Money demand by firms, labor market frictions, and optimal long-run inflation

Abstract

Optimal monetary policy, under flexible and sticky prices, and sticky nominal wages are studied in the canonical matching model of the labor market with a working capital channel. A money demand by firms is motivated by the fact that a significant amount of M1 is held by firms. As a result of the working capital, the Ramsey-optimal monetary policy calls for inflation when employment is above the socially efficient level and for deflation when it is below that level. This result holds under flexible wages as well as nominal wage rigidities. In other words, to improve labor market efficiency, deflation is necessary to “grease the wheels of the labor market.” Although there is no relationship between optimal monetary policy and the market tightness under flexible prices, a long-run and mainly negative relationship between the optimal inflation rate and the equilibrium tightness emerges under sticky prices.

1. Introduction

Evidence suggests that a substantial part of M1 (about 70%) in industrialized economies is held by firms. There is a large literature in finance documenting this and that USA and many advanced economies nonfinancial firms have been holding increasing amounts of cash in recent years. ¹ For instance, Bates et al. (2009) and Duchin (2010) discuss how the average ratio of cash to assets for US industrial firms has massively increased from 10.5% in 1980 to 23.2% in 2006. Figure 1 reports the evolution of checkable deposits and currency held by US nonfinancial corporations across the economy from 1945/10/01 to 2020/01/01. It documents the fact that since the 2008 financial crisis, the magnitude of cash held by US nonfinancial firms has increased immensely. ²

Figure 1. US nonfinancial firms liquidity demand.

In light of this evidence, our objective in this paper is to introduce a working capital channel into the conventional model of unemployment due to Mortensen and Pissarides and study normative implications of long-run relationship between monetary policy, as measured by nominal interest rates or inflation, and labor market performance, as measured by the market tightness. The working capital channel comes from the assumption that firms’ variable labor must be financed by cash. ³ Our long-run focus in this paper differs from that of a related relatively large literature on optimal monetary policy in frameworks with frictional labor markets which is mainly interested in understanding the short-run implications of optimal policy. Since this literature abstracts from a demand for fiat money, the optimal deterministic inflation rate that it features is always zero, regardless of whether real wages are fully flexible or quite rigid. ⁴ When the literature does assume a money demand, it is motivated for households, that, in itself, calls for a negative inflation rate between the negative of the real rate of interest and zero; see Carlsson and Westermark (2016) in this regard. ⁵ Our main contribution is to demonstrate how the mere presence of a working capital channel in the form of a cash constraint on the firm’s wage bill can significantly change the policy prescription for the optimal steady-state inflation rate under both flexible and sticky prices when the economy is in an inefficient equilibrium. ⁶

To derive clean analytical results that explain how the optimal monetary policy operates in our model, we assume that real wages are rigid á la Hall (2005). We provide some quantitative results for a wide range of values for the real wage. We also provide numerical results for the case that wages are fully flexible as well as the case of nominal wage rigidity. We show that our main mechanism remains almost qualitatively intact. As Gali (2010) discusses in detail, labor market frictions by themselves have not much effects on the dynamics of an economy’ s equilibrium. It is their combinations with wage rigidities that lead to inefficient responses to shocks and tradeoffs for monetary policy. However, as we show below, the combination of labor market frictions and real wage rigidities by themselves leaves no room for monetary policy in the long run. In this case, the optimal inflation rate is always zero, even if real wages are fully fixed. Monetary policy cannot correct for any labor market inefficiencies that might arise in the deterministic steady state. This no longer is the case in the presence of a working capital channel. In this case, the optimal monetary policy does have the ability to correct for labor market inefficiencies in the steady state and push the economy toward the optimum when it is away from it.

We analytically solve for the Ramsey-optimal monetary policy in a simple New Keynesian model with labor market frictions where firms hold cash because it facilitates their ability to pay for their labor hiring costs. We discuss the optimal monetary policy under flexible prices as a special case as well. We show that when the zero lower bound on the nominal interest rate does not bind the Ramsey problem under flexible prices and perfect competition delivers the first-best outcome. More precisely, we find the value of the real wage at which the efficient allocation holds and the Friedman rule (the rule that calls for setting the net nominal interest rate to zero) is optimal. We call this value the efficient real wage value. When the real wage is above its efficient level, the zero lower bound binds, and the efficient allocation is not achievable. Monetary policy cannot correct for the labor market inefficiencies in this case. When it is below its efficient level, the optimal monetary policy calls for an inflation rate that is increasing in the distance of the real wage from its efficient value. In this case, the Friedman rule ceases to be optimal despite the fact that the Ramsey allocation gives the same market tightness level and unemployment rate as their counterparts in the first-best allocation. Put another way, due to the working capital channel, when the real wage is lower than its efficient value, the Ramsey planner uses inflation to correct for this inefficiency through making working capital more costly, while the equilibrium market tightness remains at the optimum. In this case, as a result of fully flexible prices, there is no steady-state relationship between the market tightness and inflation.

Under sticky prices, the efficient allocation is generally not achievable unless the real wage is at its socially efficient level. In this case, the optimal long-run inflation is zero and the Ramsey equilibrium solution is the same as the solution to the social planning problem. We show that when the real wage is higher than its efficient level, the Ramsey-optimal monetary policy calls for deflation. The reason is that a high real wage results in a market tightness, and thus an unemployment rate, lower than its socially optimum. Therefore, the Ramsey planner finds it optimal to impose a moderate negative inflation rate that, due to the presence of the working capital channel, induces higher surplus for the firm and encourages it to post more vacancies to push the economy toward the optimum. This is an interesting result, and it is the exact opposite of what the existing literature on optimal monetary policy, either the short-run studies or the long-run ones, recommends. In the above-mentioned literature on short-run dynamics in footnote 4, ⁷ if there is any optimal monetary policy deviation from price stability, the deviation is an increase in inflation to “grease the wheels of the labor market” in response to, for instance, adverse productivity shocks. This is also a long-run policy recommendation that arises in the related work of Carlsson and Westermark (2016). Carlsson and Westermark (2016) study the possibility of a positive deterministic optimal inflation rate in a DSGE monetary framework with matching frictions, sticky prices, and nominal wages that are not continuously rebargained and some newly hired workers that enter into an existing wage structure. They show that the Ramsey-optimal policy calls for increases in inflation when the steady-state unemployment rate is inefficiently high. Due to the presence of nominal wage rigidity, inflation in their framework affects real wage profiles and redistributes surplus toward workers. As we mentioned above, in our framework, in such a scenario that the market tightness is inefficiently low, it is deflation that is the optimal monetary policy, which through the working capital channel redistributes the surplus toward firms and encourages them to post more vacancies.

When the real wage is below its efficient value, which features an equilibrium market tightness above its efficient level, and an unemployment rate below its socially optimal rate, the optimal inflation rate is quite positive but its relationship with the real wage and the market tightness gap (i.e. defined as the difference between the equilibrium market tightness and its optimum that the social planner would choose) is nonlinear. More specifically, given the baseline calibration, there is a real wage threshold that below it the optimal inflation rate is increasing in the real wage, and beyond it, inflation is decreasing. In other words, the optimal inflation is initially sharply rising as the equilibrium market tightness rises above its optimum but then it starts to fall as the tightness gap gets larger. The reason behind this nonlinearity is because of the tradeoff that here exists between using inflation to correct for labor market inefficiencies when market tightness is significantly above it efficient level, and the fact that changes in the rate of inflation come at a cost since firms face nominal rigidities. When the market tightness deviation from its first best gets large, the Ramsey planner no longer finds it optimal to increase inflation and starts to lowering it as the real wage gets smaller and smaller.

We show that our results qualitatively hold when wages are assumed to be flexible and determined through a generalized Nash bargaining mechanism. As under fixed wages, when the market tightness is below the optimum, deflation is optimal since it lowers the adjusted workers’ outside option and the expected present value of hiring costs and thus provides incentive for firms to post more vacancies. The inverse happens when the equilibrium market tightness is larger than the optimum a social planner would choose.

We then introduce quadratic nominal wage rigidity into the model and find similar results in line with Carlsson and Westermark (2016) regarding the importance of nominal wage rigidities in inducing highly positive inflation rate in the steady state when the working capital channel is shut down. However, in the presence of working capital we show that the optimal policy results substantially change and are dominated by the borrowing channel.

The paper proceeds by describing the baseline monetary model with a working capital channel in the form of a cash-in-advance constraint on wages. Section 3 discusses the first-best allocation. Section 4 states the Ramsey problem. Section 5 discusses the main results by studying optimal monetary policy under both flexible and sticky prices and constant wages. Section 6 shows that the main results of the paper hold qualitatively with period-by-period Nash bargaining. Section 7 discusses the results under nominal wage rigidity as well. Section 8 demonstrates that the main results under sticky prices hold when we replace Rotemberg pricing with Calvo pricing. Section 9 offers some brief conclusions.

