1.3.3

Index

1. More than One Variable Factor Cost Function

2. One Fixed Factor Cost Function
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As with the production function, the cost function or C = f (P_KK + P_LL) can be calculated for one or more than one variable factors of production. The one variable factor cost function corresponds to the short-run in which at least one factor is fixed. In effect, the more than one variable factor cost function corresponds to the long-run in which all factors are variable and a firm can choose at what level or scale to produce (M&Y 10th Fig. 8.2; M&Y 11th Fig. 7.2; B&B Fig. 7.2; B&Z Fig. 8.4).

To maximize profit a firm can either: (a) maximize output for a given cost; or, (b) minimize cost for a given level of output.

Let us assume a firm wants to maximize output for a given cost - C. Assuming that P_K and P_Lare fixed we can calculate: (i) the maximum amount of K; (ii) the maximum amount of L; and, (iii) all the various combinations of K & L that the firm can afford, that is, we can calculate an isocost curve (M&Y 10th Fig. 8.1; M&Y 11th Fig. 7.1; B&B Fig. 7.1; B&Z Fig. 8.4). The slope of the resulting curve reflect the negative of the price ratio, i.e. -(P_L/P_K).

From the Production Function for more than one variable factors (see 1.3.2) isoquants can be determined, that is, the various fixed levels of output that can be produced using varying quantities of inputs. The slope of the isoquant represents the Marginal Rate of Technical Substitution (MRTS) or dL/dK. As with the consumer's indifference curve and budget line, the optimal combination of inputs will be found on the highest attainable isoquant where the MRTS = the slope of the isocost curve (M&Y 10th Fig. 8.2; M&Y 11th Fig. 7.2; B&B Fig. 7.2; B&Z Fig. 8.4). At this point MP_L/MP_K = - (P_L/P_K) or MP_L/P_L = MP_K/P_K, that is at the point of tangency:

Alternatively if a firm wants to maximize profit by minimizing the cost of producing a given level of output (assuming fixed factor prices P_L & P_K) then having selected the specific isoquant it chooses the lowest attainable isocost curve (M&Y 10th Fig. 8.3; M&Y 11th Fig. 7.3; B&B Fig. 7.2; B&Z Fig. 8.4).

Assuming technology, P_K & P_L remain fixed, it is possible (in a way analogous to deriving the income-consumption curve for the consumer, see 1.2.5) to derive the expansion path for a firm. For each level of production (isoquant) there will be a corresponding tangency with an isocost curve. The set of these tangency will trace out the expansion path for the firm (M&Y 10th Fig. 8.18; M&Y 11th Fig. 7.21; B&Z Fig. 8.4).

From the expansion path, it is possible (in a way analogous to the derivation of the Engel Curve from the income-consumption curve for the consumer (see 1.2.5) to derive the long-term cost curve for the firm. Each tangency point provides the cost and the corresponding output level. These two bits of information can then be plotted on a graph with output on the x-axis and cost on the y-axis (M&Y 10th Fig. 8.19; M&Y 11th Fig. 7.22; B&B Fig. 8.1; B&Z Fig. 8.6).

In turn, from the long-run total cost curve it is possible to derive the long-run average cost curve by dividing long-run cost by the corresponding output. Depending on whether or not constant, increasing or decreasing returns to scale are present, the long-run average cost curve will have a different shape. To the degree that increasing returns are present, the average long-run cost curve will tend to decline as output increases (M&Y 10th Fig. 8.20; M&Y 11th Fig. 7.23; B&B Fig. 8.9; B&Z Fig. 8.6). In most cases, however, even in the presence of increasing returns to scale, long-run average cost will eventually increase in response to 'inefficiencies of management' (see M&Y 10th p. 249) as the firm become simply to large to effective control.

The relevant public policy question is whether declining long-run average cost occurs over the effective market demand. If it does then a large firm will be significantly more cost-effective than many small firms. Accordingly, as we will see in the next part of the course, a 'natural monopoly' will tend to arise.