PRODUCTION COSTS. Econ 311 Microeconomics 1 Lecture Material Prepared by Dr. Emmanuel Codjoe

Transcription

1 PRODUCTION COSTS In this section we introduce production costs into the analysis of the firm. So far, our emphasis has been on the production process without any consideration of costs. However, production activities do involve costs implicit and explicit. But cost is a rather complicated concept. It is a term that is open to more interpretations. Note for example the difference between consumers cost and producers. 1

2 PRODUCTION COSTS Social Costs of production, refer to costs to society when its resources are employed to make a given commodity. Since economic resources are limited, when resources are used to produce a certain product, less can be produced of some other product that can be made with those resources. Private costs, are in contrast with social costs. Private costs are defined to be costs to the individual firm or producer. Take case of pollution! 2

3 PRODUCTION COSTS Explicit costs include the ordinary items that an accountant would include as the firms Expenses, e.g., payroll, payments for raw materials, etc. Implicit costs (alternative costs or opportunity cost) include opportunity costs of resources owned and used by the firm s owner. This type is often omitted in calculating the firm s costs. These costs generally come under private costs 3

4 PRODUCTION COSTS The economic cost of any activity is the value of the best forsaken alternative. The firm in order to attract the resources or factors necessary to engage in production, must pay resource owners amounts sufficient to induce them to sacrifice their best alternatives (whether employment or leisure). But it is important to note that in production, cost and price are different. 4

5 PRODUCTION COSTS As we shall see later, costs are important in determining a firm s optimum profit position. They are also the basis for a firm s supply curve. Having noted all these, it is worth stating the goal of the firm. Economists usually assume that the firm maximises profit, which is defined as the difference between revenue and cost. This section on costs assumes that the firm under analysis is a competitive or price-taking firm. 5

6 PRODUCTION COSTS Distinguishing between short-run and long-run costs This is related to the same concept in production theory. In the short-run some costs are fixed, whilst in the long-run they become variable. This is the fundamental difference between the two. Nevertheless, the distinction is a matter of degree. 6

7 PRODUCTION COSTS Distinguishing between short-run and longrun costs The longer the run contemplated, the greater the range of costs regarded as variable rather than fixed. Hence, if a firm is not committed to any outlays, it is in the long-run. In the long-run all options are open! The situation changes to the short-run once a commitment to some factor of production has been undertaken. 7

8 PRODUCTION COSTS Distinguishing between short-run and long-run costs Thus, the short-run is characterised by fixed and variable costs. In the long-run, all costs are variable since all costs depend on the volume of output. 8

9 PRODUCTION COSTS Total, Average and Marginal Costs The total cost can be regarded as the sum, taken over all resources employed, of factor prices times factor quantities. In other words, it is the sum of all costs incurred by the firm to produce a given level of output. From the total cost, two other measures emerge: average and marginal costs. 9

10 PRODUCTION COSTS Total, Average and Marginal Costs The average cost (AC) is defined as the cost per unit of output. Formally, this is defined as: AC TC Q Where TC is total cost and Q is output 10

11 PRODUCTION COSTS Total, Average and Marginal Costs The marginal cost (AC) is defined as the change in total cost resulting from a unit change in output. Formally, this is defined as: MC TC Q Where TC is total cost and Q is output 11

12 PRODUCTION COSTS Total, Average and Marginal Costs Some important conclusions are worth making! We have noted earlier that in the short-run, the firm faces both fixed and variable costs. But as the firm alters its output, only the variable costs change (why?). The marginal cost that a firm experiences as it expands output from given fixed resources are entirely due to its variable costs. 12

13 PRODUCTION COSTS Total, Average and Marginal Costs This therefore leads to a major conclusion! Decisions about output are based entirely on marginal costs; fixed costs are totally irrelevant to any output decisions. 13

14 PRODUCTION COSTS Total, Average and Marginal Costs. But based on our understanding from production, we know that cost is a multivariable function, that is, it is determined by many factors. Thus, in the short-run, C f ( X, T, P f, K ) 14

15 PRODUCTION COSTS Total, Average and Marginal Costs Where C = total costs; X = output; P f = prices of factors; T = technology; and K = fixed factors. In the long-run C f ( X, T, Pf ) 15

16 PRODUCTION COSTS Total, Average and Marginal Costs We have earlier made the assumption that the firm is a competitive firm. Thus, it seeks to be efficient in production, aiming to produce at the minimum cost of production for any given output level, Q, when factor prices are P f. If we also assume that firms are price takers in the factor markets, then P f is fixed. 16

17 PRODUCTION COSTS Total, Average and Marginal Costs Thus, we can write our cost function as dependent on output, Q, alone. This can be written as C c(q) Hence total costs can be written as C c( Q) FC 17

18 PRODUCTION COSTS Total, Average and Marginal Costs The marginal cost function can be written as Q FC Q Q c Q Q c Q MC v ) ( ) ( ) ( VC Q Q c Q MC v ) ( ) ( 18