2. Model

The starting point of the analysis is a simple real business cycle model with sticky prices and matching frictions in the labor market, where firms are required to hold cash to pay for their wage payments. We assume that prices are sticky á la Rotemberg (1982) where firms face a convex cost of price adjustment. To derive clean analytical solutions, we first assume that real wages are constant á la Hall (2005), but we will also explore the robustness of our optimal monetary policy results by assuming flexible real wages, determined via a Nash bargaining mechanism. Furthermore, we will show that our results are robust to the introduction of nominal wage rigidity and under Calvo pricing.

2.1. The household

There is a representative household in the economy consisting of a measure one of family members. Each family member is either working or willing to work. If employed, the worker will receive the real wage $w_{t}$ . If unemployed, he will receive a fixed amount $z$ of household production units. As is standard in the related literature, we abstract from labor force participation choices and assume that all workers unemployed at the beginning of the period search for a job. Thus, the total number of unemployed workers searching for a job is $u_{t}=1-n_{t-1}$ , where $u$ is the rate of unemployment, and $n$ employment rate (the timing is the same as that in Monacelli et al. (2010), among others). Following convention, full consumption insurance is assumed, coming from the assumption that all family members belong to a representative household that pools income and shares consumption. The family members receive a utility flow $u$ from consuming $c_{t}\gt 0,$ where the function $u$ is strictly increasing and strictly concave in consumption.

Let $0\lt \beta \lt 1$ be the discount factor, the objective of the representative household is to maximize

(1) \begin{equation} E_{0}\sum _{t=0}^{\infty }\beta ^{t}u\!\left(c_{t}\right), \end{equation}

subject to period-t budget constraint,

(2) \begin{equation} P_{t}c_{t}+B_{t}=P_{t}w_{t}n_{t}+R_{t-1}B_{t-1}+P_{t}zu_{t}+P_{t}\varPhi _{t}-P_{t}t_{t}, \end{equation}

where $P_{t}$ is the aggregate price level, $R_{t}$ is the one-period gross nominal interest rate, $B_{t}$ represents one-period risk-free nominal bonds, $\varPhi _{t}$ denotes profits returned to the household through dividends, and $t_{t}$ denotes real lump-sum taxes. Condition (2) is supplemented with a solvency condition which prevents the household from engaging in Ponzi schemes.

The representative household maximizes (1) subject to (2) by choosing the set of processes $ \{ c_{t},B_{t} \} _{t=0}^{\infty }$ , taking as given $ \{ P_{t},w_{t},R_{t} \} _{t=0}^{\infty }$ . This maximization problem results in the following standard Euler equation:

(3) \begin{equation} u^{\prime}\!\left(c_{t}\right)=\beta R_{t}E_{t}\frac{u^{\prime}\!\left(c_{t+1}\right)}{\pi _{t+1}}, \end{equation}

where $\pi _{t}=\frac{P_{t}}{P_{t-1}}$ denotes the gross consumer price inflation between periods $t-1$ and $t$ .

2.2. The firm, search, vacancies, matching, and wages

The production side of the economy features a two-sector structure: final goods sector and intermediate goods sector. In the final goods sector, the representative firm aggregates a continuum of intermediate inputs according to a CES technology and produces a final homogeneous good, $y_{t}$ . The firm’s profit maximization problem gives rise to the following familiar first-order condition for its ith input

(4) \begin{equation} y_{it}=\left(\frac{P_{it}}{P_{t}}\right)^{-\xi }y_{t}, \end{equation}

where $\xi \gt 1$ denotes the elasticity of substitution between different intermediate inputs, $P_{it}$ is the nominal price of input $i$ at time $t$ , and $P_{t}=\left[\int _{0}^{1}P_{it}^{1-\xi }\right]^{\frac{1}{1-\xi }}$ is the nominal price index.

In the intermediate goods sector, each good’ s variety $i\in [0,1 ]$ is produced by a single firm in a monopolistically competitive environment, using labor $n_{it}$ as the sole factor input through the production technology:

\begin{equation*} y_{it}=a_{t}n_{it}, \end{equation*}

where $a_{t}$ is the exogenous total factor productivity, which in the long run is equal to unity. That is, $a=1$ . As is standard in the New Keynesian literature, it is assumed that the firm must satisfy demand at the posted price, which means

(5) \begin{equation} y_{it}=\left(\frac{P_{it}}{P_{t}}\right)^{-\xi }y_{t}=a_{t}n_{it}. \end{equation}

Each firm $i$ has a liquidity need to pay for its wage bill. This liquidity need is assumed to be covered by cash holdings. More specifically, firm $i$ faces the following liquidity constraint in real terms:

(6) \begin{equation} \phi w_{t}n_{it}\leq m_{it}, \end{equation}

where $m_{it}=\frac{M_{it}}{P_{t}}$ stands for the demand for real money balances in period $t$ , $M_{it}$ is nominal money holdings of the firm in period $t$ , and $\phi$ is a parameter denoting the fraction of the wage payment that must be backed with cash.

To hire new workers, firm $i$ posts $v_{it}$ vacancies, which is a costly process. Vacancy posting costs are assumed to be linear in the number of vacancies. That is, the firm must pay a fixed cost $\gamma$ for each posted job. Its period $t$ law of motion for employment level $n_{it}$ is:

(7) \begin{equation} n_{it}=\left(1-\rho \right)n_{it-1}+q_{t}v_{it}. \end{equation}

where $\rho$ stands for the fraction of workers employed at $t-1$ that exogenously separate from the firm, and $q_{t}$ is the probability that a vacancy is filled by the firm. As is the case in most of the related literature, in our framework new workers start their job right away, rather than with a one-period delay.

To introduce nominal wage rigidity, we follow Arseneau and Chugh (2008) and assume that firm $i$ pays nominal wages $W_{t}$ and faces a quadratic cost of adjusting nominal wages $\frac{\varphi ^{w}}{2}\left(\frac{W_{t}}{W_{t-1}}-1\right)^{2}$ for each of its workers, where $\varphi ^{w}$ measures the size of the wage adjustment cost. Notice that $w_{t}=\frac{W_{t}}{P_{t}}$ .

The objective of firm $i$ is to choose contingent plans for $ \{ P_{it},n_{it},v_{it},m_{it} \}$ to maximize the following present discounted value of profits:

\begin{align*} & E_{0}\sum _{t=0}^{\infty }\beta ^{t}\frac {u^{\prime}\!\left(c_{t}\right)}{u^{\prime}\!\left(c_{0}\right)}\frac {1}{P_{t}}\left\{\left(1+\tau ^{s}\right)P_{it}y_{it}+P_{t}\frac {m_{it-1}}{\pi _{t}}-W_{t}n_{it}-P_{t}m_{it}\right.\\ & \quad \left. -P_{t}\gamma v_{it}-P_{t}\frac {\varphi ^{w}}{2}\left(\frac {W_{t}}{W_{t-1}}-1\right)^{2}n_{it}-P_{t}\frac {\varphi }{2}\left(\frac {P_{it}}{P_{it-1}}-1\right)^{2}\right\} \end{align*}

subject to (5)−(7). Here, $\varphi$ measures the degree of price stickiness; $\frac{\varphi }{2}\!\left(\frac{P_{it}}{P_{it-1}}-1\right)^{2}$ represents price adjustment costs; and $\gamma v_{it}$ represents vacancy posting expenditures. ⁸ In order to ensure that no steady-state inefficiency arises from monopolistic competition, we assume that firms revenues are subsidized at the constant rate $\tau ^{s}.$ The production subsidies provided by the government are such that $\tau ^{s}=\frac{1}{\xi -1}$ , where they are assumed to be included in $g_{t}$ . ⁹

Following the literature, our attention is restricted to symmetric equilibria, where all firms charge the same price, produce the same amount of goods, and hire an equal amount of labor. Therefore, we can drop the index $i$ . Taken together, the above maximization problem yields the following first-order conditions (these conditions are associated with the case that the liquidity constrain is binding. As shown in Appendix C, in the Ramsey equilibrium because of the presence of price adjustment costs this is the case):

(8) \begin{equation} u^{\prime}\!\left(c_{t}\right)\pi _{t}\!\left(\pi _{t}-1\right)=\beta E_{t}u^{\prime}\!\left(c_{t+1}\right)\pi _{t+1}\!\left(\pi _{t+1}-1\right)+\frac{u^{\prime}\!\left(c_{t}\right)\xi a_{t}n_{t}}{\varphi }\left(\frac{\left(1+\tau ^{s}\right)\left(1-\xi \right)}{\xi }+mc_{t}\right), \end{equation}

(9) \begin{equation} \frac{\gamma }{q_{t}}=a_{t}mc_{t}-w_{t}\!\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)-\frac{\varphi ^{w}}{2}\left(\frac{w_{t}}{w_{t-1}}\pi _{t}-1\right)^{2}+\beta \!\left(1-\rho \right)E_{t}\frac{u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac{\gamma }{q_{t+1}}, \end{equation}

where $mc_{t}$ is the marginal cost of the firm. Equation (8) is the well-known expectations augmented Phillips curve. Equation (9) is the job creation condition where, relative to the standard one, due to cash holdings, the contemporaneous marginal profit from the match is distorted by the nominal interest rate. Besides, because of the presence of nominal wage rigidity, the contemporaneous marginal profit from the match includes also nominal wage adjustment costs. It states that at the optimal choice, the vacancy creation cost incurred by the firm is equated to the discounted expected value of profits from a match.

