Economic growth and economic development 119

Introduction to Modern Economic Growth ˙ rather than x, we would need to make further assumpIf we are interested in A/A tions For example, if we assume that the production function takes the standard labor-augmenting form Y (t) = F˜ [K (t) , A (t) L (t)] , then we would have A˙ = [g − αK gK − αL gL ] , A αL ˙ but this equation is not particularly useful, since A/A is not something we are inherently interested in The economically interesting object is in fact xˆ in (3.3), since it measures the effect of technological progress on output growth directly In continuous time, equation (3.3) is exact because it is defined in terms of instantaneous changes (derivatives) In practice, instead of instantaneous changes, we look at changes over discrete time intervals, for example over a year (or sometimes with better data, over a quarter or a month) With discrete time intervals, there is a potential problem in using (3.3); over the time horizon in question, factor shares can change; should we use beginning-of-period or end-of-period values of αK and αL ? It can be shown that the use of either beginning-of-period or end-of-period values might lead to seriously biased estimates of the contribution of TFP to output growth, xˆ This is particularly likely when the distance between the two time periods is large (see Exercise 3.1) The best way of avoiding such biases is to use as high-frequency data as possible For now, taking the available data as given, let us look at how one could use the growth accounting framework with data over discrete intervals The most common way of dealing with the problems pointed out above is to use factor shares calculated as the average of the beginning of period and end of period values Therefore in discrete time, for a change between times t and t + 1, the analogue of equation (3.3) becomes (3.4) ¯ K,t,t+1 gK,t,t+1 − α ¯ L,t,t+1 gL,t,t+1 , xˆt,t+1 = gt,t+1 − α 105