# Firms¶

The production side of the OG-Core model is populated by a unit measure of identical perfectly competitive firms that rent private capital $$K_t$$ and public capital $$K_{g,t}$$ and hire labor $$L_t$$ to produce output $$Y_t$$. Firms also face a flat corporate income tax $$\tau^{corp}$$ as well as a tax on the amount of capital they depreciate $$\tau^\delta$$.

## Production Function¶

Firms produce output $$Y_t$$ using inputs of private capital $$K_t$$, public capital $$K_{g,t}$$, and labor $$L_t$$ according to a general constant elasticity (CES) of substitution production function,

(27)$\begin{split} \begin{split} Y_t &= F(K_t, K_{g,t}, L_t) \\ &\equiv Z_t\biggl[(\gamma)^\frac{1}{\varepsilon}(K_t)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g,t})^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(e^{g_y t}L_t)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \quad\forall t \end{split}\end{split}$

where $$Z_t$$ is an exogenous scale parameter (total factor productivity) that can be time dependent, $$\gamma$$ represents private capital’s share of income, $$\gamma_{g}$$ is public capital’s share of income, and $$\varepsilon$$ is the constant elasticity of substitution among the two types of capital and labor. We have included constant productivity growth rate $$g_y$$ as the rate of labor augmenting technological progress.

A nice feature of the CES production function is that the Cobb-Douglas production function is a nested case for $$\varepsilon=1$$.1

(28)$Y_t = Z_t K_t^\gamma K_{g,t}^{\gamma_{g}}(e^{g_y t}L_t)^{1-\gamma-\gamma_{g}} \quad\forall t \quad\text{for}\quad \varepsilon=1$

## Optimality Conditions¶

The profit function of the representative firm is the following.

(29)$PR_t = (1 - \tau^{corp}_t)\Bigl[F(K_t,K_{g,t},L_t) - w_t L_t\Bigr] - \bigl(r_t + \delta\bigr)K_t + \tau^{corp}_t\delta^\tau_t K_t \quad\forall t$

Gross income for the firms is given by the production function $$F(K,K_g,L)$$ because we have normalized the price of the consumption good to 1. Labor costs to the firm are $$w_t L_t$$, and capital costs are $$(r_t +\delta)K_t$$. The government supplies public capital to the firms at no cost. The per-period interest rate (rental rate) of capital for firms is $$r_t$$. The per-period economic depreciation rate for private capital is $$\delta$$. The $$\delta^\tau_t$$ parameter in the last term of the profit function governs how much of capital depreciation can be deducted from the corporate income tax.

Taxes enter the firm’s profit function (29) in two places. The first is the corporate income tax rate $$\tau^{corp}_t$$, which is a flat tax on corporate income. Corporate income is defined as gross income minus labor costs. This will cause the corporate tax to only distort the firms’ capital demand decision.

The tax policy also enters the profit function (29) through depreciation deductions at rate $$\delta^\tau_t$$, which then lower corporate tax liability. When $$\delta^\tau_t=0$$, no depreciation expense is deducted from the firm’s tax liability. When $$\delta^\tau_t=\delta$$, all economic depreciation is deducted from corporate income.

Firms take as given prices $$w_t$$ and $$r_t$$ and the level of public capital supply $$K_{g,t}$$. Taking the derivative of the profit function (29) with respect to labor $$L_t$$ and setting it equal to zero (using the general CES form of the production function (27)) and taking the derivative of the profit function with respect to capital $$K_t$$ and setting it equal to zero, respectively, characterizes the optimal labor and capital demands.

(30)$w_t = e^{g_y t}(Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[(1-\gamma-\gamma_{g})\frac{Y_t}{e^{g_y t}L_t}\right]^\frac{1}{\varepsilon} \quad\forall t$
(31)$r_t = (1 - \tau^{corp}_t)(Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[\gamma\frac{Y_t}{K_t}\right]^\frac{1}{\varepsilon} - \delta + \tau^{corp}_t\delta^\tau_t \quad\forall t$

Note that the presence of the public capital good creates economic rents. However, given perfect competition, any economic profits will be competed away. For this reason, the optimality condition for capital demand (31) is only affected by public capital $$K_{g,t}$$ through the $$Y_t$$ term.