As is standard in the literature, we assume a constant returns to scale matching function which converts the number of unemployed workers $u$ and vacancies $v$ into matches $e (u_{t},v_{t} )$ . Given this, we can write the aggregate law of motion for employment as:

(10) \begin{equation} n_{t}=\left(1-\rho \right)n_{t-1}+e\!\left(u_{t},v_{t}\right). \end{equation}

The number of workers searching for jobs can also be written as:

(11) \begin{equation} u_{t}=1-n_{t-1}. \end{equation}

Next, we turn to discuss the determination of real wages, where the baseline wage determination mechanism is Nash bargaining which is fairly a standard assumption in the related literature. Let $p_{t}$ be the probability of finding a job by an unemployed worker. Since we assume a matching function that exhibits CRS, $p_{t}$ and $q_{t}$ , which as we defined above is the vacancy-filling probability, are defined as:

(12) \begin{equation} p_{t}=\frac{e\!\left(u_{t},v_{t}\right)}{u_{t}}, \end{equation}

(13) \begin{equation} q_{t}=\frac{e\!\left(u_{t},v_{t}\right)}{v_{t}}. \end{equation}

We adopt a constant returns to scale matching function of the form introduced by den Haan et al. (2000) that has the desirable property that job-finding and vacancy-filling probabilities fall between 0 and 1. That is:

(14) \begin{equation} e\!\left(u_{t},v_{t}\right)=\frac{u_{t}v_{t}}{\left(u_{t}^{\alpha }+v_{t}^{\alpha }\right)^{\frac{1}{\alpha }}} \end{equation}

in which $\alpha \gt 0$ .

The market tightness $\theta _{t}$ is defined as the ratio of vacancies to unemployment, $\theta _{t}=\frac{v_{t}}{u_{t}}$ , and is a variable at the heart of search and matching models. Furthermore, under constant returns to scale in matching, we have that $\theta _{t}=\frac{p_{t}}{q_{t}}$ , as is clear from definitions (12) and (13).

2.3. The equilibrium wage

The equilibrium wage is determined endogenously by a generalized period-by-period Nash bargaining between the worker and the firm. Under nominal wage rigidity, the Nash wage outcome is the following (see Appendix A for a formal derivation):

(15) \begin{align} \left(1-\frac{\kappa \!\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)}{\left(1-\kappa \right)\varDelta _{t}}\right)w_{t} & = \frac{-\kappa }{\left(1-\kappa \right)\!\varDelta _{t}}\left[a_{t}mc_{t}-\frac{\varphi ^{w}}{2}\left(\frac{w_{t}}{w_{t-1}}\pi _{t}-1\right)^{2}\right]+z\nonumber\\ & \quad +\frac{\kappa }{1-\kappa }\gamma E_{t}\beta \frac{u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac{1}{q_{t+1}}\!\left(\frac{1-\rho -p_{t+1}}{\varDelta _{t+1}}-\frac{1-\rho }{\varDelta _{t}}\right) \end{align}

where

(16) \begin{align}\Delta _{t} & =-\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)-\frac{\varphi ^{w}\pi _{t}}{w_{t-1}}\left(\frac{w_{t}}{w_{t-1}}\pi _{t}-1\right)\nonumber\\&\quad + E_{t}\!\left(1-\rho \right)\beta \frac{u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac{\varphi ^{w}w_{t+1}\pi _{t+1}}{w_{t}^{2}}\left(\frac{w_{t+1}}{w_{t}}\pi _{t+1}-1\right) \end{align}

Under flexible nominal wages, the above expressions reduce to

\begin{equation*} w_{t}=\frac {\kappa a_{t}mc_{t}}{\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)}+\left(1-\kappa \right)z+E_{t}\beta \frac {u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac {\kappa \gamma \!\left[1-\left(1-p_{t+1}\right)\frac {\left(1+\phi \left(1-R_{t}^{-1}\right)\right)}{\left(1+\phi\left(1-R_{t+1}^{-1}\right)\right)}\right]}{q_{t+1}\!\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)}. \end{equation*}

where the first two terms of the wage equation are a convex combination of the contemporaneous values of the marginal product of a new employee $\frac{a_{t}mc_{t}}{\left(1+\phi\left(1-R_{t}^{-1}\right)\right)}$ , which is distorted by the presence of the working capital constraint, and the value of the productivity of workers in home production $z$ . The last term captures the forward-looking aspect of employment where it is affected by both period-t and expectations of period-(t + 1) nominal interest rates.

When there is no working capital channel, the last expression can be written as

\begin{equation*} w_{t}=\kappa a_{t}mc_{t}+\!\left(1-\kappa \right)z+E_{t}\beta \frac {u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\kappa \gamma \theta _{t+1}, \end{equation*}

which is the standard Nash wage equation.

2.4. Closing the model

The final steps are to specify the government budget constraint and the aggregate resource constraint. The government imposes lump-sum taxes, $t_{t}$ , issues one-period nominally risk-free bonds, $B_{t}$ , and prints money, $M_{t}$ , to finance its exogenous spending, $g_{t}$ . Thus, the period- $t$ government budget constraint can be written as

(17) \begin{equation} B_{t}+M_{t}+P_{t}t_{t}=R_{t-1}B_{t-1}+M_{t-1}+P_{t}g_{t}, \end{equation}

Finally, the resource constraint of the economy is

(18) \begin{equation} c_{t}+g_{t}+\gamma v_{t}+\frac{\varphi }{2}\left(\pi _{t}-1\right)^{2}+\frac{\varphi ^{w}}{2}\left(\frac{w_{t}}{w_{t-1}}\pi _{t}-1\right)^{2}=a_{t}n_{t}+\left(1-n_{t}\right)z. \end{equation}

2.5. Equilibrium

A competitive equilibrium in this economy is a sequence of prices and allocations $R_{t}$ , $P_{t}$ , $M_{t}$ , $c_{t}$ , $n_{t}$ , $w_{t}$ , $\Delta _{t}$ , $u_{t}$ , $v_{t}$ , $B_{t}$ for $t=0,\cdots,\infty$ that satisfy conditions (3), (8)−(11), (15)−(18), and

(19) \begin{equation} \phi w_{t}n_{t}\leq m_{t}, \end{equation}

(20) \begin{equation} R_{t}\geq 1, \end{equation}

given the definitions of $q_{t}$ and $p_{t}$ , (12) and (13), respectively, and given $ \{ a_{t},g_{t} \}$ , and the initial condition $R_{-1}B_{-1}+M_{-1}.$ In equilibrium, it must be the case that the nominal interest rate is non-negative; that is, condition (20) must hold.

3. Social planner problem

Before moving to present and discuss the Ramsey-optimal policy results, it would be helpful to characterize the efficient allocation in this section. As shown in appendix B, the first-best solution in our search and matching model for a general constant returns matching technology is the following

(21) \begin{equation} \frac{\gamma }{e_{v}\!\left(u_{t},v_{t}\right)}=(a_{t}-z)+E_{t}\beta \frac{u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac{\gamma }{e_{v}\!\left(u_{t+1},v_{t+1}\right)}\left[1-\rho -e_{u}\!\left(u_{t+1},v_{t+1}\right)\right], \end{equation}

in which $e_{v} (. )$ and $e_{u} (. )$ are the marginal products of the matching function with respect to vacancies and unemployment, respectively. This condition is an intertemporal dimension of efficiency, and it corresponds to the RBC model’ s Euler equation for efficient capital accumulation (see Arseneau and Chugh (2012)). It shows that at optimum the intertemporal MRS between $ (c_{t},c_{t+1} )$ is equal to the intertemporal MRT between $ (c_{t},c_{t+1} )$ . In the steady state, it takes the following form (given that in the long-run $a=1$ ):

(22) \begin{equation} \frac{\gamma }{e_{v}\!\left(.\right)}=\frac{1-z}{1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)}. \end{equation}

Note that both $e_{v} (. )$ and $e_{u} (. )$ are functions of market tightness, $\theta =\frac{v}{u}$ , only

4. Ramsey problem

In the present model, the Ramsey-optimal monetary policy is to choose the path of the nominal interest rate that is associated with the competitive equilibrium that yields the highest level of welfare to households. Formally, the Ramsey problem is to choose state-contingent processes $ \left\{ R_{t},\pi _{t},m_{t},c_{t},n_{t},w_{t},u_{t},\Delta _{t},v_{t},mc_{t} \right\} _{t=0}^{t=\infty }$ to maximize (1) subject to (3), (8)−(11), (15), (16), and (18)−(20). Note that since the government has access to lump-sum taxes its budget constraint is always satisfied. Thus, condition (17) is not a constraint in the Ramsey planner problem.

