Aggregation

In each event history s^t, the differentiated labor skills {L(i, s^t)}_i∈[0,1] are aggregated into a composite skill L(s^t) according to

, and the differentiated goods {X(j, s^t)}_j∈[0,1] are aggregated into a composite good X(s^t) according to

, where θ_l∈(1, ∞) and θ_x∈(1, ∞) denote, respectively, the elasticities of substitution between the differentiated skills and between the differentiated goods. The aggregation activity is perfectly competitive. The resulting demands for labor skill i and good j are given by

where the wage rate

for the composite skill and the wage rates {W(i, s^t)}_i∈[0,1] for the differentiated skills have the relation

, and the price

for the composite good and the prices {P(j, s^t)}_j∈[0,1] for the differentiated goods satisfy the relation

.

The defining feature of our model is that, while the composite skill can only serve as an input for the production of each differentiated good, the composite good can serve either as a final good consumed by households or as an intermediate production input used by firms.

Firms

The technology for producing a good j is represented by a constant-returns-to-scale production function,

where Γ(j, s^t) and L(j, s^t) denote the intermediate input and the composite labor, respectively, and ϕ∈(0, 1) is the share of the intermediate input in production.

All firms are price- and wage-takers in the input markets and imperfect competitors in the output markets, where they set prices for their own outputs, taking the demand schedule (2) as given. At each date t, a fraction 1/N_p of the firms chooses new prices after the realization of the shock s_t. Once a price is set, it remains effective for N_p periods. The firms are divided into N_p cohorts based on the timing of their pricing decisions. A firm j in the cohort that can choose a new price at date t solves the problem

where D(s^τ|s^t) is the price of one dollar at s^τ in units of dollars at s^t, and Q(s^τ) is the unit cost of production at s^τ (which is also the marginal cost, given the constant-returns-to-scale technology) given by

The solution to the problem stated in (4) is

According to (6), the optimal price is a markup over a weighted average of the marginal costs during the periods in which the price will remain in effect. The solution to the cost-minimization problem (5) yields the unit cost function

, where

is a constant determined by ϕ, and the demand functions for the intermediate input and the composite labor in producing X^d(j, s^τ) are given by

Note that, even if a firm cannot choose a new price at a given date, it would still need to choose the inputs of the intermediate good and the composite labor in order to minimize its unit production cost. In other words, it has to solve the problem stated in (5) and, therefore, (7) and (8) hold for all firms at any time.

Households

Each household i has a subjective discount factor β∈(0, 1) and a utility function

where

is a CES composite of consumption good and real money balances, and L(i, s^t) denotes the household's hours worked. The period-utility function U is strictly increasing and the period-disutility function V is strictly decreasing, and both U and V are strictly concave and twice continuously differentiable. The budget constraint facing the household in event history s^t is

where B(i, s^t+1) is the household's holdings of a nominal bond, each unit of which costs D(s^t+1|s^t) dollars at s^t and pays off one dollar in period t+1 if s^t+1 is realized, W(i, s^t) is a nominal wage rate for its type of labor skill, Π(i, s^t) is its share of firms' profits, and T(i, s^t) is a lump-sum transfer it receives from the government.

All households are price-takers in the good markets and imperfect competitors in the labor markets, where they set nominal wages for their labor skills, taking the demand schedule (1) as given. At each date t, upon the realization of the shock s_t, a fraction 1/N_w of the households chooses new wages. These wages, once set, will remain in effect for N_w periods. The households are divided into N_w cohorts based on the timing of their wage decisions. Each household maximizes (9) subject to (10) and a borrowing constraint B(i, s^t+1)≥−B, for some large positive number B. The initial conditions on bond- and moneyholdings are taken as given. At date t, if a household i can set a new wage, then the optimal choice of its nominal wage is

where V_l(i, s^τ) and U_c(i, s^τ) denote its marginal disutility of labor and its marginal utility of consumption, respectively. According to (11), the optimal wage is a markup over a weighted average of dollarized marginal rates of substitution between leisure and consumption during the periods in which the wage will remain in effect. Regardless of whether they can adjust their wages at a given time, all households need to make decisions on consumption, money balances, and bondholdings in any period. We have used the standard first-order condition for bondholdings in deriving (11).

