_t − p_t−1)

which is the same as (7). I'm not sure there isn't a simpler, one-step derivation that follows the log-linear approximation pattern in Sims more closely; this one uses two power series approximations, one after another. Note once more that this again assumes we are near equilibrium.

"[Equation (7)] makes it clear that, in the present setup, inflation results from the fact that firms reoptimizing in any given period choose a price that differs from the economy's average price in the previous period."

3.2.2 Optimal Price Setting

When a firm reoptimizes in this model, it chooses the price P*_t to maximize "the current market value of the profits generated while that price remains effective. Formally, it solves the problem"

(J)      max_P*t Σ θ^k E_t{Q_t,t+k[P*_tY_t+k|t − Ψ_t+k(Y_t+k|t )]}

subject to the demand constraints (from (1))

(8)      Y_t+k|t(i) = (P*_t(i)/P_t+k)^−εC_t, ∀ k = 0, 1, ...

where

(K)      Q_t,t+k ≡ β^k (C_t+k/C_t)^−σ(P_t/P_t+k)

"is the stochastic discount factor for nominal payments."

(L)      Ψ(⋅)

"is the cost function"

and

(M)      Y_t+k|t

"denotes output in period t+k for a firm that last reset its price in period t."

The fraction of firms retaining the price P* at step k is θ^k, which accounts for the factor θ^k. To explain Q_t,t+k, notice the similarity between (K) and (2.7), which is derived from the intertemporal optimality condition:

(2.7)      Q_t = βE_t[(C_t+1/C_t)^−σ(P_t/P_t+1)] = E_t[ β(C_t+1/C_t)^−σ(P_t/P_t+1)]

Q_t,t+k is therefore the price at time t of a bond that pays off one unit of money at time t+k, and so represents the (stochastic) measure of how much future payments in nominal units should be discounted.

Substituting the constraints (8) into (J) and setting the derivative with respect to P*_t to zero yields the first-order optimality condition

(9)      Σ θ^k E_t{Q_t,t+kY_t+k|t[P*_t − Mψ_t+k]} = 0

where ψ_t+k|t = Ψ '_t+k(Y_t+k|t ) is the marginal cost for a firm that last reset its price in period t, and M = ε/(ε−1). Notice that Gali suppresses the argument i of P*_t and Y_t+k|t ; this optimality condition must hold for all i∈[0,1]. (I checked the algebra.)

Note that in the limiting case of no price rigidities (θ = 0), the previous condition collapses to the familiar optimal price-setting condition under flexible prices (see §2.2)

     P*_t = M ψ_t|t

which allows us to interpret M as the desired markup in the absence of constraints on the frequency of price adjustment. Henceforth, M is referred to as the desired or frictionless markup.

We linearize this around the zero-inflation steady state. To do this, rewrite it in terms of variables that have a well-defined value in that steady state. Dividing by P_t−1 gets us

(10)      Σ θ^k E_t{Q_t,t+kY_t+k|t [P*_t/P_t−1 − M MC_t+k|tΠ_t−1,t+k]} = 0

where MC_t+k|t ≡ ψ_t+k|t/P_t+k is the real marginal cost, and Π_t,t+k ≡ P_t+k/P_t. I think this means that Π_t−1,t+k ≡ P_t+k/P_t−1, and I use that below.

In the zero inflation steady state, P*_t/P_t−1 = 1 and Π_t−1,t+k = 1. Furthermore, constancy of the price level implies that P*_t = P_t+k in that steady state, from which it follows that Y_t+k|t = Y and MC_t+k|t = MC, because all firms will be producing the same quantity of output. In addition, Q_t,t+k = β^k must hold in that steady state. Accordingly MC = 1/M. A first-order Taylor expansion of (10) around the zero inflation steady state yields

(11)      p*_t − p_t−1 = (1 − βθ) Σ (βθ)^k E_t{ mc_t+k|t + (p_t+k − p_t−1)}

where mc_t+k|t ≡ mc_t+k|t − mc denotes the log deviation of marginal cost from its steady state value mc = −μ, and where μ ≡ log M is the log of the desired gross markup (which, for M close to one, is approximately equal to the net markup M−1).

To derive (11) from (10), write (10) as

(N)      Σ θ^k E_t{Q_t,t+kY_t+k|t f(x,y,z)} = 0

where x = P*_t/P_t−1, y = Π_t−1,t+k ≡ P_t+k/P_t, z = M MC_t+k|t, and f(x,y,z) = Q_t,t+kY_t+k|t[x − zy]. From the preceding, the equilibrium is at (x*,y*,z*) = (1,1,1), and f(x*,y*,z*) = 0. We use the three-variable Taylor series like the two-variable series in Sims,

(O)      f(x,y,z) ≈ f(x*,y*,z*) + f_x(x*,y*,z*)(x − x*) + f_y(x*,y*,z*)(y − y*) + f_z(x*,y*,z*)(z − z*) = f_x(1,1,1)(x − 1) + f_y(1,1,1)(y − 1) + f_z(1,1,1)(z − 1) = 0

For x near 1, x−1 ≈ log(x), and similarly for y and z, and therefore

(P)      f(x,y,z) ≈ f_x(1,1,1)(p*_t − p_t−1) + f_y(1,1,1)(p_t+k − p_t−1) + f_z(1,1,1) log M MC_t+k|t