To derive clean analytical results that help us to see how the working capital channel operates and influences the optimal rates of nominal interest rate and inflation, we follow Hall (2005) and first assume that real wages are perfectly rigid. We then show numerically in Sections 6 and 7 that our main results remain qualitatively the same under fully flexible wages and nominal wage rigidity, respectively.

5. Main results

To gain intuition for the results presented below, it is important to note that market tightness does two things in the canonical matching framework: through the matching function it drives job creation and affects the expected hiring costs of workers, $\frac{\gamma }{q\left(\theta _{t+1}\right)}$ . As discussed by Pissarides (2009), these are the two central interconnected and identifying features of the canonical DMP model. ¹⁰ Now, to see how not only today’s inflation rate but also the expected future values of inflation, due to the presence of the working capital channel, matter to the expected hiring costs of the firm iterate the job creation equation (9) forward to obtain (for convenience consider the case that there is no nominal wage rigidity):

(23) \begin{equation} \frac{\gamma }{q_{t}}=E_{t}\sum _{k=0}^{\infty }\beta ^{k}\!\left(1-\rho \right)^{k}\frac{u^{\prime}\!\left(c_{t+k}\right)}{u^{\prime}\!\left(c_{t}\right)}\!\left(a_{t+k}mc_{t+k}-w_{t+k}\!\left(1+\phi \!\left(1-\beta \pi _{t+k}^{-1}\right)\right)\right). \end{equation}

The important thing to note about equation (23) is that it highlights how changes in inflation affect working capital and thus in turn hiring costs. The canonical matching framework assumptions are such that workers expected hiring costs increase in proportion to the increase in the mean duration of vacancies. The increase in average hiring costs checks the growth in vacancies and so reduces the firm’s incentive to post more vacancies.

Another way of seeing the importance of working capital channel to the firm’s marginal search cost and thus employment is to use equations (10)−(13) and derive the job-finding rate $p_{t}$ as a function of past and present employment and then the cost of hiring a worker in terms of $p_{t}$

(24) \begin{equation} p_{t}=\frac{n_{t}-\left(1-\rho n_{t-1}\right)}{1-n_{t-1}}\quad\qquad \end{equation}

(25) \begin{equation} \frac{\gamma }{q_{t}}=\gamma \!\left(\frac{1}{1-p_{t}^{\alpha }}\right)^{\frac{1}{\alpha }}=\gamma \!\left(\frac{1}{1-\left(\frac{n_{t}-\left(1-\rho n_{t-1}\right)}{1-n_{t-1}}\right)^{\alpha }}\right)^{\frac{1}{\alpha }}=\bar{\gamma }\!\left(n_{t-1},n_{t}\right) \end{equation}

Abstracting from the intertemporal dimension of hiring condition (25) reduces to

\begin{equation*} \bar {\gamma }\!\left(n,n\right)=\gamma \!\left(\frac {1}{1-\left(\frac {\left(1+\rho \right)n-1}{1-n}\right)^{\alpha }}\right)^{\frac {1}{\alpha }} \end{equation*}

Next, using the steady state of condition (9) under no nominal wage rigidity (and the fact that $a=1$ ) we have

\begin{equation*} mc-\left(1-\beta \!\left(1-\rho \right)\right)\bar {\gamma }\!\left(n,n\right)=w\!\left(1+\phi \!\left(1-\beta \pi ^{-1}\right)\right) \end{equation*}

which shows that the marginal search cost that is increasing in employment, all else equal, is decreasing in the inflation rate. Notice also that our assumption of a non-Cobb−Douglas matching function implies that the relationship between the cost of hiring $\bar{\gamma } (n_{t-1},n_{t} )$ and the job-finding rate $p_{t}$ is nonlinear.

Now, we turn to our analytical Ramsey policy implications. As mentioned before, to present clean analytical results we first assume that real wages are fixed over time. More precisely, suppose that the real wage is a constant $\bar{w}$ inside the equilibrium bargaining set, that is $w_{t}=\bar{w}$ , where $\bar{w}\in [z,1)$ . Notice that, we rule out a zero real wage, $\bar{w}=0$ , as an equilibrium outcome. This is because a positive valuation of leisure by workers, here captured by a positive outside option $z\gt 0$ , imposes a positive lower bound on an equilibrium $\bar{w}$ . Given a constant equilibrium real wage, we have the following proposition.

Proposition 1. The solution to the Ramsey problem under a working capital channel, sticky prices, and constant real wages in the steady state is

(26) \begin{equation} \varphi \pi ^{2}\!\left(\pi -1\right)=\dfrac{\phi \bar{w}\!\left\{ \dfrac{\gamma }{e_{v}\!\left(.\right)}-\dfrac{(1-z)}{\!\left[1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)\right]}\right\} }{\dfrac{\beta \rho \dfrac{\gamma e_{u}\!\left(.\right)}{qe\!\left(.\right)}+\dfrac{1}{1-u}\!\left(\dfrac{\!\left(1+\tau ^{s}\right)\!\left(1-\xi \right)}{\xi }+mc\right)}{1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)}+\dfrac{\gamma \rho }{e_{v}\!\left(.\right)e\!\left(.\right)}\!\left[1-\dfrac{e_{v}\!\left(.\right)}{q}\right]}. \end{equation}

Proof. See Appendix B.

The Ramsey condition (26) along with the steady states of New Keynesian Phillips curve (8), the job creation condition (11), the aggregate law of motion for employment (10), and the aggregate resource constraint (8), which are, respectively,

(27) \begin{equation} \frac{\gamma }{q}=\frac{mc-\bar{w}\!\left(1+\phi \!\left(1-\frac{\beta }{\pi }\right)\right)}{1-\beta \!\left(1-\rho \right)}, \end{equation}

(28) \begin{equation} \frac{\left(1+\tau ^{s}\right)\left(1-\xi \right)}{\xi }+mc=\frac{\left(1-\beta \right)\varphi \pi \!\left(\pi -1\right)}{\xi \!\left(1-u\right)}, \end{equation}

(29) \begin{equation} \rho \!\left(1-u\right)=ue\!\left(1,\theta \right), \end{equation}

(30) \begin{equation} c+\gamma u\theta +\frac{\varphi }{2}\left(\pi -1\right)^{2}=\left(1-u\right)\left(1-z-\alpha \right)+z, \end{equation}

give us a system of 5 equations in 5 variables $\pi$ , $u$ , $\theta$ , $mc$ , and $c$ . Note that above we used the facts that $u=1-n$ , $\pi =\beta R$ , and that $g=\alpha n$ , where $\alpha$ is the government spending to GDP ratio.

From the optimality condition (26), we observe that the optimal long-run (net) inflation rate is always zero if one of the following two conditions holds: first, if there is no working capital channel, $\phi =0$ . In this case, regardless of the wage mechanism (whether it is fully rigid or quite flexible) and the wage value, monetary policy cannot affect the marginal cost (benefit) of creating a vacancy and the only possible solution is $\pi =1$ . ¹¹ Therefore, it should be clear that our results on the optimal inflation rate have qualitatively nothing to do with the wage mechanism. Notice that, under $\phi =0$ and with Rotemberg pricing assumption, inflation works like a fixed operational cost independent of the size of the firm. Therefore, setting inflation to any positive or negative number only imposes costs to the firm, but does not change the marginal cost (benefit) of vacancy posting. Second, when the labor market is at its efficient state; that is, when the following condition holds:

(31) \begin{equation} \frac{\gamma }{e_{v}\!\left(.\right)}=\frac{(1-z)}{\left[1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)\right]}, \end{equation}

which is simply the first-best allocation presented by (22). Thus, under $\phi,\varphi \gt 0$ , the optimal rate of inflation is zero only if the equilibrium delivers the first-best allocation. When $\phi \gt 0$ and the economy is not at the optimum, the optimal inflation rate is different from zero. In this case, as is clear from (26) to (28), and we will also numerically discuss later in detail, the relationship between the optimal inflation and the real wage is highly nonlinear.

Ramsey-optimal outcome (26) shows that, in the presence of working capital channel, the key endogenous variables and structural parameters determining the optimal inflation rate are the following: the level of the real wage $\bar{w}$ ; the steady-state unemployment rate $u$ ; the marginal products of matching, $e_{u} (. )$ and $e_{v} (. )$ which are functions of market tightness, the probability of filling a vacancy $q$ ; the exogenous job separation rate $\rho$ ; the subjective discount factor $\beta$ ; the degree of price stickiness $\varphi$ ; the elasticity of substitution between different intermediate inputs $\xi$ ; and production subsidies $\tau ^{s}$ . With the exception of the last four parameters, the other factors affecting $\pi$ are related to the labor market. Observe from (26) that the optimal inflation rate is closely related to the market tightness $\theta$ and therefore the unemployment rate $u$ . Such a relation, as we will discuss shortly in the next subsection, does not exist under flexible prices.