Monetary Policy

We are interested in the dynamic effect of monetary policy shocks on the fluctuations of real GDP. To serve this purpose, we assume that monetary policy is conducted via lump-sum transfers to households so that the sum of the transfers is equal to the change in money supply. That is, we assume that ∫¹₀T(i, s^t)=M^s(s^t)−M^s(s^t−1)=[μ(s^t)−1]M^s(s^t−1). We also assume that the log of the growth rate of money supply lnμ(s^t) follows a stationary stochastic process given by

where 0<ρ<1, and ε_t is a white-noise process, with a zero mean and a finite variance σ²_ε.

Equilibrium

An equilibrium for this economy consists of allocations C(i, s^t), M(i, s^t), B(i, s^t+1), and wage W(i, s^t) for household i, for all i∈[0, 1], allocations Γ(j, s^t), L(j, s^t), and price P(j, s^t) for firm j, for all j∈[0, 1], together with prices D(s^t+1|s^t),

, and wage

, that satisfy the following conditions: (i) taking the wages and all prices but its own as given, each firm's allocations and price solve its profit-maximization problem; (ii) taking the prices and all wages but its own as given, each household's allocations and wage solve its utility-maximization problem; (iii) markets for money, bonds, the composite labor, and the composite good clear; (iv) monetary policy is as specified.

We assume that there are complete financial markets that allow households to transfer their nominal wealth across dates and states and to insure one another against their exposure to idiosyncratic income risks that may arise from differences in the timing of their wage-setting decisions under staggered wage contracts. We suppose that, with such financial arrangements, the equilibrium consumption, and holdings of real money balances are identical across households, assuming an identically distributed initial wealth. It follows that C(i, s^t)=∫¹₀C(i, s^t)di=Y(s^t) and M(i, s^t)=∫¹₀M(i, s^t)di≡M(s^t), for all i and for every s^t, where Y(s^t) denotes real GDP.

and the market-clearing condition for the composite labor ∫¹₀L(j, s^t)dj=L(s^t) implies that

where

. Equations (13) and (14), together with the price-setting equation (6), the wage-setting equation (11), and the money-market-clearing condition M(s^t)=M^s(s^t), characterize an equilibrium.

Endogenous Nominal Rigidity

As revealed by the third term on the right-hand side of (15), the amount of endogenous nominal rigidity is inversely related to γ_z, which measures the response of wages or prices to a given change in expected aggregate demand conditions within a contract length. A smaller γ_z implies greater endogenous nominal rigidity for given exogenous nominal stickiness. Inspecting (16) and (17) reveals that whether wage and price staggering may have the same implication for endogenous nominal rigidity depends on whether γ_w and γ_p are equal.

Before looking into this possibility, we first provide some intuitions for why γ_w and γ_p are linked to the fundamental parameters via the functional forms in (16) and (17), especially, why γ_w is independent of ϕ whereas γ_p is strictly increasing in ϕ. The intuitions can best be illustrated by considering what would happen following a 1% shock to aggregate demand. With nominal staggering of either type, part of the shock transmits into a higher demand for labor and a higher real income. In consequence, the marginal disutility of working rises and the marginal utility of consumption falls. This would induce a household to raise its nominal wage whenever possible. This is why ξ_l (which determines how rapidly the marginal disutility of working rises as hours worked increase) and ξ_c (which determines how fast the marginal utility of consumption falls as consumption rises) enter the numerator of γ_w and of γ_p.

If it is wage-setting decisions that are staggered, then an increase in a household's nominal wage will lead to a rise in its relative wage. This creates an incentive for firms to substitute away from its labor service toward those of the other households. Such substitution tends to reduce the household's hours worked and marginal disutility of working, which would discourage it from raising its wage by too much. In consequence, the optimal increment in the household's wage is less than ξ_c+ξ_l. This is why ξ_l and θ_l (which determines how rapidly the amount the household works falls as its nominal wage rises) enter the denominator of γ_w.

As noted above, γ_w is independent of ϕ, the share of the intermediate input in production. This independence is one part of the key to establishing the equality between γ_w and γ_p under reasonable parameter values. To understand this independence of γ_w on ϕ, note that, with synchronization in pricing decisions, the price level, and thus the intermediate-input price as well, coincide with the price of each individual good that can be adjusted in any period. The price of the intermediate input is thus a flexible part of firms' marginal cost. Since the firms set their prices equal to a markup over the marginal cost, rigidity in their pricing decisions arise entirely from rigidity in the nominal wage index, regardless of the share of the intermediate input. In other words, the presence of the intermediate input does not affect the pricing behavior of the firms or the property of output dynamics. In fact, the optimal adjustment of each firm's price is given by γ_w, which is equal to the exact change in the nominal wage index, just as in the case with no intermediate input at all.

