1 COST FUNCTIONS Reference : Chapter 10 ;Nicholson and Snyder (10 th Edition)

2 Definitions of Costs It is important to differentiate between accounting cost and economic cost – the accountant’s view of cost stresses out-of-pocket expenses, historical costs, depreciation, and other bookkeeping entries – economists focus more on opportunity cost – Opportunity costs are what could be obtained by using the input in its best alternative use

3 Definitions of Costs Labour Costs – To both economist and accountants, labour costs are very much the same thing: labour costs of production (hourly wage)

4 Definitions of Costs Capital Costs (accountants and economists differ…) – accountants use the historical price of the capital and apply some depreciation rule to determine current costs – the cost of the capital is what someone else would be willing to pay for its use (and this is what the firm is forgoing by using the machine) we will use v to denote the rental rate for capital

5 Definitions of Costs Costs of Entrepreneurial Services – accountants believe that the owner of a firm is entitled to all profits revenues or losses left over after paying all input costs – economists consider the opportunity costs of time and funds that owners devote to the operation of their firms

6 Example… IT programmer, she does a new software on her free time and sells it by £5000. Accounting profits=£5000. This seems like a good project Economist profits=£5000 minus what she could have earned working for a firm in her time. Might not seem such a good project any more… part of accounting profits would be considered as entrepreneurial costs by economists

7 Economic Cost The economic cost of any input is the payment required to keep that input in its present employment – the remuneration the input would receive in its best alternative employment

8 Another example A shop owner If her accounting profits are smaller than the rental price of the physical shop, it means that she is having losses as she could obtain more money by no running the shop but renting it

9 Two Simplifying Assumptions There are only two inputs – homogeneous labor ( l ), measured in labor-hours – homogeneous capital (k), measured in machine- hours entrepreneurial costs are included in capital costs Firms cannot influence the input prices, they are given… (they do not depend on firms decisions on the inputs to be used…)

10 Economic Profits Total costs for the firm are given by total costs = C = w l + vk Total revenue for the firm is given by total revenue = pq = pf(k, l ) Economic profits (  ) are equal to  = total revenue - total cost  = pq - w l - vk  = pf(k, l ) - w l - vk

11 Economic Profits Economic profits are a function of the amount of capital and labor employed – we could examine how a firm would choose k and l to maximize profit – We will do later on…

12 Minimizing costs… For now… – we will assume that the firm has already chosen its output level (q 0 ) and wants to minimize its costs – We will examine the inputs that the firm will choose in order to minimize costs but produce q 0 (kind of a compensated demand…)

13 Cost-Minimizing Input Choices Mathematically, we seek to minimize total costs given q = f(k, l ) = q 0 Setting up the Lagrangian: L = w l + vk + [q 0 - f(k, l )] First order conditions are  L/  l = w - (  f/  l ) = 0  L/  k = v - (  f/  k) = 0  L/  = q 0 - f(k, l ) = 0

14 Cost-Minimizing Input Choices Dividing the first two conditions we get The cost-minimizing firm should equate the RTS for the two inputs to the ratio of their prices This is subject to the same reservations as with utility (if RTS is strictly decreasing, no corner solutions…)

15 Cost-Minimizing Input Choices Cross-multiplying, we get For costs to be minimized, the marginal productivity per dollar spent should be the same for all inputs The solution to this minimization problem for an arbitrary level of output q 0 give us the contingent demand functions for inputs: l c =l c (w,v,q 0 ); k c =k c (w,v,q 0 )

16 Contingent Demand for Inputs The contingent demand for input would be analogous to the compensated demand function in consumer theory Notice that the contingent demand for input is based on the level of firm’s output. So, it is a derived demand.

17 q0q0 We fix the isoquant of output q 0. C1C1 C2C2 C3C3 Costs are represented by parallel lines with a slope of - w/v Graphically….Cost-Minimizing Input Choices l per period k per period C 1 < C 2 < C 3

18 C1C1 C2C2 C3C3 q0q0 The minimum cost of producing q 0 is C 2 Graphically…. Cost-Minimizing Input Choices l per period k per period k* l*l* The optimal choice is l *, k* This occurs at the tangency between the isoquant and the total cost curve

19 Total Cost Function The total cost function gives the minimum cost incurred by the firm to produce any output level with given input prices. C = C(v,w,q) We can compute it as: C = C(v,w,q)=w*l c (v,w,q)+ v*k c (v,w,q)

20 Average Cost Function The average cost function (AC) is found by computing total costs per unit of output

21 Marginal Cost Function The marginal cost function (MC) is found by computing the change in total costs for a change in output produced