Before turning to discuss quantitatively the Ramsey-optimal inflation rate under sticky prices and under the working capital channel in detail, we discuss the optimal policy under flexible prices as a special case.

5.1. The case of flexible prices

An special case of the Ramsey condition (26) is when prices are fully flexible, $\varphi =0$ , and there is no market power (or that the government provides subsidy to offset steady-state distortions caused by the market power, that means $mc=1$ ). It is clear from (26) that under no price stickiness, the condition reduces to (31) which is the social planner’s solution. However, as we make it clear from the Ramsey first-order conditions presented in appendix B, this first-best solution follows as long as the inequality constraint on the gross nominal interest rate, $R_{t}\geq 1$ , does not bind. Suppose that $\theta ^{s}$ is the socially efficient value of market tightness that we get from (22), given the structural parameters. Also define $\bar{w}^{*}$ as that level of the real wage at $\theta ^{s}$ that gives $R=1$ from (27). That is:

(32) \begin{equation} \bar{w}^{*}=1-\frac{\gamma }{q\!\left(\theta ^{s}\right)}\left(1-\beta \!\left(1-\rho \right)\right). \end{equation}

Given (32), and using (27), we can therefore write down the optimal value of nominal interest rate as:

(33) \begin{equation} R^{-1}=1-\frac{1}{\phi }\left(\frac{\bar{w}^{*}}{\bar{w}}-1\right)\,\,\,\text{for}\,\bar{w}\leq \bar{w}^{*}. \end{equation}

Note also that $R^{-1}$ must be between 0 and 1. This means that the following condition must always hold:

(34) \begin{equation} z\leq \frac{\bar{w}^{*}}{1+\phi }\leq \bar{w}\leq \bar{w}^{*}. \end{equation}

Given what we have derived in this subsection on the optimal nominal interest rate under flexible prices, we can have the following proposition:

Proposition 2. Under fully flexible prices, $\varphi =0$ , and perfect competition, the optimal monetary policy is independent of market tightness and there is no long-run relationship between the optimal inflation and unemployment. In this case, we have that:

a. if $\bar{w}\gt \bar{w}^{*}$ , then the lower bound on the nominal interest rate is binding, $R=1$ , and market tightness is determined by

(35) \begin{equation} q\!\left(\theta \right)=\frac{\gamma \!\left(1-\beta \!\left(1-\rho \right)\right)}{1-\bar{w}} \end{equation}

b. if $z\leq \frac{\bar{w}^{*}}{1+\phi }\leq \bar{w}\leq \bar{w}^{*},$ the zero lower bound is slack, $R\gt 1$ , and the optimal allocation is the same as that of the social planner.

c. if $\bar{w}\lt \frac{\bar{w}^{*}}{1+\phi }$ then there is no Ramsey equilibrium, since $R=\infty$ , which leads to a real interest rate equal to infinity.

The fact that the optimal inflation rate under flexible prices is independent of market tightness, and therefore unemployment, is evident from condition (33). In the case of $\bar{w}\gt \bar{w}^{*}$ (part a), it should be clear that the optimal allocation is different from the first-best allocation and $\theta \lt \theta ^{s}$ . In other words, while the Friedman rule (R = 1) is in place the Ramsey solution is different from the efficient one, which in this case cannot be achieved.

Regarding the second part, note that the Friedman rule is optimal only when the real wage is at the level implied by (32). When the wage is below $\bar{w}^{*}$ deviations from the Friedman rule become optimal and monetary policy can correct for labor market inefficiencies. In this case, a lower than $\bar{w}$ is associated with a larger inflation rate that sets the Ramsey equilibrium at the optimum. Note that this is the case as long as $\bar{w}\geq \frac{\bar{w}^{*}}{1+\phi }\geq z$ . Because, as part c emphasizes, when $\bar{w}\lt \frac{\bar{w}^{*}}{1+\phi }$ the real wage is well below worker’s outside option, and thus, there is no equilibrium associated with that rate. Under the assumption of zero outside option, we can also say that since the wage is too low even imposing an extremely high cost on the firm through setting the nominal interest rate at infinity cannot correct for the inefficiency.

This outcome that the Ramsey planner may find it optimal to deviate from the Friedman rule when the real wage is lower than its optimum, while the economy is at its socially efficient situation, is an interesting result. It is all the more important when we recall that in DSGE monetary frameworks with perfect competition and neoclassical labor markets the Friedman rule is the only socially optimal outcome under lump-sum taxation. It is also important to note that the reason behind the non-optimality of the Friedman rule here is different from that raised by Phelps (1973). Phelps mentioned that the reason behind the optimality of the Friedman rule is because the government is assumed to have access to lump-sum taxes. When this assumption is relaxed and the government has to finance its spending via distortionary taxation, then the Friedman rule ceases to be optimal. As a matter of fact, following the Phelps’ criticism, a large literature has emerged that shows that the Friedman rule is always optimal regardless of whether the government finances its budget through lump-sum taxes or via distortionary taxation (see Schmitt-Grohé and Uribe (2010) for a review). As we observed above in Proposition 2, a positive nominal interest rate follows despite the presence of lump-sum taxes. It should be clear that this result is not related to the fiscal policy and the absence or presence of distortionary taxes. It is a consequence of the presence of a working capital channel and the labor market outcomes. Part b of the proposition establishes the non-optimality of the Friedman is in fact a socially optimal outcome when the real wage is below its efficient magnitude.

It is also important to mention that the assumption of a working capital channel that leads to this result that the optimal monetary policy calls for inflation when the real wage is below its socially efficient level is quite the opposite of what we see in the related work of Carlsson and Westermark (2016). There, due to the assumption of discontinuously rebargained nominal wages the Ramsey-optimal policy is to increase inflation when vacancy postings are inefficiently low to push the equilibrium closer to the optimum. Also note that, in Carlsson and Westermark (2016), as is the case in all the existing literature on optimal monetary policy with a fiat money demand by households, the presence of the monetary friction simply calls for an optimal inflation rate that falls between the negative of the real interest rate and zero.

As we already observed in Proposition 1, and will also shortly discuss quantitatively in subsection 5.3, similar results emerge under sticky prices as well. That is, the optimal inflation rate can significantly change in response to deviations of labor market equilibrium from its socially optimum, despite the fact that price changes come with a cost. The key difference, as is evident from the Ramsey condition (26), is that under sticky prices a long-run relationship between the optimal rate of inflation and the market tightness emerges that is absent under flexible prices.

Figure 2 summarizes the optimal monetary policy outcomes under sticky prices and no working capital channel, and the case of flexible prices and a working capital channel. More specifically, the graph is a schematic depiction of the optimal nominal interest rate under the above-discussed scenarios. The red line is the optimal monetary policy under sticky prices and no working capital channel. As we demonstrated above in Proposition 1, the optimal net inflation rate in this case is always zero. This means that the gross nominal interest rate equals $R=\frac{1}{\beta }$ . The blue line in the graph shows the optimal monetary policy under both flexible prices and the working capital constraint. On the graph, $\bar{w}^{*}$ is the real wage at which the Friedman rule is optimal condition (32). For any value below that level and above $\frac{\bar{w}^{*}}{1+\phi }$ , the optimal nominal interest rate is above zero, and it is higher the lower is the real wage relative to $\bar{w}^{*}$ . It then follows that there is a wage level in between these two values that at which the optimal inflation rate is zero in the presence of both the working capital channel and sticky prices. We show this level by $w_{0}$ in the figure, which is determined by condition (27), evaluated at the efficient market tightness level. We get back to this value in our analysis in subsection 5.3 frequently.

Figure 2. A schematic depiction of optimal monetary policy under different scenarios.

5.2. Calibration

To illustrate how the introduction of a capital constraint can fundamentally change the optimal long-run monetary policy implications under sticky prices in the matching framework, in this subsection, we calibrate the model to provide some quantitative results on the Ramsey steady state in the next subsection. The time unit is assumed to be a quarter. We adopt the following form for the period utility function: $u (c_{t} )=\text{ln}c_{t}$ . As in Carlsson and Westermark (2016), the matching function parameter $\alpha$ is assumed to be 1.27. The subjective discount factor is set to $\beta =0.99$ . The fraction of the wage bill $\phi$ that must be backed with monetary assets is normalized to 1. Given that the assumed form of price stickiness leads to exactly the same form for the new Keynesian Phillips curve as that in Schmitt-Grohé and Uribe (2004), we adopt their calibration of the price elasticity of demand, $\xi$ , and the degree of price stickiness, $\varphi$ , and set them, respectively, to 6 and 17.5. The value of $\xi$ implies to a value-added markup of prices over marginal costs equal 0.2. The calibration of $\varphi$ , as pointed out by Schmitt-Grohé and Uribe (2004), is equivalent to assuming that s that firms change their price on average every 9 months in a Calvo–Yun staggered-price setting model.

