Calibration of Firms Parameters

Aggregate Production Function and Capital Accumulation

The OG-Core firm theory documentation outlines the constant returns to scale, constant elasticity of substitution production function of the representative firm. This function has two parameters; the elasticity of substitution and capital’s share of output.

The production function is given as:

(1)\[\begin{split} \begin{split} Y_{m,t} &= F(K_{m,t}, K_{g,m,t}, L_{m,t}) \\ &\equiv Z_{m,t}\biggl[(\gamma_m)^\frac{1}{\varepsilon_m}(K_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + (\gamma_{g,m})^\frac{1}{\varepsilon_m}(K_{g,m,t})^\frac{\varepsilon_m-1}{\varepsilon_m} + \\ &\quad\quad\quad\quad\quad(1-\gamma_m-\gamma_{g,m})^\frac{1}{\varepsilon_m}(e^{g_y t}L_{m,t})^\frac{\varepsilon_m-1}{\varepsilon_m}\biggr]^\frac{\varepsilon_m}{\varepsilon_m-1} \quad\forall m,t \end{split}\end{split}\]

This production function has the following parameters:

• \(\varepsilon_m\) is the elasticity of substitution between capital, labor, and infrastructure in sector \(m\).

• \(\gamma_m\) is the share of capital in sector \(m\).

• \(\gamma_{g,m}\) is the share of government capital in sector \(m\).

• \(Z_{m,t}\) is the total factor productivity in sector \(m\) at time \(t\).

Elasticity of substitution

OG-ZAF’s default parameterization has an elasticity of substitution of \(\varepsilon=1.0\), which implies a Cobb-Douglas production function.

Factor shares of output

In the default calibration, we set infrastructure’s share of output to \(\gamma_{g,m}=0.0\) for all sectors. This parameter is hard to identify from national accounts data and would entail an empirical study to tease out the relationship between infrastructure and output.

We use a default value of \(\gamma =0.40\), which corresponds to one minus labor’s share of output, where labor’s share of output is found as 0.60 in in the UN ILOSTAT database.

Total factor productivity

In the case of the single production sector, we can normalize \(Z_{m,t}=1.0\). In the case of multiple production sectors, we use {cite}`PRS2020 who identify TFP for various sectors in South Africa.