# ESTIMATED U.S. MANUFACTURING CAPITAL AND PRODUCTIVITY BASED ON AN ESTIMATED DYNAMIC ECONOMIC MODEL

*Baoline Chen* and
*A. Zadrozny*

Additional contact information

A. Zadrozny: Bureau of Labor Statistics

No 133, Computing in Economics and Finance 2000 from Society for Computational Economics

**Abstract:**
Two fundamental sources of output growth are production capital and technological knowledge. We consider technological knowledge in the form of total-factor productivity (henceforth, "productivity") in a constant-elasticity-of-substitution (CES) production function. But capital and productivity cannot be directly observed, except perhaps at disaggregated levels of activity. To use capital and productivity in a quantitative analysis, then, one must first construct or estimate them. We claim that conventional estimates of capital and productivity are based on unnecessarily limited theoretical and sample information. We develop a method for estimating capital and productivity based on far better information that involves an estimated structural dynamic model of a representative firm and the application of a Kalman smoother.Our representative firm solves a dynamic optimization problem. Current levels of capital and productivity are determined by net-of-depreciation levels carried over from the previous period, current rates of investment and research, and current disturbances. The firm sets optimal rates of investment and research according to the solution of the optimization problem. To allow numerical solution, an approximate linear-quadratic form is used. This solution is used to construct the reduced form from a maximum likelihood estimation of the model. The approximation comes from the CES production function considered as a quadratic approximation to its dual variable cost function. The production function has four "inputs:" capital, productivity, labor, and materials, which trade off along convex-to-the-origin isoquants, and three "outputs:" production of saleable output, investment in capital, and research in productivity, which trade off along concave-to-the-origin transformation surfaces. Concavity of the transformation surfaces imposes internal adjustment costs on investment and research, hence, makes capital and productivity quasi-fixed inputs.The model is estimated with annual U.S. total manufacturing data from 1947 to 1997. Estimating capital and productivity involves two major steps. The structural parameters of the model are estimated by maximum likelihood, and the missing-data Kalman filter is used to compute the likelihood function in the face of completely unobserved (latent) capital and productivity. Then, for given parameter estimates, the Kalman smoother computes smoothed estimates of capital and productivity. MLE requires the parameters be identified (negative definite Hessian matrix), and smoothing requires a system-theoretic reconstructibility condition to hold. Both conditions are verified numerically. The Kalman smoother also produces covariance estimates that allow probability bounds to be placed over the capital and productivity estimates. The resulting capital and productivity estimates, which embody wider-than-usual theoretical and sample information, are evaluated and compared in several ways to official U.S. government estimates.

**Date:** 2000-07-05

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