INTERMEDIATE MICROECONOMICS 3. Producer Theory (cont'd) 

3.4 Cost & Supply Curves
As with the production function, the cost function or C = f (P_{K}K + P_{L}L) can be calculated for one or more than one variable factors of production. The one variable factor cost function corresponds to the shortrun in which at least one factor is fixed. In effect, the more than one variable factor cost function corresponds to the longrun 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).
1. More than One Variable Factor Cost Function To maximize profit a firm can either: (a) maximize output for a given cost; or, (b) minimize cost for a given level of output. a) Maximize Output for Given Cost Let us assume a firm wants to maximize output for a given cost  C. Assuming that P_{K} and P_{L }are 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: i  MRTS = dL/dK ii  MRTS =  (P_{L}/P_{K}) and iii  MP_{L}/P_{L} = MP_{K}/P_{K} b) Minimize Cost for a Given Output 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). c) Expansion Path & LongRun Cost Assuming technology, P_{K} & P_{L} remain fixed, it is possible (in a way analogous to deriving the incomeconsumption 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 incomeconsumption curve for the consumer (see 1.2.5) to derive the longterm 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 xaxis and cost on the yaxis (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 longrun total cost curve it is possible to derive the longrun average cost curve by dividing longrun cost by the corresponding output. Depending on whether or not constant, increasing or decreasing returns to scale are present, the longrun average cost curve will have a different shape. To the degree that increasing returns are present, the average longrun 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, longrun 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 longrun average cost occurs over the effective market demand. If it does then a large firm will be significantly more costeffective than many small firms. Accordingly, as we will see in the next part of the course, a 'natural monopoly' will tend to arise.