The quarterly job separation rate $\rho$ is set at 0.1, a standard calibration in search and matching models (see, e.g. Christiano et al. (2016) and Ljungqvist and Sargent (2017)). In line with Ljungqvist and Sargent (2017), we set the value of the outside option $z$ to 0.6. To calibrate the per-vacancy posting cost $\gamma$ , we follow Petrosky-Nadeau et al. (2018) who find that for the DMP model to generate a realistic unemployment rate of 6.3% for the US economy, based on postwar US data, the total unit costs of vacancy should on average be around 0.7. Thus, we initially set $\gamma =0.7$ .

Suppose that $\theta ^{\text{FB}}$ stands for the value of market tightness that we get from the first-best allocation (condition (22)) and $\theta ^{\text{RA}}$ for what we get from the Ramsey problem. Define the tightness gap as the difference between $\theta ^{\text{RA}}$ and $\theta ^{\text{FB}}$ ; that is $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . Given the baseline calibration, the value of the real wage $w_{0}$ in the above graph, that gives the efficient allocation in the presence of the working capital channel and sticky prices and results in the optimality of zero inflation rate, is $w_{0}=0.8837$ . Table 1 summarizes the calibration of the economy.

Table 1. Calibration

Table 2. Ramsey steady state when $\phi \gt 0$ and $\varphi \gt 0$ when the cost of posting a vacancy changes

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

5.3. Quantitative results on the general case (Sticky Prices)

Tables 2 and 3 present the long-run Ramsey-optimal inflation under sticky prices and the cash-in-advance constraint (in percent per annum) in the monetary model under consideration for different values of the per-vacancy posting cost $\gamma$ and the real wage $\bar{w}$ , respectively. The economy is initially at its socially efficient position (shown by the fourth row of each table), and then, we start changing either $\gamma$ or $\bar{w}$ , while all the other structural parameters are kept fixed at their baseline calibration. For the real wage, the baseline is 0.8837 and for $\gamma$ it is 0.7 that gives together the optimum (see our discussion of calibration in the previous subsection). The fourth row of each table reports the Ramsey results for the case that the market tightness gap is zero; that is, the Ramsey-optimal allocation delivers a level of welfare that is as high as the level of welfare associated with the s social planner’ s problem (the highest attainable level of welfare). In this case, as we discussed above, the Ramsey-optimal inflation rate is zero. Notice that, to compute the deterministic Ramsey steady-state equilibrium, we simply use a nonlinear Matlab numerical solution algorithm, which gives an exact solution to the above equations.

Table 3. Ramsey steady state when $\phi \gt 0$ and $\varphi \gt 0$ and the real wage changes

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

Table 2 documents the Ramsey results for a range of values for the cost of posting a vacancy $\gamma$ above and below its baseline calibration, keeping all other structural parameters and the real wage at their baseline values. ¹² The main result emerging from this table is that the optimal inflation rate is strictly decreasing in hiring costs. When the hiring costs are above the level that gives the efficient outcome ( $\gamma =0.7$ ) deflation emerges as the optimal monetary policy. When they fall below that level, all else equal, inflation becomes optimal. More precisely, higher hiring costs make the firm’s surplus too small relative to the optimum and result in a smaller market tightness compared to its efficient level. Since due to the working capital channel the firm’s surplus is decreasing in inflation, the Ramsey planner finds it optimal to impose negative inflation to persuade the firm to create more vacancies to push the equilibrium closer to the optimum. The logic is the same for $\gamma \lt 0.7$ . In this case, the optimal monetary policy calls for higher inflation to make vacancy creation more costly for firms to lower the gap between the equilibrium market tightness and the socially efficient level of tightness.

This mechanism is in the exact opposite of what we see in the existing related literature on optimal monetary policy under downward nominal wage rigidity or under sticky prices with/without wage rigidity, and with/without frictional labor markets (see, e.g. Thomas (2008); Faia (2009); Blanchard and Gali (2010); Gali (2010); Ravenna and Walsh (2011, 2012); and Kim and Ruge-Murcia (2009, 2011)). First, it is important to recall again that in this literature the optimal deterministic inflation rate is always zero. In contrast, the focus of this paper is to explore how the Ramsey-optimal steady-state inflation rate can be positive in the presence of a working capital channel. Second, that literature finds that over the business cycle a positive rate of inflation may “grease the wheel of the labor market” and be welfare improving. Here, in contrast, as we observe from Table 2, a positive rate of inflation is optimal when employment is above the social planner solution, and it is deflation that is optimal in “greasing the wheels of the labor market.”

The results and the mechanism here are also the opposite of what Carlsson and Westermark (2016) find in a closely related paper. More precisely, Carlsson and Westermark (2016) study optimal monetary policy in a DSGE model with search and matching frictions where some newly hired workers enter into an existing wage structure. They motivate a money demand friction by households that in itself calls for an on overage negative optimal inflation rate. They find that under nominal wage rigidity, the Ramsey planner has incentive to raise the steady-state rate of inflation to affect firms’ incentives to create more vacancies and therefore lower unemployment rate. As we analytically discussed in the previous subsections and quantitatively showed in Table 2, in the presence of a working capital channel, when the firm’s surplus due to larger hiring costs is too low relative to the optimum, the Ramsey-optimal policy calls for deflation and not inflation to correct for this inefficiency in the labor market.

Next, we turn to discuss the optimal monetary policy in our model for a range of real wage values while keeping $\gamma$ and all the other parameters fixed at their baseline calibration. First note that, as we discussed above, the optimality condition (26) shows that when condition (31) holds, which means that the labor market is at the optimum, the optimal rate of inflation is zero. In Figure 2, we denote the efficient real wage associated with this case by $w_{0}$ , which is defined as:

(36) \begin{equation} w_{0}=\frac{1-\dfrac{\gamma }{q\left(\theta ^{s}\right)}\left(1-\beta \!\left(1-\rho \right)\right)}{1+\phi \!\left(1-\beta \right)}=\frac{\bar{w}^{*}}{1+\phi \!\left(1-\beta \right)}. \end{equation}

Now, using (36) along with (27)–(29), we have that: ¹³

(37) \begin{equation} \bar{w}\!\left(1+\phi \!\left(1-\frac{\beta }{\pi }\right)\right)=1+\frac{\left(1-\beta \right)\varphi \pi \!\left(\pi -1\right)}{\xi \!\left(1-u\right)}-\frac{q\!\left(\theta ^{s}\right)}{q\!\left(\theta \right)}\left(1-\bar{w}^{*}\right). \end{equation}

Next, use (28) to modify (26) as:

(38) \begin{equation} \varphi \pi ^{2}\left(\pi -1\right)=\dfrac{\phi \bar{w}\!\left\{ \dfrac{\gamma }{e_{v}\!\left(.\right)}-\dfrac{1-z}{1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)}\right\} }{\dfrac{\beta \rho \dfrac{\gamma e_{u}\!\left(.\right)}{q\!\left(\theta \right)e\!\left(.\right)}+\dfrac{\left(1-\beta \right)\varphi \pi \!\left(\pi -1\right)}{\xi \!\left(1-u\right)^{2}}}{1-\beta \!\left(1-\rho -e_{u}\!\left(.\right)\right)}+\dfrac{\gamma \rho }{e_{v}\!\left(.\right)e\!\left(.\right)}\left[1-\dfrac{e_{v}\!\left(.\right)}{q\!\left(\theta \right)}\right]}. \end{equation}

When the real wage $\bar{w}$ is not equal to $w_{0}$ , the optimal inflation rate is different from zero. The reason is that in this case condition (31) ceases to be the case, and therefore, the numerator of (38) is no longer zero. Thus, $\pi =1$ is not a solution to this equation. In this case, conditions (29), (37), and (38) together give us the optimal rate of inflation and the equilibrium values of unemployment $u$ and market tightness $\theta$ . ¹⁴

Table 3 documents the Ramsey results for a range of values for the real wage above and below the efficient level, $w_{0}=0.8837$ , all else equal. Figure 3 plots the optimal relationship between $\pi$ and $\bar{w}$ for a wide possible range of values for $\bar{w}$ . First, as the table shows, and expected to be seen, there is an inverse (relatively) linear relationship between the real wage and the tightness gap; the lower is the real wage, the lager is the firm surplus, and therefore, the higher is the tightness gap, $\theta ^{\text{GAP}}$ , (the higher is the market tightness relative to its optimum). Regarding the optimal monetary policy, what the panel and the figure together illustrate is that for real wages above $w_{0}$ the Ramsey-optimal policy calls for deflation. That is, when $\bar{w}\gt w_{0}$ the market tightness is below its socially efficient level and the Ramsey planner uses negative inflation to move the economy toward the optimum. When $\bar{w}\lt w_{0}$ , the optimal inflation rate can be significantly positive and high, despite the fact that price changes are costly. More specifically, in the case of $\bar{w}\lt w_{0}$ , except for a range of values for $\bar{w}$ close to its efficient value, the optimal inflation rate is increasing in the real wage. As the figure displays, the optimal inflation rate is increasing in $0.6\leq \bar{w}\leq 0.75$ and then starts to monotonically fall as $\bar{w}$ rises more and reaches zero at $w_{0}=0.8837$ . It becomes negative as $\bar{w}$ increases above the efficient level (0.8837), as highlighted by the red line. Note that the real wage cannot increase beyond 0.9 because otherwise the gross nominal interest $R$ falls below 1.

