There exist two intermediate goods: cash and credit goods. We assume that both intermediate-goods firms are competitive. The production function of cash-goods firms is

where y_1,t denotes cash goods and h_1,t denotes labor input for the production of cash goods. For simplicity, we assume that the only factor for the production is labor. The analogue of credit-goods firms is

where y_2,t denotes credit goods, H_2,t denotes the aggregate labor input of credit goods, and h_2,t denotes individual firm’s labor input for the production of credit goods. We assume that γ > 0, that means that positive externalities in the production of credit goods.

Households supply labor to intermediate-goods firms and earn wages. They buy cash and credit goods from intermediate-goods firms at prices P_tp_1,t and P_tp_2,t, respecttively, and sell them to final-goods firms at prices P_tr_1,t and P_tr_2,t, respectively. They buy final goods from final-goods firms c_t and possess money M_t and risk-free nominal bonds B_t as assets. The budget constraint of households is

(3)

where w_t denotes real wage, R_t–1 denotes nominal interest rate, and T_t denotes monetary injection.

We assume that there is a cash-in-advance constraint for the purchase of cash goods:

Finally, the utility function is

where σ > 0 denotes the relative risk aversion and β Î (0, 1) denotes the discount factor of households. In this economy, we assume that total labor supply is constant.

Competitive final-goods firms buy intermediate goods from households at prices P_tr_1,t and P_tr_2,t. They produce and sell final goods to households at price P_t. The production function is constant elasticity of substitution:

where 1/ρ>0 denotes the elasticity of substitution between cash and credit goods and η Î (0,1) denotes the share of cash goods in the production of final goods.

We consider that the monetary authority sets a nominal interest rate R_t. The market clearing conditions are as follows:

For simplicity, we consider total labor supply h to be constant.

At the steady state, the equilibrium system is summarized as

Equation (10) shows that the monetary authority controls gross inflation π by setting R. The Friedman rule implies that π = β. Equation (11) is the optimization condition of labor input among two intermediate-goods sectors. Finally, by Equation (12), the output is determined.

In this paper, we focus on the steady-state relationship between inflation and the social welfare. Since the social welfare depends only on output, we investigate how output is affected by inflation in the following analyses.

Two lemmas are useful for the analyses. The first lemma is on the relationship between output and labor supply in the cash-goods sector.

Lemma 1. The steady-state output y is decreasing in the steady-state labor supply in cash-goods sector h₁ if and only if

If, y is increasing in h₁.

Proof. Taking total differentiation of (11) yields

.

By the steady-state relationship (12), we obtain

.

A necessary and sufficient condition for is Equation (13).□

The second lemma is on the relationship between labor supply in the cash-goods sector and inflation.

Lemma 2. The steady-state labor supply in cashgoods sector h₁ is decreasing in the steady-state inflation π if and only if

Proof. Taking total differentiation of Equation (12) yields

where

By the steady-state relationship (12), we obtain

.

A necessary and sufficient condition for is Equation (14).□

Using these two lemmas, we provide two propositions. In the first, no externalities exist: γ = 0.

Proposition 1. If no externalities exist, γ = 0, the Friedman rule is optimal.

Proof. If γ = 0, Equation (14) holds. By Lemmas 1 and 2, it is shown that y is decreasing in π for π ³ β. □

Increasing the nominal interest rate generates inefficiency losses for society since there exists a wedge between the private marginal cost of holding money, which is the nominal interest rate, and the social marginal cost of producing money, which is zero, as in standard models. Therefore, the Friedman rule is optimal in our model without externalities.

If there exist externalities, the optimality of the Friedman rule does not hold under suitable conditions.

The main result in this paper is as follows.

Proposition 2. Assume that γ > 0 and. The Friedman rule is not optimal, and the social welfare is maximized at a steady state with and R = γ.

Proof. Since, Equation (14) holds at a steady state with π ³ β. Then, h₁ is decreasing in π for all π ³ β. By Lemma 1, it is shown that for and for. Since the utility function implies that the social welfare is increasing in y, the optimal inflation level is

. □

In the case with positive externalities in the production of credit goods, increasing the nominal interest rate has positive effects on the economy. By increasing the nominal interest rate, money holding incentive reduces and labor input shifts from the cash-goods sector to the credit-goods sector. Since there exist positive externalities in the production of credit goods, this labor shift has positive effects on the economy. Proposition 2 implies that the benefit of increasing the nominal interest rate is greater than the cost if the nominal interest rate is lower than the threshold.

We verify this result by numerical simulations. The model is annual. The discount factor is β = 0.96, which implies that the real interest rate is four percent. The relative risk aversion is σ = 2, following the standard literature. The degree of externalities γ is set such that the optimal inflation is two percent: γ = 0.0625, since the stylized fact shows that economic performance is good under mild inflation rates. We set ρ = 0.065, which satisfies condition (14) and ensures high elasticity of substitution between cash and credit goods. We also set η such that at a steady state where inflation is two percent: η = 0.5011. In our numerical simulations, the assumption of the interim solution of labor input is satisfied under this value of η.

Figure 1 shows the effect of inflation on steady-state output. We change the steady-state inflation from π = β, that means the case of the Friedman rule, to ten percent. We normalize the output level at π = β to be one hundred. As shown in Proposition 2, output is maximized at a steady state with two percent inflation. The output level at the optimal inflation rate is 1.5 percent higher than that with the Friedman rule. The welfare is also maximized since it is monotone in output in our model.

For simplicity, we assume that total labor supply is constant in the model. Here, we relax this assumption. We employ the following utility function:

where σ > 0 and ψ > 0. Other settings are the same as in Section 2.

We investigate the effects of inflation at the steady state by numerical simulations. We set σ so that the

steady-state total labor supply is 0.3 and ψ = 2. Other parameter values are the same as in the case of inelastic total labor supply.

Figure 2 shows output, welfare (defined by the utility), and total labor supply given steady-state inflation. We normalize output and total labor level at π = β to be one hundred and welfare to be minus one hundred. In Figure 2, it is shown that even in a model with elastic total labor supply, the Friedman rule is not optimal and there is an optimal inflation level that maximizes the welfare. The optimal inflation is approximately two percent as in the case of inelastic total labor supply. We also find that inflation that maximizes output is less than optimal in this case.