Production-planning horizon, production smoothing, and convexity of the cost functions

One of the productions planning models economists have used for the past four decades is the production-smoothing model. The model's conclusion is simple: if cost functions are convex, under certain conditions, firm's production plans may show smaller variability than the firm's sales. This is a result of the first-order condition for cost minimization over the production-planning horizon. In a 1986 paper Blinder raised doubts regarding the ability of the production-smoothing model to explain the observed data. Some researchers (e.g. Ramey, Journal of Political Economy 99(2) (1991) 306) explained the inconsistency between the theory and the data by providing evidence that the cost functions are non-convex. In this paper, I point out that the shapes of the cost functions depend on the length of the time horizon under consideration. It is possible that while short-run cost functions are upward sloping, long-run cost functions may be downward sloping. Because the planning horizon for firms facing seasonal demand has been shown to be one seasonal cycle, the short-run cost function, whose shape determines the firm's production plan over the horizon, need to be estimated using single planning horizon data, i.e., a single seasonal cycle. Researchers who use long time series data are, in effect estimating long-run cost functions, the slopes of which may be non-positive. I show, further, that convexity of the production cost function is neither necessary nor sufficient condition for production smoothing to be an optimal plan.