# Derivations#

This appendix contains derivations from the theory in the body of this book.

## Household first order condition for industry-specific consumption demand#

The derivation for the household first order condition for industry-specific consumption demand (13) is the following:

(181)#$\begin{split} \tilde{p}_{i,t} = \tilde{p}_{j,s,t}\alpha_m(c_{i,j,s,t} - c_{min,i})^{\alpha_i-1}\prod_{u\neq i}^I\left(c_{u,j,s,t} - c_{min,u}\right)^{\alpha_u} \\ \tilde{p}_{i,t}(c_{i,j,s,t} - c_{min,i}) = \tilde{p}_{j,s,t}\alpha_i(c_{i,j,s,t} - c_{min,i})^{\alpha_i}\prod_{u\neq i}^I\left(c_{u,j,s,t} - c_{min,u}\right)^{\alpha_u} \\ \tilde{p}_{i,t}(c_{i,j,s,t} - c_{min,i}) = \tilde{p}_{j,s,t}\alpha_i\prod_{i=1}^I\left(c_{i,j,s,t} - c_{min,i}\right)^{\alpha_i} = \alpha_i \tilde{p}_{j,s,t}c_{j,s,t}\end{split}$

## Properties of the CES Production Function#

The constant elasticity of substitution (CES) production function of capital and labor was introduced by and further extended to a consumption aggregator by . The CES production function of private capital $$K$$, public capital $$K_g$$ and labor $$L$$ we use in Chapter Firms is the following,

(182)#$\begin{split} Y &= F(K, K_g, L) \\ &\equiv Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_g)^\frac{1}{\varepsilon}(K_g)^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_g)^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1}\end{split}$

where $$Y$$ is aggregate output (GDP), $$Z$$ is total factor productivity, $$\gamma$$ is a share parameter that represents private capital’s share of income in the Cobb-Douglas case ($$\varepsilon=1$$), $$\gamma_g$$ is public capital’s share of income, and $$\varepsilon$$ is the elasticity of substitution between capital and labor. The stationary version of this production function is given in Chapter Stationarization. We drop the $$m$$ and $$t$$ subscripts, the $$\:\,\hat{}\,\:$$’’ stationary notation, and use the stationarized version of the production function for simplicity.

The Cobb-Douglas production function is a nested case of the general CES production function with unit elasticity $$\varepsilon=1$$.

(183)#$Y = Z(K)^\gamma(K_{g})^{\gamma_{g}}(L)^{1-\gamma-\gamma_{g}}$

The marginal productivity of private capital $$MPK$$ is the derivative of the production function with respect to private capital $$K$$. Let the variable $$\Omega$$ represent the expression inside the square brackets in the production function (182).

(184)#$\begin{split} MPK &\equiv \frac{\partial F}{\partial K} = \left(\frac{\varepsilon}{\varepsilon-1}\right)Z\left[\Omega\right]^\frac{1}{\varepsilon-1}\gamma^\frac{1}{\varepsilon}\left(\frac{\varepsilon-1}{\varepsilon}\right)(K)^{-\frac{1}{\varepsilon}} \\ &= Z\left[\Omega\right]^\frac{1}{\varepsilon-1}\left(\frac{\gamma}{K}\right)^\frac{1}{\varepsilon} = \frac{Z\left[\Omega\right]^\frac{1}{\varepsilon-1}}{Z^\frac{1}{\varepsilon-1}\left[\Omega\right]^\frac{1}{\varepsilon-1}}\left(\frac{\gamma}{K}\right)^\frac{1}{\varepsilon}Y^\frac{1}{\varepsilon} \\ &= (Z)^\frac{\varepsilon-1}{\varepsilon}\left(\gamma\frac{Y}{K}\right)^\frac{1}{\varepsilon}\end{split}$

The marginal productivity of public capital $$MPK_g$$ is the derivative of the production function with respect to public capital $$K_g$$.

(185)#$\begin{split} MPK_g &\equiv \frac{\partial F}{\partial K_g} = \left(\frac{\varepsilon}{\varepsilon-1}\right)Z\left[\Omega\right]^\frac{1}{\varepsilon-1}\gamma_g^\frac{1}{\varepsilon}\left(\frac{\varepsilon-1}{\varepsilon}\right)(K_g)^{-\frac{1}{\varepsilon}} \\ &= Z\left[\Omega\right]^\frac{1}{\varepsilon-1}\left(\frac{\gamma_g}{K_g}\right)^\frac{1}{\varepsilon} = \frac{Z\left[\Omega\right]^\frac{1}{\varepsilon-1}}{Z^\frac{1}{\varepsilon-1}\left[\Omega\right]^\frac{1}{\varepsilon-1}}\left(\frac{\gamma_g}{K_g}\right)^\frac{1}{\varepsilon}Y^\frac{1}{\varepsilon} \\ &= (Z)^\frac{\varepsilon-1}{\varepsilon}\left(\gamma_g\frac{Y}{K_g}\right)^\frac{1}{\varepsilon}\end{split}$

The marginal productivity of labor $$MPL$$ is the derivative of the production function with respect to labor $$L$$.