22 Graphical Analysis of Total Costs Suppose that k 1 units of capital and l 1 units of labor input are required to produce one unit of output C(q=1) = vk 1 + w l 1 To produce m units of output (assuming constant returns to scale) C(q=m) = vmk 1 + wm l 1 = m(vk 1 + w l 1 ) C(q=m) = m  C(q=1)

23 Graphical Analysis of Total Costs Output Total costs C With constant returns to scale, total costs are proportional to output AC = MC Both AC and MC will be constant

24 Graphical Analysis of Total Costs Suppose instead that total costs start out as concave and then becomes convex as output increases – one possible explanation for this is that there is a third factor of production that is fixed as capital and labor usage expands – total costs begin rising rapidly after diminishing returns set in

25 Graphical Analysis of Total Costs Output Total costs C Total costs rise dramatically as output increases after diminishing returns set in

26 Graphical Analysis of Total Costs Output Average and marginal costs MC MC is the slope of the C curve AC If AC > MC, AC must be falling If AC < MC, AC must be rising min AC

27 Shifts in Cost Curves The cost curves are drawn under the assumption that input prices and the level of technology are held constant – any change in these factors will cause the cost curves to shift

28 Properties of Cost Functions 1.Homogeneity: If the input prices are multiplied by an amount t, the total cost is multiplied by the same amount cost minimization requires that the ratio of input prices be set equal to RTS, a doubling of all input prices will not change the levels of inputs purchased 2. Nondecreasing in q, v, and w

29 3. Concave in input prices – The cost function increases less than proportionally when one input price increases because the firm can substitute it by other inputs – As the expenditure function in consumer theory

30 C(v,w,q 1 ) TC is concaveity of Cost Function w Costs The cost function C(v,w 1,q 1 ) is concave in input prices. C(v,w1,q1)C(v,w1,q1) w1w1 Pseudo Cost

31 Reaction to input prices increases If the price of an input increases, the cost will increase If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs It is important to measure the substitution of inputs in order to predict how much costs will be affected by an increase in the price of an input (possibly due to a tax increase)

32 Input Substitution A change in the price of an input will cause the firm to alter its input mix We wish to see how k/ l changes in response to a change in w/v, while holding q constant

33 Input Substitution Rather than the derivative, we will use the elasticity: gives an alternative definition of the elasticity of substitution –in the two-input case, s must be nonnegative –large values of s indicate that firms change their input mix significantly if input prices change. Hence, costs will not change so much –It can me estimated using econometrics

34 Shephard’s Lemma Shephard’s lemma (a trick to obtain the contingent demand functions from the cost function) the contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that input’s price See example 10.4 in the book As we obtained the compensated demand from the expenditure function in consumer theory

35 Short-Run, Long-Run Distinction Economic actors might not be completely free to change the amount of inputs Economist usually assume that it takes time to change capital levels, while labor can be changed quickly Economist say that the “short run” is the period of time in which some of the inputs cannot be changed “Long-run” is when the period of time when all the inputs can be changed

36 Short-Run, Long-Run Distinction Assume that the capital input is held constant at k 1 and the firm is free to vary only its labor input The production function in the short run becomes q = f(k 1, l )

37 Short-Run Total Costs Short-run total cost for the firm is SC = vk 1 + w l There are two types of short-run costs: – short-run fixed costs are costs associated with fixed inputs (vk 1 ) – short-run variable costs are costs associated with variable inputs (w l )

38 Short-Run Total Costs In the Short-run: – Cannot decide the amount of fixed inputs. The firm does not have the flexibility of input choice – to vary its output in the short run, the firm must use nonoptimal input combinations – the RTS will not be equal to the ratio of input prices – Consequently, short-run costs will be equal to or larger than long-run costs

39 Short-Run Total Costs l per period k per period q0q0 q1q1 q2q2 k1k1 l1l1 l2l2 l3l3 Because capital is fixed at k 1, the firm cannot equate RTS with the ratio of input prices. Notice that short run costs will be Equal or larger than the long run cost

40 Short-Run and Long-Run costs The short run cost depend on the amount of fixed capital available There is no a unique short run cost curve. There will be as many as possible levels of capitals are The short run cost of producing q 1 will be equal to the long run cost when the available capital in the short run is the same as the optimal level of capital for the long run (see q 1 and k 1 in the previous graph).

41 Short-Run Marginal and Average Costs Remember that short run costs are given by: – SC = vk 1 + w l The short-run average total cost (SAC) function is SAC = total costs/total output = SC/q The short-run marginal cost (SMC) function is SMC = change in SC/change in output =  SC/  q

42 Relationship between Short-Run and Long-Run Costs Output Total costs SC (k 0 ) SC (k 1 ) SC (k 2 ) The long-run C curve is the minimum of Short-run ones q0q0 q1q1 q2q2 C

43 Some Illustrative Cost Functions