If it is pricing decisions that are staggered, then increases in households' nominal wages, while not affecting their relative wages, given synchronization in wage-setting decisions, will cause the real wage to rise in the face of the sticky intermediate-input price. A rise in the real wage creates an incentive for firms to substitute away from the composite labor toward the intermediate input. This would discourage the households from raising their nominal wages by too much. Therefore, the intermediate-input price being a rigid part of firms' marginal cost results in endogenous nominal wage rigidity. As a consequence, the optimal increase in the households' nominal wages, which in the present case coincides with the nominal wage index, is less than ξ_c+ξ_l. This is why f(ϕ, θ_x) and ξ_l enter the denominator of γ_p.

To understand the term f(ϕ, θ_x), we note that it corresponds to the elasticity of demand for the composite labor with respect to the real wage, with the real aggregate demand being the shift variable in the demand schedule for the composite labor.¹⁰

On the other hand, for any given ϕ∈(0, 1), the term f(ϕ, θ_x) is decreasing in θ_x. This is so because a larger θ_x corresponds to a greater elasticity of substitution between differentiated intermediate goods and firms would be able to rely less on substitution between the composite labor skill and the composite intermediate good while more on the substitution between the differentiated intermediate goods. As a consequence, when θ_x increases, firms' demand for the composite labor skill becomes less sensitive to changes in the real wage, and thus f(ϕ, θ_x) falls. This tension between the two dimensions of factor substitutions is present only if both labor and intermediate goods are used in production. This is why θ_x enters the term only if ϕ ≠ 0 and ϕ≠1.

To understand why the term (1−ϕ) appears in the numerator of γ_p, we observe that, with ϕ share of the intermediate input in production, a firm's marginal cost is composed of ϕ fraction of the price level and of (1−ϕ) fraction of the nominal wage index. Thus, exogenous nominal stickiness in the form of staggered price setting translates directly into the sluggishness in the movement of this former part of the marginal cost. Since the firm's optimal price is a markup over a weighted average of the marginal costs within its contract duration, we can scale down ϕ fraction of the rise in the nominal wage index to get the firm's optimal price response.

The fact that f(ϕ, θ_x) is increasing in ϕ and (1−ϕ) is decreasing in ϕ implies that γ_p is a decreasing function of ϕ. This monotone relation is the other part of the key to establishing the equality between γ_w and γ_p and thus the equivalent implications of wage and price staggering for endogenous nominal rigidity under reasonable parameter values.

Endogenous Real Persistence

We turn now to endogenous real persistence. Naturally, the amount of output persistence is negatively related to γ_z. The smaller γ_z is, the smaller and the less rapid are the responses of nominal prices to an aggregate demand shock, and thus the larger and the more persistent are the responses of real output to the shock. We illustrate here this inverse relationship between γ_z and output persistence by obtaining explicit solutions of the equilibrium dynamics for the case with N=2.

To proceed, we note that, when N=2, equation (15) simplifies to

Substituting into (18) the static money demand equation and the equation defining the price level,

, we obtain a second-order difference equation in p_t as follows:

Assuming that the money stock m_t follows a random-walk process, we obtain from (19) a closed-form solution for equilibrium output dynamics:

The autoregressive coefficient a_z in (20) determines how gradually the initial response of y_t to a money supply shock will die out over time, that is, it determines how persistent the output response will be. The larger a_z is, the more persistent is the output response. The second equation in (20) shows that a_z is a strictly decreasing function of γ_z. Therefore, the amount of output persistence is greater, the smaller is γ_z. For any γ_z between 0 and 1, a_z is also between 0 and 1, and there exists endogenous output persistence. That is, the initial response of output dies out monotonically, but not completely even at the end of the initial contract duration. If γ_z=0, then a_z=1, so that output dynamics follow a random-walk process. If γ_z=1, then a_z=0, and there is no endogenous output persistence. In the case that γ_z>1, we have a_z<0, and there is oscillation rather than persistence in output dynamics. The one-to-one inverse correspondence between γ_z and a_z indicates that, whether wage and price staggering are equally capable of generating real persistence depends on whether γ_w and γ_p are equal.¹¹

Theoretical Possibility

In light of the above analysis, the ability of staggered wage setting relative to staggered price setting in propagating a monetary policy shock is tied to the magnitude of γ_w relative to γ_p. We summarize here the theoretical implications of our analysis on the relative ability of the two types of nominal staggering in generating endogenous nominal rigidity and real persistence under general parameter values. This summary also helps connect our results to the literature.