Figure 3. The y-axis shows the optimal inflation rate per annum and the x-axis stands for the real wage rate.

Figure 4. The y-axis shows the optimal inflation rate per annum, and the x-axis displays the market tightness gap. A positive deviation means that the tightness is above its socially efficient level.

The intuition for this nonlinear relationship is as follows. Because price adjustment is costly, the Ramsey planner faces a tradeoff. On the one hand, the planner would like to use higher inflation to correct for labor market inefficiencies when the market tightness gap gets larger. On the other hand, price changes are brought about at a cost, therefore, the Ramsey planner has incentives to stabilize the price level in order to minimize the cost. Thus, the price adjustment costs are greater the more inflation gets higher with larger market tightness gap. What the table and the figures below illustrate is that when the market tightness gap increases beyond a level, the tradeoff starts to get resolved in favor of price stability and inflation starts to fall as the gap widens.

Figure 4 plots the optimal inflation rate as a function of the market tightness gap for the range of values of the real wage in Figure 3. As we observed from Table 3, the market tightness gap varies inversely (linearly) with the real wage and the optimal inflation varies nonlinearly with $\bar{w}$ . Therefore, it is natural to expect a nonlinear relation between inflation and the tightness gap, which is what Figure 4 displays. More specifically, the figure shows that negative inflation (deflation) is optimal when the tightness is below its first-best level. A positive inflation rate is optimal when the economy operates above its first-best capacity, by that we mean the market tightness is excessive relative to the optimum a social planner would choose. More specifically, for the initial positive deviations of the tightness, which are associated with small falls in the real wage relative to its efficient level, the optimal inflation rate sharply rises, but then steadily falls as the real wage falls sharply below its efficient value.

A relatively qualitatively similar nonlinear relationship as that between inflation and tightness gap arises when we plot the optimal inflation rate for the unemployment gap, which is defined as the unemployment rate featured by the Ramsey problem minus the efficient unemployment rate.

Figure 5 plots this relationship. The unemployment gap is zero when the real wage is at its efficient level, which is the point where the blue line crosses the red line. At that point, the optimal inflation rate is zero. The figure makes it clear that the optimal inflation is nonlinear in the unemployment rate. When unemployment is above its efficient level, negative inflation rate is optimal. When it is below the efficient rate, positive inflation rate is optimal. In this case, inflation is negatively related to the unemployment rate once the unemployment gap is roughly between −0.063 and 0 and is positively rising with unemployment gap when it is between −0.084 and −0.063.

Figure 5. The y-axis shows the optimal inflation rate per annum, and the x-axis displays the unemployment gap. The unemployment gap is the difference between the unemployment rate and its efficient rate.

6. Flexible wages

In this section, we drop the assumption of fixed real wages and assume that wages are fully flexible and determined by the Nash wage outcome (15), keeping everything else as before, including keeping our parameter values at their baseline calibrations. We have a new parameter to calibrate, workers’ bargaining power $\kappa$ . We calibrate $\kappa$ initially such that the Ramsey economy delivers the first-best allocation., That is, it gives us a level of market tightness that is the same as that we obtain from (31). All else equal, this is the case under $\kappa =0.3479$ .

Under Nash bargaining, the expected cost of hiring a worker in the steady state can be obtained from the long-run versions of job creation condition (9) and the Nash wage outcome (15) which, respectively, are

(39) \begin{equation} w=\frac{mc-\dfrac{\gamma }{q\!\left(\theta \right)}\left(1-\beta \!\left(1-\rho \right)\right)}{1+\phi \!\left(1-\frac{\beta }{\pi }\right)}, \end{equation}

(40) \begin{equation} w=\frac{\kappa mc}{\left(1+\phi \!\left(1-\frac{\beta }{\pi }\right)\right)}+\left(1-\kappa \right)z+\beta \frac{\kappa \gamma p\!\left(\theta \right)}{q\!\left(\theta \right)\left(1+\phi \!\left(1-\frac{\beta }{\pi }\right)\right)}. \end{equation}

Putting these two conditions together, we have that

(41) \begin{equation} \frac{\gamma }{q\!\left(\theta \right)}=\frac{\left(1-\kappa \right)\left(mc-z\!\left(1+\phi \!\left(1-\frac{\beta }{\pi }\right)\right)\right)}{1-\beta \!\left(1-\rho -\kappa p\!\left(\theta \right)\right)}, \end{equation}

where, in line with what we discussed before, in equilibrium the worker’s hiring cost is increasing in the working capital channel. More precisely, inflation (deflation) by making working capital more (less) costly increases (decreases) the expected present value of hiring costs, reducing (increasing) the firm’s incentive to create vacancies. More specifically, condition (41) shows that, all else equal, higher inflation rates raise adjusted outside option and decrease the market tightness $\theta$ .

Table 4 reports the optimal monetary policy results under period-by-period Nash bargaining wages and a range of values for workers’ bargaining weight $\kappa$ . The fourth row is the case that the Ramsey solution coincides with the first-best allocation. As is evident, the Ramsey results are qualitatively similar to those under fixed wages. The table thus confirms that the main results of the paper remain robust under flexible wages. More specifically, it documents that the optimal inflation rate rises above zero when the economy is operating beyond its socially optimal level and the market tightness gap becomes positive. This is a symmetric result, which means that the Ramsey planner calls for deflation when the market tightness is below its efficient level. The lower is the tightness relative to its optimum, the higher is the optimal deflation rate. The intuition is the same as before. Due to the working capital channel, the firm’s surplus is increasing in deflation. A higher worker bargaining power reduces the firm’s surplus and its incentives to post vacancies. Deflation increases the surplus and vacancy postings and pushes the equilibrium closer to the optimum. Therefore, according to this model, it is a modest level of deflation and not inflation that serves to “grease the wheels of the labor market” and reduces the frictions.

Table 4. Ramsey steady state when $\phi \gt 0$ and $\varphi \gt 0$ when wages are fully flexible

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

7. Nominal rigid wages

Now we turn to the case of sticky nominal wages and illustrate that a tradeoff regarding the optimal rate of inflation under working capital channel and under nominal wage rigidity arises in the model. We present the results for the case that nominal wages are assumed to be rigid for three quarters, which means that $\varphi ^{w}=5.88$ , as in Arseneau and Chugh (2008).

Panel B of Table 5 displays the Ramsey-optimal inflation rate under both sticky price and sticky nominal wages when the working capital channel is shut down. Panel C reports the results under the presence of the channel. Panel B confirms the main result already establish in the literature regarding the importance of nominal wage rigidity in featuring a positive inflation rate when unemployment is inefficiently high. ¹⁵ The first five rows of the table show that when employment is below its socially efficient rate the optimal rate of inflation is quite positive. The larger the distance of employment form its efficient level, the higher the optimal inflation rate.

Panel C shows that the results substantially change under working capital channel as well. The first row of the table shows that the optimal inflation rate falls from about 4.8% per year to about −1.7%. That is deflation becomes optimal. Although, given the baseline calibration, low positive inflation rates are still optimal when the market tightness is below its socially optimal level. However, inflation remains significantly lower than the case with no working capital channel. The last rows of the table establish the fact that when employment is notably above the social planner solution in the presence of working capital channel optimal policy calls for very high inflation rate, despite the fact that nominal wage rigidity in isolation calls for very high deflation rates. Notice that in the last two rows, the optimal rate of inflation under no working capital channel becomes so negative that it violates the ZLB.

Table 5. Ramsey steady state under both nominal price and nominal wages rigidities

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

8. Robustness check-calvo pricing

This section explores the robustness of the main results under sticky prices, presented in Tables 2 and 3, by dropping the assumption of Rotemberg pricing and assuming that firms are facing Calvo-type dynamic price rigidities. More specifically, now it is assumed that each period a fraction $\alpha ^{p}\in [0,1)$ of randomly picked firms is not allowed to change the nominal price of the good it produces; that is, each period, a fraction $\alpha$ of firms, must charge the same price as in the previous period. The remaining $ (1-\alpha ^{p} )$ firms choose prices optimally.

Since all firms that change their price are assumed to be identical, therefore, they all charge the same price. Suppose $\hat{P_{it}}$ is the price that firm $i$ gets to choose in period $t$ . This price is set to maximize the expected present discounted value of profits. That is, $\hat{P_{it}}$ maximizes:

\begin{equation*} E_{0}\sum _{k=0}^{\infty }(\alpha ^{p}\beta )^{k}\left[\frac {u^{\prime}\!\left(c_{t+k}\right)}{u^{\prime}\!\left(c_{t}\right)}\left(\frac {\left(1+\tau ^{s}\right)\hat {P_{it}}-P_{t+k}mc_{t+k}}{P_{t+k}}\right)y_{t+k}\!\left(i\right)\right], \end{equation*}

subject to the demand for good $i$ :

\begin{equation*} y_{t+k}\!\left(i\right)=\left(\frac {\hat {P_{it}}}{P_{t+k}}\right)^{-\xi }y_{t+k}. \end{equation*}

Again, it is assumed that the government provides a subsidy to factor inputs to dismantle the inefficiency introduced by imperfect competition in product and factor markets. Thus, no steady-state inefficiency arises due to imperfect competition.

