# Derivations¶

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

## 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 aggregate capital $$K_t$$ and aggregate labor $$L_t$$ we use in Chapter Firms is the following,

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

where $$Y_t$$ is aggregate output (GDP), $$Z_t$$ 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 capita’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 $$t$$ subscripts, the $$\:\,\hat{}\,\:$$’’ stationary notation, and use the stationarized version of the production function (82) for simplicity.

(124)$Y= Z\biggl[(\gamma)^\frac{1}{\varepsilon}(K)^\frac{\varepsilon-1}{\varepsilon} + (\gamma_{g})^\frac{1}{\varepsilon}(K_{g})^\frac{\varepsilon-1}{\varepsilon} + (1-\gamma-\gamma_{g})^\frac{1}{\varepsilon}(L)^\frac{\varepsilon-1}{\varepsilon}\biggr]^\frac{\varepsilon}{\varepsilon-1} \quad\forall t$

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

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

### 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 $$w_t$$.

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

Solving for the firm’s first order conditions for capital and labor demand from profit maximization (83) gives the following equations in their respective stationarized forms from Chapter Stationarization.

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

As can be seen from (128) and (129), the wage $$w$$ and interest rate $$r$$ are functions of $$Y/L$$ and $$Y/K$$, respectively. Equations (126) and (127) 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:

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

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