19 PRODUCTION COSTS Total, Average and Marginal Costs But note that the marginal cost measures the rate of change, hence we can define the marginal cost function as dtc MC( Q) dq d[ c( Q)] dq Refresh your memory on the relationship between MC, AVC and AC. 19

20 PRODUCTION COSTS 20

21 PRODUCTION COSTS Total, Average and Marginal Costs The marginal cost curve, average variable cost curve and average total cost curves are generally U-shaped. The U-shape in the short run is attributed to increasing and diminishing returns from a fixed-size plant, because the size of the plant is not variable in the short run. 21

22 PRODUCTION COSTS Total, Average and Marginal Costs The marginal cost and average cost curves are related: When MC exceeds AC, average cost must be rising When MC is less than AC, average cost must be falling This relationship explains why marginal cost curves always intersect average cost curves at the minimum of the average cost curve. 22

23 Optimal Input Combinations How will a competitive firm combine inputs to produce a given quantity of output? As a first approximation, we assume that firms are out-and-out profit maximizers; that is to maximize the difference between revenue (R) and (economic) costs (C) incurred. 23

24 Optimal Input Combinations Thus, our typical firm seeks to maximize profits; max R C The assumption of profit maximization implies that a firm will seek to minimize the costs of producing a given output or seek to maximize the output derived from a given level of cost. 24

25 Optimal Input Combinations Also remember the assumption of perfect factor markets, such that firms are price takers in the input markets. Thus, if we suppose there are two inputs, labour (L) and capital (K), then what combinations of L and K should the firm choose? 25

26 Optimal Input Combinations If the wage rate (w) is the cost of labour and the rental rate (r) is the cost of capital, then the total cost outlay, C, is given by: C K wl rk C w L r r 26

27 Optimal Input Combinations Thus, the various combinations of L and K that can be purchased, given input prices and total outlays can be represented by a straight line. K C/r Slope = (w/r) 0 C/w L 27

28 Optimal Input Combinations A family of Isocost lines can be illustrated below K C (3) > C (2) > C (1) C (1) C (2) C (3) 0 L 28

29 Maximization of output for given cost K K * 0 L * R L 29

30 Optimization Condition The firm maximizes output at point R, by choosing L* and K* of labour and capital respectively. At point R, the isoquant is tangent to the isoquant. Thus, MRTS L, K MP MP L K w r 30

31 Optimization Condition It follows that the optimal combination of inputs is where. MP MP L K w r Or MP w L MP r K 31

32 Optimization Condition More generally, a firm will choose an input combination such that. MP P a a MP b... P b MP Where MP a, MP b,......, MP n are the marginal products of inputs, a, b,..., n, and P a, P b,......, P n are input prices. P n n 32

33 K Minimization of Cost for a Given Output Level K * Z L * L 33

34 Optimization Condition To minimize the cost of producing the output level, Q=400, the firm chooses point Z. Here too, the firm must equate the MRTS to the ratio of input prices MRTS L, K MP MP L K w r 34

35 Constrained Optimization The firm s decision, as with the case of consumers, can be represented as a constrained optimization problem. We first consider the case of constrained output maximization. Thus, given Q = f(k, L) and C = rk + wl, we can set up the Lagrangian function: Z f ( K, L) ( rk wl C) 35

36 Constrained Optimization Z symbolises the Lagrangian function. Here is an undetermined Lagrange multiplier 0 Also note that another formulation of the Lagrangian function is: Z f ( K, L) ( C rk wl) 36

37 Constrained Optimization We set the first-order conditions, which are to set to the partial derivatives of K, L, and λ equal to zero. Z K f k r 0 (1) Z L f l w 0 (2) 37

38 Constrained Optimization Z rk wl C 0 (3) From (1) and (2): moving the price terms to the right and dividing (2) by (1): f f l k r w (4) 38

39 Constrained Optimization From (4) the first order conditions state that the ratio of marginal products must be equated with their price ratios. Solving (1) and (2) for λ, yields: f r k f l w (5) 39

40 Constrained Optimization (5) states that the contribution to output of the last money outlay expended on each input must equal λ. The multiplier, λ, is the derivate of output with respect. 40

41 Constrained Cost Minimization The firm may desire to minimize the cost of producing a prescribed level of output. As with our earlier analysis, we form the Lagrangian function, and set the partial derivatives to zero for K, L, and λ. Z rk wl ( Q 0 f ( K, L) 41

42 Constrained Cost Minimization An alternative formulation of the Lagrangian function is: Z rk wl ( f ( K, L) 0 Q ) In what follows, we set the various partial derivatives to zero and obtain the optimal conditions for input combination in production 42

43 Constrained Cost Minimization Z K r f k 0 (1') Z L w f l 0 (2') Z Q 0 f ( K, L) 0 (3') 43

44 Constrained Cost Minimization Since r and w are both positive, moving the price terms and dividing (2 ) by (1 ), we obtain: w r f f l k MRTS The first order conditions for the minimization of cost subject to an output constraint are similar to those for the maximization of output subject to a cost constraint. 44

