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:

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

Properties of the CES Production Function

The constant elasticity of substitution (CES) production function of capital and labor was introduced by [Solow, 1956] and further extended to a consumption aggregator by [Armington, 1969]. The CES production function of private capital \(K\), public capital \(K_g\) and labor \(L\) we use in Chapter Firms is the following,

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

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 \(\varepsilon=1\).

(190)\[ Y = Z(K)^\gamma(K_{g})^{\gamma_{g}}(L)^{1-\gamma-\gamma_{g}}\]

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 (189).

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

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

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

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

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

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 \(w_t\).

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

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

Solving for the firm’s first order conditions for capital and labor demand from profit maximization (134) gives the following equations in their respective stationarized forms from Chapter Stationarization.

(196)\[ w = (Z_t)^\frac{\varepsilon-1}{\varepsilon}\left[(1-\gamma-\gamma_{g})\frac{Y}{L}\right]^\frac{1}{\varepsilon}\]

(197)\[ r = (1 - \tau^{corp})(Z)^\frac{\varepsilon-1}{\varepsilon}\left[\gamma\frac{Y}{K}\right]^\frac{1}{\varepsilon} - \delta + \tau^{corp}\delta^\tau\]

As can be seen from (196) and (197), the wage \(w\) and interest rate \(r\) are functions of \(Y/L\) and \(Y/K\), respectively. Equations (194) and (195) 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:

(198)\[ \text{if}\:\:\,\varepsilon=1:\quad w = (1-\gamma-\gamma_g)Z\left(\frac{K}{L}\right)^\gamma \left(\frac{K_{g}}{L}\right)^{\gamma_{g}}\]

(199)\[ \text{if}\:\:\:\varepsilon=1:\quad r = (1 - \tau^{corp})\gamma Z\left(\frac{K_{g}}{K}\right)^{\gamma_{g}}\left(\frac{L}{K}\right)^{1-\gamma-\gamma_{g}} - \delta + \tau^{corp}\delta^\tau\]

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

Sparsity of the household equation Jacobian

Holding fixed the prices and policies a type-\(j\) cohort faces, its \(2S\) stationarized necessary conditions (128), (129), and (130) in the \(2S\) unknowns \(\{n_{j,s},\hat b_{j,s+1}\}_{s=E+1}^{E+S}\) have a banded Jacobian. From the budget constraint (127), stationarized consumption at age \(s\) depends on only three unknowns,

(200)\[ \hat c_{j,s} = \frac{1}{p}\Bigl[(1+r_p)\hat b_{j,s} + \hat w\,e_{j,s}\,n_{j,s} - \widehat{tax}_{j,s} - e^{g_y}\hat b_{j,s+1}\Bigr] + X_{j,s},\]

where \(\widehat{tax}_{j,s}\) depends only on \((\hat b_{j,s}, n_{j,s})\) through labor and capital income (already in the active set, so it adds no further coupling), and \(X_{j,s}\) collects terms fixed in the inner solve (bequests \(\hat{bq}_{j,s}\), remittances \(\hat{rm}_{j,s}\), government transfers \(\hat{tr}_{j,s}\), UBI \(\hat{ubi}_{j,s}\), the pension benefit \(\theta_j\), and the \(\hat c_{min,i}\) terms). The labor Euler equation (128) at age \(s\) therefore depends on \(\{\hat b_{j,s},\hat b_{j,s+1},n_{j,s}\}\) alone, and the savings Euler equation (129)—which links \(\hat c_{j,s}\) to \(\hat b_{j,s+1}\) and \(\hat c_{j,s+1}\)—depends on \(\{\hat b_{j,s},\hat b_{j,s+1},\hat b_{j,s+2},n_{j,s},n_{j,s+1}\}\). The marginal tax rates \(\tau^{mtrx}_s\) and \(\tau^{mtry}_{s+1}\) are functions of own-age income (already in these sets), so they add no further coupling, and the terminal condition (130) is sparser still. Each of the \(2S\) equations therefore depends on at most five of the \(2S\) unknowns, regardless of \(S\), so the Jacobian has at most \(10S\) nonzero entries rather than the \((2S)^2 = 4S^2\) of a fully coupled system. This is the per-cohort counterpart to the dense \(2JS\) system noted at the start of Chapter Equilibrium: cohorts couple only through prices, which are held fixed in the inner solve. A finite-difference Jacobian can then be built from a number of evaluations set by the bandwidth—about seven at \(S = 80\)—rather than \(2S\), by probing together unknowns that affect no common equation (Figure Fig. 4).