Exercise 3.1. Suppose that output is given by the neoclassical production function Y (t) = F [K (t), L(t), A (t)] satisfying Assumptions 1 and 2, and that we observe output, capital and labor at two dates t and t + T. Suppose that we estimate TFP growth between these two dates using the equation xˆ (t, t + T) = g (t, t + T) − αK (t) gK (t, t + T) − αL (t) gL (t, t + T), where g (t, t + T) denotes output growth between dates t and t + T, etc., while αK (t) and αL (t) denote the factor shares at the beginning date. Let x (t, t + T) be the true TFP growth between these two dates. Show that there exists functions F such that xˆ (t, t + T) /x (t, t + T) can be arbitrarily large or small. Next show the same result when the TFP estimate is constructed using the end date factor shares, i.e., as xˆ (t, t + T) = g (t, t + T) − αK (t + T) gK (t, t + T) − αL (t + T) gL (t, t + T).

Explain the importance of differences in factor proportions (capital-labor ratio) between the beginning and end dates in these results.