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. (Reference Monacelli, Perotti and Trigari2010), 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 (Reference Arseneau and Chugh2008) 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.Footnote 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 $g_{t}$ .Footnote 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. (Reference den Haan, Ramey and Watson2000) 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.

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 (Reference Phelps1973). 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 (Reference Schmitt-Grohé and Uribe2010) 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 (Reference Carlsson and Westermark2016). 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 (Reference Carlsson and Westermark2016), 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 (Reference Carlsson and Westermark2016), 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 (Reference Schmitt-Grohé and Uribe2004), 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 (Reference Schmitt-Grohé and Uribe2004), 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. (Reference Christiano, Eichenbaum and Trabandt2016) and Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017)). In line with Ljungqvist and Sargent (Reference Ljungqvist and Sargent2017), 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. (Reference Petrosky-Nadeau, Zhang and Kuehn2018) 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.Footnote 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 (Reference Thomas2008); Faia (Reference Faia2009); Blanchard and Gali (Reference Blanchard and Gali2010); Gali (Reference Gali2010); Ravenna and Walsh (Reference Ravenna and Walsh2011, Reference Ravenna and Walsh2012); and Kim and Ruge-Murcia (Reference Kim and Ruge-Murcia2009, Reference Kim and Ruge-Murcia2011)). 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 (Reference Carlsson and Westermark2016) find in a closely related paper. More precisely, Carlsson and Westermark (Reference Carlsson and Westermark2016) 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:Footnote 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 $\theta$ .Footnote 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.