(186)#$\begin{split} MPL &\equiv \frac{\partial F}{\partial L} = \left(\frac{\varepsilon}{\varepsilon-1}\right)Z\left[\Omega\right]^\frac{1}{\varepsilon-1}(1-\gamma-\gamma_g)^\frac{1}{\varepsilon}\left(\frac{\varepsilon-1}{\varepsilon}\right)(L)^{-\frac{1}{\varepsilon}} \\ &= Z\left[\Omega\right]^\frac{1}{\varepsilon-1}\left(\frac{1-\gamma-\gamma_g}{L}\right)^\frac{1}{\varepsilon} = \frac{Z\left[\Omega\right]^\frac{1}{\varepsilon-1}}{Z^\frac{1}{\varepsilon-1}\left[\Omega\right]^\frac{1}{\varepsilon-1}}\left(\frac{1-\gamma-\gamma_g}{L}\right)^\frac{1}{\varepsilon}Y^\frac{1}{\varepsilon} \\ &= (Z)^\frac{\varepsilon-1}{\varepsilon}\left([1-\gamma-\gamma_g]\frac{Y}{L}\right)^\frac{1}{\varepsilon}\end{split}$

### Wages as a function of interest rates#

The below shows that with the addition of public capital as a third factor of production, wages and interest rates are more than a function of the capital labor ratio. This means that in the solution method for OG-Core we will need to guess both the interest rate $$r_t$$ and wage $$w_t$$.

(187)#$\begin{split}\begin{split} Y &= Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g})^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \\ &= Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon}\left(\frac{L^\frac{\varepsilon-1}{\varepsilon}}{L^\frac{\varepsilon-1}{\varepsilon}}\right) + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g})^\frac{\varepsilon-1}{\varepsilon}\left(\frac{L^\frac{\varepsilon-1}{\varepsilon}}{L^\frac{\varepsilon-1}{\varepsilon}}\right) + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \\ &= ZL\biggl[(\gamma)^\frac{1}{\varepsilon}\left(\frac{K}{L}\right)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{K_{g}}{L}\right)^\frac{\varepsilon-1}{\varepsilon}+ (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1}\\ \Rightarrow\quad \frac{Y}{L} &= Z\biggl[(\gamma)^\frac{1}{\varepsilon}\left(\frac{K}{L}\right)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{K_{g}}{L}\right)^\frac{\varepsilon-1}{\varepsilon}+ (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \end{split}\end{split}$
(188)#$\begin{split}\begin{split} Y &= Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g})^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \\ &= Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g})^\frac{\varepsilon-1}{\varepsilon}\left(\frac{K^\frac{\varepsilon-1}{\varepsilon}}{K^\frac{\varepsilon-1}{\varepsilon}}\right) + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\left(\frac{K^\frac{\varepsilon-1}{\varepsilon}}{K^\frac{\varepsilon-1}{\varepsilon}}\right)\biggr]^\frac{\varepsilon}{\varepsilon-1}\\ &= ZK\biggl[(\gamma)^\frac{1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{K_{g}}{K}\right)^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{L}{K}\right)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \\ \Rightarrow\quad \frac{Y}{K} &= Z\biggl[(\gamma)^\frac{1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{K_{g}}{K}\right)^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}\left(\frac{L}{K}\right)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \end{split}\end{split}$

Solving for the firm’s first order conditions for capital and labor demand from profit maximization (127) gives the following equations in their respective stationarized forms from Chapter Stationarization.

(189)#$w = (Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[(1-\gamma-\gamma_{g})\frac{Y}{L}\right]^\frac{1}{\varepsilon}$
(190)#$r = (1 - \tau^{corp})(Z)^\frac{\varepsilon-1}{\varepsilon}\left[\gamma\frac{Y}{K}\right]^\frac{1}{\varepsilon} - \delta + \tau^{corp}\delta^\tau$

As can be seen from (189) and (190), the wage $$w$$ and interest rate $$r$$ are functions of $$Y/L$$ and $$Y/K$$, respectively. Equations (187) and (188) show that both $$Y/L$$ and $$Y/K$$ are functions of the capital-labor ratio $$K/L$$, the public-capital-labor ratio, $$K_{g}/L$$, and the public-private capital ratio, $$K/K_{g}$$. We cannot solve these equations for $$r$$ and $$w$$ solely as functions of the same ratios.

In the Cobb-Douglas unit elasticity case ($$\varepsilon=1$$) of the CES production function, the first order conditions are:

(191)#$\text{if}\:\:\,\varepsilon=1:\quad w = (1-\gamma-\gamma_g)Z\left(\frac{K}{L}\right)^\gamma \left(\frac{K_{g}}{L}\right)^{\gamma_{g}}$
(192)#$\text{if}\:\:\:\varepsilon=1:\quad r = (1 - \tau^{corp})\gamma Z\left(\frac{K_{g}}{K}\right)^{\gamma_{g}}\left(\frac{L}{K}\right)^{1-\gamma-\gamma_{g}} - \delta + \tau^{corp}\delta^\tau$

Again, even if this simple case, we cannot solve for $$r$$ as a function of $$w$$ for the reasons above.