The nominal price index is the same as before, that is:

(42) \begin{equation} P_{t}=\left[\int _{0}^{1}P_{it}^{1-\xi }\right].^{\frac{1}{1-\xi }} \end{equation}

The first-order condition with respect to $\hat{P_{it}}$ gives rise to the following standard optimality conditions in a recursive form in a symmetric equilibrium ( $\hat{P_{it}}=\hat{P_{t}}$ ):

(43) \begin{equation} \frac{\xi }{\left(1+\tau ^{s}\right)\left(\xi -1\right)}\varPsi _{t}^{1}=\varPsi _{t}^{2}, \end{equation}

(44) \begin{equation} \varPsi _{t}^{1}=\hat{p_{t}}^{-1-\xi }a_{t}n_{t}mc_{t}+\alpha ^{p}\beta E_{t}\frac{\lambda _{t+1}}{\lambda _{t}}\pi _{t+1}^{\xi }\left(\frac{\hat{p_{t}}}{\hat{p_{t+1}}}\right)^{-1-\xi }\varPsi _{t+1}^{1}, \end{equation}

(45) \begin{equation} \varPsi _{t}^{2}=\hat{p_{t}}^{-\xi }a_{t}n_{t}+\left(1+\tau ^{s}\right)\alpha ^{p}\beta E_{t}\frac{\lambda _{t+1}}{\lambda _{t}}\pi _{t+1}^{\xi -1}\left(\frac{\hat{p_{t}}}{\hat{p_{t+1}}}\right)^{-\xi }\varPsi _{t+1}^{2}, \end{equation}

where $\hat{p_{t}}=\frac{\hat{P_{t}}}{P_{t}}$ stands for the relative price of any good whose price was adjusted in period $t$ in terms of the aggregate price level. $\varPsi _{t}^{1}$ and $\varPsi _{t}^{2}$ are two auxiliary variables which stand for an output weighted present discounted value of marginal costs and marginal revenues, respectively. Note that, the aggregate price index (42) can also be written as:

(46) \begin{equation} 1=\alpha ^{p}\pi _{t}^{\xi -1}+\left(1-\alpha ^{p}\right)\hat{p_{t}}^{1-\xi }. \end{equation}

As is well known, due to the existence of relative price dispersion across varieties, which arises because of the assumed nature of price stickiness, we have output loss. This loss is captured by:

(47) \begin{equation} f_{t}=\int _{0}^{1}\left(\frac{P_{it}}{P_{t}}\right)^{-\xi }di=\left(1-\alpha ^{p}\right)\hat{p_{t}}^{-\xi }+\alpha ^{p}\pi _{t}^{\xi }f_{t-1}. \end{equation}

We are now ready to define a competitive equilibrium for this economy. A competitive equilibrium with a cash-in-advance constraint on wages is a set of processes $R_{t},P_{t},$ $B_{t},M_{t},mc_{t},c_{t},n_{t},f_{t},\theta _{t}$ , $\varPsi _{t}^{1}$ , $\varPsi _{t}^{2}$ , and $\hat{p_{t}}$ that satisfies conditions (43)−(47), and the following ones:

(48) \begin{equation} u^{\prime}\!\left(c_{t}\right)=\beta R_{t}E_{t}\frac{u^{\prime}\!\left(c_{t+1}\right)}{\pi _{t+1}}, \end{equation}

(49) \begin{equation} u_{t}=1-n_{t-1}, \end{equation}

(50) \begin{equation} n_{t}=\left(1-\rho \right)n_{t-1}+e\!\left(u_{t},v_{t}\right), \end{equation}

(51) \begin{equation} \frac{1}{f_{t}}a_{t}n_{t}=c_{t}+g_{t}+\gamma v_{t}-\left(1-n_{t}\right)z, \end{equation}

(52) \begin{equation} \phi \bar{w}n_{t}\leq m_{t}. \end{equation}

(53) \begin{equation} \frac{\gamma }{q_{t}}=a_{t}mc_{t}-\bar{w}\!\left(1+\phi \!\left(1-R_{t}^{-1}\right)\right)+\beta \!\left(1-\rho \right)E_{t}\frac{u^{\prime}\!\left(c_{t+1}\right)}{u^{\prime}\!\left(c_{t}\right)}\frac{\gamma }{q_{t+1}}, \end{equation}

(54) \begin{equation} B_{t}+M_{t}+P_{t}t_{t}=R_{t-1}B_{t-1}+M_{t-1}+P_{t}g_{t}, \end{equation}

(55) \begin{equation} R_{t}\geq 1. \end{equation}

given the exogenous processes $ \{ g_{t},a_{t} \}$ , subsidy $\tau ^{s}$ , and the initial condition $R_{-1}B_{-1}+M_{-1}$ .

Table 6. Ramsey steady state under Calvo pricing and when $\phi \gt 0$ , $\varphi \gt 0$ , and $\gamma$ changes

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

The Ramsey-optimal monetary policy problem in this model consists in choosing the path of the nominal interest rate that is associated with the competitive equilibrium and gives the maximum level of welfare to the representative household. Formally, the Ramsey problem consists in choosing sequences $R_{t},\pi _{t},mc_{t},m_{t},c_{t},n_{t},s_{t},\theta _{t},ss_{t}$ , $\varPsi _{t}^{1}$ , $\varPsi _{t}^{2}$ , and $\hat{p_{t}}$ to maximize the objective function (1) subject to equations (43)−(53).

In reporting the quantitative results, the only new parameter that we have and need to calibrate is the probability that a firm is not able to optimally set the price it charges in a particular quarter, $\alpha ^{p}$ . We set it at a common value, 0.75 (Gali (2010) and Schmitt-Grohé and Uribe (2010)). We keep all the other parameters at their baseline calibration. It is straightforward to see from conditions (36) and (37) that, all else equal, the real wage level, $w_{0}$ , at which full price stability is optimal and the Ramsey market tightness level equals its counterpart from the social planning problem remains unchanged under Calvo pricing. That is, $w_{0}=0.8837$ . Both Tables 6 and 7 confirm qualitatively similar results regarding the relationships between the Ramsey-optimal inflation rate and vacancy posting costs and the level of real wage, respectively, under Calvo pricing, as those under Rotemberg pricing, reported in Tables 2 and 3. Nevertheless, there are some differences. All else equal, while we see a nonlinear relationship between real wage and the optimal inflation rate under Rotemberg pricing, this relationship is linear with Calvo pricing. Furthermore, the optimal rate of inflation is quantitatively lower under Calvo pricing than Rotemberg pricing. The reason behind these differences could be because the Rotemberg pricing model features one fewer state variable than the Calvo model (see, Sims and Wolff (2017).)

Table 7. Ramsey steady state under Calvo pricing and when $\phi \gt 0$ , $\varphi \gt 0$ , and real wage changes

Note: $\theta ^{\text{GAP}}$ denotes the distance between the market tightness in the Ramsey economy and its counterpart in the planner’s problem; $\theta ^{\text{GAP}}=\theta ^{\text{RA}}-\theta ^{\text{FB}}$ . $\pi -1$ denotes the net rate of inflation expressed in percent per annum.

9. Conclusion

Cash holdings of US nonfinancial firms are enormous and have increased hugely over the past decade. Motivated by this fact, this paper introduces a working capital channel into a standard New Keynesian framework with matching frictions and studies optimal monetary policy under both sticky and flexible prices. The main contribution of the paper is to argue that some of the optimal monetary policy results in the existing literature can significantly change as a result of the assumed working capital constraint. We first assume constant real wages and analytically solve for the Ramsey-optimal policy under flexible and sticky prices. We show that under flexible prices, the Ramsey planner solution coincides with that of the social planning problem where it calls for deviation from the Friedman rule when the real wage is not at its efficient level. The Ramsey planner uses higher inflation rates to compensate for lower real wages. Nonetheless, there is no relationship between optimal inflation and market tightness in this case.

We then show that under sticky prices, unless the labor market outcome is socially efficient, which features the optimality of zero inflation rate, there is an optimal long-run relationship between the equilibrium market tightness and inflation. When the employment rate is above its efficient rate and the economy operates above its capacity the Ramsey planner calls for inflation to discourage firms from posting more vacancies. The reverse holds when the market tightness is below the optimum. In this case, the optimal monetary policy is to generate negative inflation to lower firms’ expected hiring costs and encourage them to post more vacancies, and therefore to push the equilibrium toward the optimum. This basically means “to grease the wheels of the labor market” deflation and not inflation is optimal. The main mechanism is shown to hold under flexible wages as well as nominal wage rigidity.