According to (16) and (17), the intermediate-input share ϕ enters into γ_w and γ_p in different ways, implying that the use of the intermediate input in production interacts differently with the two types of nominal staggering: while it makes no difference in equilibrium dynamics under staggered wage setting (for γ_w is independent of ϕ), it improves the ability of staggered price setting in generating endogenous nominal rigidity and real persistence (for γ_p is a decreasing function of ϕ).

This observation suggests that the ability of price staggering relative to wage staggering in generating persistence depends critically on whether the use of intermediate input is taken into account. In the extreme case that ϕ is set to zero, γ_p is larger than γ_w, and it is less likely to generate persistence under staggered price setting than under staggered wage setting [e.g., Andersen (1998) and Huang and Liu (1998, 2002)]. With a positive value of ϕ, γ_p can be equal to or less than γ_w, and thus persistence can be the same or even more under staggered price setting than under staggered wage setting. In particular, for any values of the other parameters in the model, there exist values of ϕ between 0 and 1 such that γ_p=γ_w. In the extreme case that ϕ is set arbitrarily close to one, γ_p is arbitrarily close to zero, and arbitrary persistence obtains under staggered price setting, while the implication on persistence of staggered wage setting remains the same, just as in the case when ϕ is set to zero.

The above discussion indicates that, as long as the value of ϕ is larger than zero, it is theoretically possible that staggered price setting is as able as staggered wage setting in generating persistence. The next question is whether the equivalence of price and wage staggering can arise under empirically reasonable parameter values.

Empirical Plausibility

We now show that, once the use of intermediate input is taken into account, the equivalence of price and wage staggering in generating endogenous nominal rigidity and real persistence is not only theoretically possible, but also empirically plausible. For concreteness, we provide examples in which γ_p and γ_w can be equal under some reasonable parameter values, so that the two types of nominal staggering can be equally important transmission mechanisms of monetary shocks.

To proceed, we first note that, by virtue of (16) and (17), γ_p and γ_w are equal if and only if

We next discuss whether this equality can hold with empirically reasonable parameter values. According to the estimates by Griffin (1992, 1996), a plausible range of θ_l, the elasticity of substitution between differentiated labor skills, is from 2 to 4. A reasonable range of θ_x, the elasticity of substitution of differentiated goods, is from 4 to 10, in light of the empirical studies by Domowitz et al. (1986), Shapiro (1987), Basu (1996), Basu and Kimball (1997), Basu and Fernald (1994, 1995, 1997), and Linnemann (1999). To pinpoint a range for the relative-risk-aversion parameter ξ_l, one usually appeals to the estimates of the intertemporal hours elasticity of substitution based on the intensive margin. Earlier studies along this avenue mostly suggest that this elasticity is smaller than 1 [see, e.g., the survey by Pencavel (1986)]. More recent work demonstrates that these earlier estimates can be subject to large downward biases [e.g., Rupert, Rogerson, and Wright (2000)]. In particular, Mulligan (1999) finds that the elasticity is instead in the range from 1 to 2. Recent studies also show that the employment elasticity based on the extensive margin can be much greater than 1 [e.g., Kimmel and Kniesner (1998)]. We consider the value of ξ_l from 0.5 to 2, corresponding to the intertemporal labor elasticity of substitution between 2 and 0.5.

Table 1 displays the intermediate-input share ϕ (between 0 and 1) that solves (21) under the above values of θ_l, θ_x, and ξ_l. Some of these values of ϕ are consistent with the empirical evidence. For example, the estimates by Jorgenson et al. (1987) suggest that the value of ϕ is at least 0.5 for U.S. manufacturing industries over the period 1947–1979. Nevo (2001) finds that the share of raw materials in the U.S. food industry (SIC 20) is 0.634, based on the Annual Survey of Manufacturers over the period 1988–1992. The recent empirical study by Hanes (1999) reveals that the input–output connections in the actual economy have become more sophisticated in the recent years than in the further past. This suggests an increasing intermediate-input share over time. Basu (1995) argues that the value of ϕ can lie between 0.8 and 0.9 if fixed costs are taken into account [see also Bergin and Feenstra (2000)]; yet, the implied markup drawn from Domowitz et al. (1988) is often viewed as somewhat large.

In light of the above discussion, we consider 0.5–0.7 as a reasonable range for ϕ. We then note that many values in Table 1 lie in this range or in its close vicinity. Therefore, for some empirically plausible values of the fundamental parameters, price and wage staggering can be two equivalent mechanisms in generating endogenous nominal rigidity and real persistence.¹²