Appendix 6: Macroeconomic Variables

This Appendix describes the method underlying projections of gross domestic product (GDP) and other macroeconomic variables in the PWBM microsimulation model.

1. Production function

Aggregate economic output in the PWBM microsimulation model is derived from a neoclassical production function that calculates nominal GDP as a function of prices, productivity, and a measure of productive inputs. The production function assumes Hicks' neutral technical change and constant returns to scale. In addition, it assumes that the economy is in long-run competitive equilibrium, so that the marginal products of capital and labor are equal to their respective market prices.

In general form, the production function can be written $Y_t = P_t A_t I_t$, where $I_t= f(K_t, L_t)$.

Here, \begin{align} Y_t & = \text{nominal gross domestic product}, \\ P_t & = \text{the GDP price index}, \\ A_t & = \text{multifactor productivity}, \\ I_t & = \text{combined inputs}, \\ K_t & = \text{capital services (or capital input)}, \\ L_t & = \text{effective labor input}. \end{align}

Differentiating the generalized input function f with respect to time leads to the following expression (omitting all time subscripts and using a hat to indicate a growth rate over time):

\begin{equation} \hat{I} = (\frac{\delta Y}{\delta K}\frac{K}Y) \hat{K} + (\frac{\delta Y}{\delta L} \frac{L}Y) \hat{L}. \end{equation}


\begin{equation} \hat{I} = E_K\hat{K} + E_L\hat{L}, \end{equation}

where $E_X$ is the elasticity of output with respect to input $X$. Thus, the measure $I$ of combined productive input is defined as a weighted average of the growth rates of each input type, with weights equal to their respective output elasticities. (The PWBM production function assumes a single production technology for the entire economy and that inputs are substitutable across sectors. For example, an increase in government capital services has the same effect on output as an equivalent increase in business sector capital services.) Under the assumptions of constant returns to scale and competitive equilibrium:

\begin{align} E_K + E_L &= 1 \\ E_K &= \alpha. \end{align}

where $\alpha$ is capital’s share in cost (the share of aggregate income that is a return to capital). Denoting by $P_K$ the rental price of capital services:

\begin{equation} \alpha \equiv \frac{P_K K}Y. \end{equation}

Incorporating these two assumptions into the definition of $I$ above:

\begin{equation} \hat{I} = \alpha\hat{K} + (1-\alpha)\hat{L}. \end{equation}

This continuous-time expression can be approximated in discrete time by a Tornqvist index. The Tornqvist index is a chain-weighted average of its components’ growth rates, where the weights are equal to the arithmetic average of their respective compensation shares in two adjacent years: (The Tornqvist index is an ‘exact index’ for a homogeneous translog production function. This means that, for given changes in inputs and input prices, the resulting change in output measured by the Tornqvist index is the same as what would be obtained from a translog production function. See DIG UP DIEWERT REFERENCE)

\begin{equation} \Delta I_t = (\frac{\alpha_t + \alpha_{t-1}}{2}) \Delta K_t + (1-\frac{\alpha_t + \alpha_{t-1}}{2})\Delta L_t. \end{equation}

The complete production function can now be written as

\begin{equation} \Delta ⁡Y_t =\Delta P_t + \Delta ⁡A_t + \hat{\alpha}_t \Delta ⁡K_t + (1-\hat{\alpha}_t) \Delta L_t, \end{equation}


\begin{equation} \hat{\alpha}_t \equiv \frac{\alpha_t+\alpha_{t-1}}{2}. \end{equation}

Of these variables, $P$, $A$, and $\alpha$ are specified in the model as exogenous parameters, while $K$ and $L$ are determined endogenously by the simulation.

2. Historical Data

The historical data underlying the parametrization of the production function and projections of inputs are drawn primarily from the Bureau Labor Statistics’ (BLS) multifactor productivity (MFP) accounts. These estimates are based on the same Tornqvist aggregation method described in the previous section. However, differences in coverage in the MFP accounts necessitate some adjustments to the published MFP series.

In order to maintain consistency with the National Income and Product Accounts (NIPAs), the PWBM production function covers all economic activity included in GDP. By contrast, the main published MFP accounts cover the private business sector only, which excludes about a quarter of GDP. Fortunately, BLS also produces a set of supplementary total economy (TE) accounts, which provides a much closer conceptual match – like GDP, they include the government, owner-occupied housing, and nonprofit sectors. These TE measures form the basis of the historical estimates used in the PWBM production function. However, due to definitional differences, they cannot be applied directly and must first be adjusted to a common conceptual basis.

In the TE accounts, BLS adopts a non-standard definition of nominal output that augments GDP with imputations of net returns on government and nonprofit capital:

\begin{equation} Y^{TE} = Y + (Y_K^G-\delta^G) + (Y_K^{NP} - \delta^{NP}) \end{equation}

where $Y_K^X$ is imputed gross capital income in sector $X$, $\delta^X$ is consumption of fixed capital in sector $X$, and $X=G$ and $X=NP$ represent the government and nonprofit sectors respectively. (Compared with business sector capital services alone, the TE measure adds owner-occupied housing capital as well as government and nonprofit capital. However, no additional imputation is performed for the return on owner-occupied capital, because GDP already includes such an imputation (see NIPA Table 7.12).)

Thus, while the measures of capital and labor input in the TE accounts correspond exactly to those in our production function, TE estimates of the price level, multifactor productivity, and capital’s share in cost – the exogenous elements during simulation – are not consistent with our measure of output.

To account for this difference in output concepts, it is necessary to recalculate historical multifactor productivity A and capital’s share in cost $\alpha$. (If we take BLS’s imputations at face value, this adjustment means our production function understates the output elasticity of capital and, to a lesser extent, growth in MFP.) (Historical estimates of the GDP price index are from the NIPAs.) Since the measure of combined input is the same as in the TE accounts, the GDP-concept multifactor productivity series A follows directly from the production function:

\begin{align} I & \equiv I^{TE} = \frac{Y^{TE}}{P^{TE} A^{TE}} \\ A & = \frac{Y}{PI} \end{align}

To recalculate capital’s share in cost, the imputed net returns to government and nonprofit capital must be removed from aggregate capital income: (The same measure of consumption of fixed capital is included in both output concepts, so only the net return is subtracted.)

\begin{align} Y_K &= Y_K^{TE} - (Y_K^G - \delta^G) - (Y_K^{NP} - \delta^{NP}) \\ \alpha &= \frac{Y_K}Y \end{align}

Estimates of imputed gross capital income in the government and nonprofit sectors are from the TE accounts, while estimates of consumption of fixed capital are from NIPA Table 7.5.

The following charts plot the major production function components since the late 1980s. For comparison, corresponding estimates from the TE accounts and from the main MFP accounts for the private business sector are shown as well.