## Positive Profits from Government Infrastructure Investment¶

The CES production function in (27) exhibits constant returns to scale (CRS). A feature of CRS production functions is that gross revenue $$Y_t$$ is a sum of the gross revenue attributed to each factor of production,

(32)$Y_t = MPK_t K_t + MPK_{g,t} K_{g,t} + MPL_t L_t \quad\forall t$

where $$MPK_t$$ is the marginal product of private capital, $$MPK_{g,t}$$ is the marginal product of public capital, and $$MPL_t$$ is the marginal product of labor. Each of the terms in (32) is growing at the macroeconomic variable rate of $$e^{g_y t}\tilde{N_t}$$ (see the third column of Table 1). Firm profit maximization for private capital demand from equation (31) implies that the marginal product of private capital is the following.

(33)$MPK_t = \frac{r_t + \delta - \tau^{corp}_t\delta^{\tau}_t}{1 - \tau^{corp}_t} \quad\forall t$

Firm profit maximization for labor demand from equation (30) implies that the marginal product of labor is the following.

(34)$MPL_t = w_t \quad\forall t$

Even though firms take the stock of public capital $$K_{g,t}$$ from government infrastructure investment as given, we can still calculate the marginal product of public capital from the production function (27).

(35)$MPK_{g,t} = Z_t^{\frac{\varepsilon - 1}{\varepsilon}}\left(\frac{\gamma_g Y_t}{K_{g,t}}\right)^{\frac{1}{\varepsilon}} \quad\forall t$

If we plug the expressions for $$MPK_t$$, $$MPK_{g,t}$$, and $$MPL_t$$ from (33), (35), and (34), respectively, into the total revenue $$Y_t$$ decomposition in (32) and then substitute that into the profit function (29), we see that positive economic rents arise when public capital is positive $$K_{g,t}>0$$.

(36)$\begin{split} \begin{split} PR_t &= (1 - \tau^{corp}_t)\Bigl[Y_t - w_t L_t\Bigr] - \bigl(r_t + \delta\bigr)K_t + \tau^{corp}_t\delta^\tau_t K_t \\ &= (1 - \tau^{corp}_t)\Biggl[\biggl(\frac{r_t + \delta - \tau^{corp}_t\delta^{\tau}_t}{1 - \tau^{corp}_t}\biggr)K_t + MPK_{g,t}K_{g,t} + w_t L_t\Biggr] ... \\ &\quad\quad - (1 - \tau^{corp}_t)w_t L_t - (r_t + \delta)K_t + \tau^{corp}_t\delta^{\tau}_t K_t \\ &= (1 - \tau^{corp}_t)MPK_{g,t}K_{g,t} \\ \end{split}\end{split}$

We assume these positive economic profits resulting from government infrastructure investment are passed on to the owners of private capital through an adjusted interest rate $$r_{K,t}$$ provided by the financial intermediary (see Chapter Financial Intermediary) that zeroes out profits among the perfectly competitive firms and is a function of $$MPK_{g,t}$$ and $$K_{g,t}$$. Total payouts from the financial intermediary $$r_{K,t}K_t$$ are a function of the perfectly competitive payout to owners of private capital $$r_t K_t$$ plus any positive profits when $$K_{g,t}>0$$ from (36).

(37)$r_{K,t}K_t = r_tK_t + (1 - \tau^{corp}_t)MPK_{g,t}K_{g,t} \quad\forall t$

This implies that the rate of return paid from the financial intermediary to the households $$r_{K,t}$$ is the interest rate on private capital $$r_t$$ plus the positive profits from (36), in which the units are put in terms of $$K_t$$ (see equation (66) in Chapter Financial Intermediary).

(38)$r_{K,t} = r_t + (1 - \tau^{corp}_t)MPK_{g,t}\left(\frac{K_{g,t}}{K_t}\right) \quad\forall t$

## Footnotes¶

1

It is important to note a special case of the Cobb-Douglas ($$\varepsilon=1$$) production function that we have to manually restrict. The inputs of production of private capital $$K_t$$ and labor $$L_t$$ are endogenous and have characteristics of the model that naturally bound them away from zero. But public capital $$K_g$$, although it is a function of endogenous variables in (49) and (50), can be exogenously set to zero as a policy parameter choice by setting $$\alpha_{I,t}=0$$. In the Cobb-Douglas case of the production function $$\varepsilon=1$$ (28), $$K_g=0$$ would zero out production and break the model. In the case when $$\varepsilon=1$$ and $$K_g=0$$, we set $$gamma_g=0$$, thereby restricting the production function to only depend on private capital $$K_t$$ and labor $$L_t$$. This necessary restriction limits us from performing experiments in the model of the effect of changing $$K_{g,t}=0$$ to $$K_{g,t}>0$$ or vice versa in the $$\varepsilon=1$$ case.