The Short Run

Chapter 3 The Short-Run 3.1 Introduction In this chapter we begin the study of the firm's short run factor market behavior. As in the previous chapter, we will continue to assume that factor markets are competitive. The difference between the short run and the long run is that in the long run all factors of production are assumed variable, while in the short run some factors are assumed fixed. Generally we assume that labor is variable in both the short run and the long run, whereas capital is fixed in the short run and variable in the long run (Note: what about land?). This affects the firm's cost minimization problem because it loses one degree of freedom in organizing its production: it must make do with the installed level of capital. Notice that regardless of the behavior of the firm in the goods market, it will always strive to minimize its cost in the factor market (question: are monopolies more or less efficient than competitive 1 firms?). In this section we will derive the short run cost function which is the maximum value function derived from the cost minimization problem with a fixed level of capital.   We will also study the relationship that exists between the short-run and long-run cost function as well as the relationship between marginal cost. We will see that there are fundamental differences in the marginal cost curves that lead to Marshall's laws of derived demand. The fundamental difference is that in general there are CRS in the long run and DRS in the short run. 3.2 The Length of the Run The short run and the long run are relative terms and there is no temporal dimension to the definition. That is, the short run is not 1 month and the long run anything longer than that. In fact, the distinction between the short run and the long run depends on the problem that we are studying so that a very short period of time can be the long run and a very long period of time can be the short run. Economists draw the line between the short run and the long run by referring to the potential variability of input use. Therefore, in the short run there are some inputs that can not be varied. An extreme version of the phrase "can not be varied" takes this statement literally. However, the amount of the input may be variable but at a very high cost. Therefore, a good approximation is to treat it as fixed. In the long run, however, all factors are variable. The firm, then, has a larger choice set open to it in the long run than in the short run. Another important aspect of the long run and short run has to do with the scale of production. As a corollary, economists often assume that in the short run the production function exhibits decreasing returns to scale, while in the long run it exhibits constant returns to scale. We will see why this is the case shortly. 3.3 Optimal Choice 3.3.1 The Firm's Short Run Cost Minimization Problem As we stated in the introduction, firms seek to minimize cost subject to the constraint that they produce y units of output. That is, the firm's cost minimization problem, 1 [SCM], is given by:   min wl + rk s.t. f(k,l) = y where k denotes the fixed level of capital with which the firm can work. We will continue to ignore the non-negativity constraints on factor inputs by arguing that the firm is not endowed with any factors and so must be a net buyer in factor markets. Notice that in the [SCM] there are only two parameters: the wage rate, w, and the target level of output, y. If we were given values for these parameters we could obtain the actual quantities demanded. Since we are using arbitrary parameters, our solution will be a quantities of l as a functions of the wage and target production levels:   \[l^{sr}(w,y)\] This demand functions is known as the short run conditional or derived demand function. Notice that conditional demand functions are functions of target output y. This means that as we vary the relative factor cost, the firm's optimal quantities will always be on the same isoquant.   3.3.2 Solving the Firm's Problem First let us relabel the fixed capital cost as (F_{k}). Then, the firm's problem is given by: min F_{k}+wl s.t. f(k,l) = y and we notice that it is a minimization problem in only one variable. The solution to this problem is trivial. First, we solve the constraint for the level of labor input that, along with our installed capital, allows us to achieve output level y: \[l^{or}=l(\overline{k},y)\] The above equation simply states that short run labor demand is a function of the level of installed capital and the target output level. Notice that in this problem there is no optimality condition: the production constraint determines the intensity of factor use. If we had two variable inputs this property of the solution would not hold. In addition, notice that in this case factor prices do not enter the equation for labor demand. The reason is simple: there is no tradeoff in using one input versus another since there is only one input that we can adjust meet our production constraint.