45 Second-Order Conditions We shall leave out the details of second-order conditions. However, more generally, for output maximization subject to a given cost, and for cost minimization subject to a given output level, the slope of the marginal product curves for the two inputs must be negative. 2 K Q f 0; f 0 kk 2 2 Q 0; 2 L 0 or: ll 45

46 Second-Order Conditions The second-order conditions ensure that we are satisfied that the isoquants are convex to the origin. If the isoquant is concave, then we have a corner solution. 46

47 K Second-Order Conditions e 1 e Q=230 0 e 2 L 47

48 Second-Order Conditions In the diagram above, output, Q=230, can be produced at points e, e 1 and e 2. This indicates different costs of producing the same level of output. The lowest cost point is given by e 2. 48

49 Profit Maximization: A Formal Analysis A firm is usually free to vary the levels of both cost and output, with the ultimate objective being to maximize profits rather than a solution to a constrained maximum or constrained minimum problems. The firm we have been analysing is assumed to operate in a competitive market. Its total revenue is given by the number of units of Q sold multiplied by the fixed unit price it receives. 49

50 Profit Maximization: A Formal Analysis The firm s profit is thus defined as: P. Q C Given Q = f(k, L) and C = rk + wl, the firm s profit function is given by П = Pf(K, L) rk - wl Profit is a function of K and L and is thus maximised with respect to these variables. 50

51 Profit Maximization: A Formal Analysis Differentiating the profit function with respect to capital and labour, gives: k Pf k r 0 (6) l Pf l w 0 (7) Moving input price items to the right, we have: 51

52 Profit Maximization: A Formal Analysis Pf k r Pf l w Thus, Pf k and Pf l are the values of the marginal product of K and L respectively, and they represent the rate at which the firm s revenue would increase with further increases in K or L. 52

53 Profit Maximization: A Formal Analysis Profit maximization requires that each input be utilised up to the point at which the value of its marginal product equals its price. Profits can be increases as long as Pf k > r and Pf l > w. That is, as long as the addition to revenue from employing an additional unit of input exceeds its cost. 53

54 Profit Maximization: A Formal Analysis The second-order conditions for profit maximization require that: 2 2 L Pf ll K Pf kk 0 These suggest that profits must be decreasing with respect to further increases in L and K. 54

55 Profit Maximization: A Formal Analysis Because P > 0, this suggests that the marginal product of both L and K must be decreasing. The conditions for first- and second-order profit maximization require that the isoquant be strictly concave in the neighbourhood of the a point at which the first-order conditions are satisfied with non-negative levels of inputs (K and L). 55

56 Optimal Expansion Path in the Short-Run In the short-run, K is fixed, the firm is therefore forced to expand along a straight line parallel to the horizontal axis. With prices of factors constant, the firm does not maximise profits in the short-run, due to the constraint of given capital. The optimal path would be along OA, but the firm can only expand along K in the short run. K 56

57 Optimal Expansion Path in the Short-Run K A _ K _ K 0 L 57

58 Optimal Expansion Path in the Long-Run In the long-run all factors are variable. Output can therefore be expanded without limitation. As is always the case, the firm s objective of profit maximization, this means it chooses the least-cost combination of inputs, which is represented by the points of tangency between the isocosts curves and isoquants. 58

59 Optimal Expansion Path in the Long-Run K E L 59

60 Optimal Expansion Path in the Long-Run The expansion path indicates how, as output rate changes (but input prices remain fixed),the quantity of each input changes. If the production function is homogeneous, the expansion path will be a straight line through the origin, whose slope depends on the ratio of factor prices. With only two inputs in production, it is also easy to derive the long-run cost function from the expansion path. 60

61 Optimal Expansion Path in the Long-Run It is also worth noting that the maximum profit-input combination lies on the expansion path. Given Pf l = w and Pf k = r, we note that this is a special case of the constrained output maximization discussed earlier. 61

62 Optimal Expansion Path in the Long-Run That is, along the long-run expansion path, the condition is f w satisfied. f l k r Also the implies that profit is also maximised, that is, Pf Pf l k w r 62

63 Input Demand Functions The firm s input demands are derived from the underlying demand for the goods and services it produces. Thus, the firm s input demand functions are obtained by solving the first-order conditions for profit maximization for L and K, as functions of input prices and output price. 63

64 Input Demand Functions Therefore, more generally, given a production function of the Cobb-Douglas form, we can obtain the firm s input demand functions Q AL K,, 0: 1 PAL K wl rk 64

65 Input Demand Functions Solving for L and K, we obtain: 0 1 w K AL P l 0 1 r K AL P k ),, ( ; ),, ( * * p w r f K p w r f L 0 0; ; 0 p r w f f f 65