At this point we set Q_t,t+k = β^k and Y_t+k|t = Y (the equilibrium output), The equilibrium values, as mentioned above. Note that this ignores the dependence of these variables on the price---I guess because the dependence of (10) on Q and Y near equilibrium will be second order, but I'm not sure of this reasoning right now. With this assumption, the evaluated partials in (P) are

(Q)      f_x(1,1,1) = β^kY,      f_y(1,1,1) = −β^kY,      f_z(1,1,1) = −β^kY,

and (P) is

(R)      f(x,y,z) ≈ β^kY (p*_t − p_t−1) − β^kY (p_t+k − p_t−1), − β^kY log M MC_t+k|t

Then (10) becomes

(S)      Σ (βθ)^k E_t{ (p*_t − p_t−1) − (p_t+k − p_t−1) − log M MC_t+k|t } ≈ 0

or

(T)      Σ (βθ)^k E_t{ (p*_t − p_t+k) − log M MC_t+k|t } ≈ 0

First, we'll ignore the ignore M part of the sums and work on the P part. Using the fact that E_t{p*_t} = p*_t,

(U)      −p*_t/(1 − βθ) + Σ (βθ)^k E_t{ p_t+k) } ≈ 0

Adding and subtracting p_t−1/(1 − βθ), and then multiplying by (1 − βθ):

(W)      −(p*_t − p_t−1) + (1 − βθ) Σ (βθ)^k E_t{ p_t+k − p_t−1 } ≈ 0

Which, together with

(X)      log M MC_t+k|t = mc_t+k|t ≡ mc_t+k|t − mc

yields (11) from (10). Recall that mc = −μ, and μ = log M is the log of the desired gross markup.

To get some insight into the meaning of (11), rearrange it as

(Y)      p*_t = μ + (1 − βθ) Σ (βθ)^k E_t{mc_t+k|t + p_t+k},

which can be interpreted as "...firms resetting their prices will choose a price that corresponds to the desired markup over a weighted average of their current and expected (nominal) marginal costs, with the weights being proportional to the probability of the price remaining effective at each horizon θ^k."

3.3 Equilibrium

"Market clearing in the good market requires condition is Y_t(i) = C_t(i) for all i∈[0,1] and all t." Note that, with aggregate output index as already defined above,

(F)      Y_t ≡ (∫ Y_t(i)^(ε−1)/ε di)^ε/(ε−1)

market clearing then implies that Y_t = C_t for all t. Combining this with the "consumer's Euler equation" (2.9) (just replace c by y) yields

(12)      y_t = E_t[y_t+1] − (1/σ)(i_t − E_t[π_t+1] − ρ)

As mentioned above, "Market clearing in the labor market requires"

(D)      N_t = ∫ N_t(i)di

Using (5) (the Cobb-Douglas production function), this becomes

(Z)      N_t = ∫ (Y_t(i)/A_t)^1/(1−α) di

and using goods market clearing, Y_t(i) = C_t(i), and eq. (1),

(AA)      N_t = (Y_t/A_t)^1/(1−α) ∫ (P_t(i)/P_t)^{−ε/(1−α)} di.

Taking logs,

(BB)      (1 − α)n_t = y_t − a_t + d_t

where

(CC)      d_t ≡ (1 − α) log ∫ (P_t(i)/P_t)^{−ε/(1−α)} di

"is a measure of price (and, hence, output) dispersion across the firms." Appendix 3.3 is devoted to showing that d_t is zero up to a first approximation in a neighborhood of the steady state. To go over this, rewrite the definition of price index, eq. (A), and use a second-order Taylor's series assuming that we are near the zero-inflation steady state:

(DD)      1 = ∫ (P_t(i)/P_t)^1−ε di = ∫ exp[ (1−ε) (p_t(i) − p_t)] di ≈ 1 + (1−ε) ∫ (p_t(i) − p_t) di + (1/2)(1−ε)² ∫ (p_t(i) − p_t)² di

or, to second order,

(EE)      p_t ≈ E_i[p_t(i)] + (1/2)(1−ε) ∫ (p_t(i) − p_t)² di.

Notice the notation E_i[p_t(i)], which is defined to be ∫ p_t(i) di, the "cross-sectional mean of (log) prices".

This approximation requires that (p_t(i) − p_t) be small. Why is this true? Continue with appendix 3.3.

---

Here is a start at making a table comparing the FM and NKM.

	Flow Model	Basic New Keynesian Model
Household, consume/save split	savings ∝ i	optimizes expected utility, disutility for working
Household, consume/work split	fixed labor resource employed	optimizes expected utility, disutility for working
CG Firm, capital/labor split	optimizes productivity for a given budget	no capital factor, optimizes profit for given price and wage
CG Firm, price	competitive market	monopolistic competition, posted price optimizes expected utility, sticky prices
nominal interest rate i	endogenous, iteratively adjusted based on demand and supply of available funds	exogenous, determined by central bank policy
financing for capital	credit determined by available funds from savings	no financing of capital
shocks in basic model	monetary policy	technology, monetary policy

Christiano, Trabandt, Walentin (2010) mention four possible orthogonal shocks in their review: technology, labor supply, Phillips curve, and monetary policy.