Analytical results

To obtain analytical results, I first derive the optimal aggregate long-run inflation rate that maximizes steady-state welfare, in the limiting case in which the discount factor β approaches unity. Maximizing steady-state welfare in this case is equivalent to minimizing only one of the two aggregate distortions in the decentralized equilibrium (18) because these distortions are equal to one another. Minimizing only one aggregate distortion in-stead of maximizing in-steady-state welfare simplifies deriving analytical results.

To show that the aggregate distortions are the same in the limiting case, I use the aggregate average markup (21) to rewrite the aggregate relative price distortion (20) as

R(π) =

ψ

ρ1(π)µ1(π)+ 1−ψ ρ2(π)µ2(π)

−1

µ(π)⁻¹ .

In the limitβ →1, equations (23)–(24) imply thatρ_z(π)µ_z(π) =θ/(θ−1), wherez = 1,2.

Thus, the term in round brackets collapses to the static markup, and this yields:

R(π) =

θ/(θ−1) µ(π)

. (25)

Without loss of generality, I assume that the labor income tax perfectly offsets the static

markup, i.e., 1−τL=θ/(θ−1).¹² Equation (25) thus states that the aggregate distortions are equal to one another. The second main result in this paper follows from minimizing µ(π) (or maximizingR(π)) and shows how the government selects the aggregate long-run inflation rate to resolve the policy tradeoff from sectoral asymmetries optimally.

Proposition 2:In the limiting case β →1, the optimal aggregate long-run inflation rate that maximizes steady-state welfare solves

The weight fulfills the condition thatω(π)∈[0,1] and depends on the optimal aggregate long-run inflation rate and the parameters:

ω(π) =

A natural interpretation of Proposition 2 is that the growth rateg_z represents the optimal long-run inflation rate in a sector z, and that the optimal aggregate long-run inflation rate π is a weighted average of the optimal long-run inflation rates in both sectors.¹³ The weight ω(π) depends on the aggregate long-run inflation rate and, beyond this, on the growth rate in firm-level productivity in a sector z, the price stickiness in a sector z, the probability to survive in a sector z, and the relative sector size. In line with this interpretation, Proposition 2 shows that what matters for the steady-state welfare is the distance between the aggregate long-run inflation rate and the optimal long-run inflation rate in a sector z, (π−g_z)/g_z, instead of only the aggregate long-run inflation rate.

12In the limit β →1, I obtain the same optimal π if τ_L is selected optimally.

13Alternatively, using equation (26), π can also be expressed as a weighted harmonic average:

In contrast to the Proposition 1 in the one-sector model, the optimal aggregate long-run inflation rate in the Proposition 2 generally cannot recover the first-best planner’s solution in the two-sector model. The reason for this is the policy tradeoff that arises from a lack of policy instruments that work at the sectoral level. Namely, while the government can use the aggregate long-run inflation rate to fully offset the distortions in either sector 1 or sector 2, this instrument is not able to fully offset the distortions in both sectors at the same time.

To illustrate this policy tradeoff, I consider the case in which the firms in sector 1 grow more slowly than the firms in sector 2, g₁ < g₂, while all firms are subject to the same amount of price stickiness, κ₁ = κ₂ = κ with κ > 0, and both sectors are of equal size.

Proposition 2 shows that in this case, the optimal aggregate long-run inflation rate obeys g₁ < π < g₂ and weights g₁ higher than g₂ because ω(π) > 0.5. This is optimal because firms in sector 1 that adjust their price anticipate that π will erode their price at a rate that is aboveg₁. Therefore, they elevate their markup and, hence, the average markup in sector 1, to above the static markup. In contrast, the adjusting firms in sector 2 rather compress their markup and, hence, the average markup in this sector, to below the static markup becauseπ will erode their price at a rate that is below g₂. However, even though the optimal aggregate long-run inflation rate π weights g₁ higher than g₂ and, therefore, compresses the elevated average markup in sector 1, π cannot be set to align the average markup and the static markup in both sectors at the same time.¹⁴

In the general case in which firms are subject to various amounts of price stickiness across sectors, Proposition 2 yields that the optimal aggregate long-run inflation rate weights the sector with the stickier prices more heavily. Figure 2 shows how the weight ω(π) depends on the probabilities α₁ and α₂ of not adjusting the price. For a particular value ofα₁, reducing the value ofα₂ increases the weight on sector 1. This phenomenon is

14One special case in which the policy tradeoff disappears and the decentralized equilib-rium is first best arises when firms in both sectors grow at the same rate g. In this case, Proposition 2 yields π=g, as in the one-sector model. Accordingly, this case generalizes Proposition 1, for the limitβ →1, to a model with asymmetric price stickiness. Another special case in which the policy tradeoff disappears arises when firms in sector 2, say, have flexible prices. In this case, Proposition 2 yieldsπ =g₁.

α₁

α 2

0.01

0.1

0.3 0.5

0.7

0.9

0.99

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2: Weightω(π) as a function of the degree of price stickinessα₁ andα₂ in the two sectors. Lines indicate the combinations ofα₁ andα₂ that yield a particular value ofω(π).

The growth rates in firm-level productivity are equal to g₁ = 1.01^1/4 and g₂ = 1.03^1/4, and the aggregate long-run inflation rate is the one in Proposition 2. The calibration is described in Section 5.2.

known as the “stickiness principle” in the literature on the optimal inflation stabilization policy (e.g., Benigno (2004) and Eusepi, Hobijn, and Tambalotti (2011)). An important consequence of Proposition 2 is that this principle applies equally to the choice of the optimal aggregatelong-run inflation rate.

The optimal aggregate long-run inflation rate obeys the stickiness principle because the expected duration for a firm to keep its price unchanged increases when its price becomes more sticky.¹⁵Therefore, the firm adjusts more sensitively to a gap between the aggregate

15Increasing α or reducing the probability of firm exit δ has the same qualitative

long-run inflation rate and the growth rate in firm-level productivity because it expects the suboptimal erosion of its price to occur over a longer period of time. This adjustment exacerbates the distortions in the sector with the stickier prices to above their levels in the sector with the less sticky prices, and the optimal aggregate long-run inflation rate weights the sector with the stickier prices more heavily because thereby it partly undoes the distortions in this sector.

As a result of the government’s policy tradeoff and unlike in Proposition 1, the opti-mal aggregate long-run inflation rate in Proposition 2 depends on assuming Calvo pricing.

However, two reasons suggest that the quantitative effects of alternative pricing assump-tions are small. First, alternative pricing assumpassump-tions affect the degree of price erosion that follows from a positive aggregate long-run inflation rate. However, unlike in the basic New Keynesian model in which any positive inflation rate is detrimental to welfare, in the model with firm-level productivity growth a positive inflation rate is only detrimental to welfare to the extent to which it differs from the positive growth rate in firm-level productivity. Second, what matters for resolving the government’s policy tradeoff is not the absolute amount of price stickiness, but the relative amount of price stickiness across sectors, and the effect of the alternative pricing assumptions on this relative stickiness is likely to be small.