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KULIAH 6

KULIAH 6. Cost Functions. Dr. Amalia A. Widyasanti. Program Pasca Sarjana Ilmu Akuntansi FE-UI, 2010. Two Simplifying Assumptions. There are only two inputs homogeneous labor ( l ), measured in labor-hours homogeneous capital ( k ), measured in machine-hours

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KULIAH 6

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  1. KULIAH 6 Cost Functions Dr. Amalia A. Widyasanti Program PascaSarjanaIlmuAkuntansi FE-UI, 2010

  2. 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 • Inputs are hired in perfectly competitive markets • firms are price takers in input markets

  3. Economic Profits • Total costs for the firm are given by total costs = C = wl + 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 - wl - vk  = pf(k,l) - wl - vk

  4. Economic Profits • Economic profits are a function of the amount of k and l employed • we could examine how a firm would choose k and l to maximize profit • “derived demand” theory of labor and capital inputs • for now, we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

  5. Cost-Minimizing Input Choices • Minimum cost occurs where the RTS is equal to w/v • the rate at which k can be traded for l in the production process = the rate at which they can be traded in the marketplace

  6. Cost Minimization • Total costs = C = wl +vk • Production function: q = f(k,l) = qo • Lagrangian expression:

  7. Cost Minimization • Suppose that the production function is Cobb-Douglas: q = kl • The Lagrangian expression for cost minimization of producing q0 is ℒ = vk + wl + (q0 - k  l )

  8. Cost Minimization • The FOCs for a minimum are ℒ/k = v - k-1l= 0 ℒ/l = w - kl-1 = 0 ℒ/ = q0 - k  l  = 0

  9. Cost Minimization • Dividing the first equation by the second gives us • This production function is homothetic • the RTS depends only on the ratio of the two inputs • the expansion path is a straight line

  10. Cost-Minimizing Input Choices • The inverse of this equation is also of interest • The Lagrangian multiplier shows how the extra costs that would be incurred by increasing the output constraint slightly

  11. Given output q0, we wish to find the least costly point on the isoquant C1 C3 C2 q0 Cost-Minimizing Input Choices k per period Costs are represented by parallel lines with a slope of -w/v C1 < C2 < C3 l per period

  12. This occurs at the tangency between the isoquant and the total cost curve The optimal choice is l*, k* k* l* Cost-Minimizing Input Choices The minimum cost of producing q0 is C2 k per period C1 C3 C2 q0 l per period

  13. The Firm’s Expansion Path • The firm can determine the cost-minimizing combinations of k and l for every level of output • If input costs remain constant for all amounts of k and l, we can trace the locus of cost-minimizing choices • called the firm’s expansion path

  14. The expansion path is the locus of cost-minimizing tangencies E q1 q0 q00 The Firm’s Expansion Path The curve shows how inputs increase as output increases k per period l per period

  15. The Firm’s Expansion Path • The expansion path does not have to be a straight line • the use of some inputs may increase faster than others as output expands • depends on the shape of the isoquants • The expansion path does not have to be upward sloping • if the use of an input falls as output expands, that input is an inferior input

  16. Cost Minimization • Suppose that the production function is CES: q = (k +l)/ • The Lagrangian expression for cost minimization of producing q0 is ℒ = vk + wl + [q0 - (k +l)/]

  17. Cost Minimization • The FOCs for a minimum are ℒ/k = v - (/)(k + l)(-)/()k-1 = 0 ℒ/l = w - (/)(k + l)(-)/()l-1 = 0 ℒ/ = q0 - (k +l)/ = 0

  18. Cost Minimization • Dividing the first equation by the second gives us • This production function is also homothetic

  19. Total Cost Function • The total cost function shows that for any set of input costs and for any output level, the minimum cost incurred by the firm is C = C(v,w,q) • As output (q) increases, total costs increase

  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 k1 units of capital and l1 units of labor input are required to produce one unit of output C(q=1) = vk1 + wl1 • To produce m units of output (assuming constant returns to scale) C(q=m) = vmk1 + wml1 = m(vk1 + wl1) C(q=m) = m  C(q=1)

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

  24. Graphical Analysis of Total Costs • Suppose 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. C Total costs rise dramatically as output increases after diminishing returns set in Graphical Analysis of Total Costs Total costs Output

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

  27. Shifts in Cost Curves • 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. Some Illustrative Cost Functions • Suppose we have a Cobb-Douglas technology such that q = f(k,l) = kl • Cost minimization requires that

  29. Some Illustrative Cost Functions • If we substitute into the production function and solve for l, we will get • A similar method will yield

  30. Some Illustrative Cost Functions • Now we can derive total costs as where which is a constant that involves only the parameters  and 

  31. Some Illustrative Cost Functions • Suppose we have a CES technology such that q = f(k,l) = (k +l)/ • To derive the total cost, we would use the same method and eventually get

  32. Input Substitution • A change in the price of an input will cause the firm to alter its input mix • The change in k/l in response to a change in w/v, while holding q constant is

  33. Input Substitution • Putting this in proportional terms as • 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

  34. Partial Elasticity of Substitution • The partial elasticity of substitution between two inputs (xi and xj) with prices wi and wj is given by • Sij is a more flexible concept than  • it allows the firm to alter the usage of inputs other than xi and xj when input prices change

  35. Size of Shifts in Costs Curves • The increase in costs will be largely influenced by • the relative significance of the input in the production process • the ability of firms to substitute another input for the one that has risen in price

  36. Technical Progress • Improvements in technology also lower cost curves • Suppose that total costs (with constant returns to scale) are C0 = C0(q,v,w) = qC0(v,w,1)

  37. Technical Progress • Because the same inputs that produced one unit of output in period zero will produce A(t) units in period t Ct(v,w,A(t)) = A(t)Ct(v,w,1)= C0(v,w,1) • Total costs are given by Ct(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t) = C0(v,w,q)/A(t)

  38. Shifting the Cobb-Douglas Cost Function • The Cobb-Douglas cost function is where • If we assume  =  = 0.5, the total cost curve is greatly simplified:

  39. Shifting the Cobb-Douglas Cost Function • If v = 3and w = 12, the relationship is • C = 480 to produce q =40 • AC = C/q = 12 • MC = C/q = 12

  40. Shifting the Cobb-Douglas Cost Function • If v = 3and w = 27, the relationship is • C = 720 to produce q =40 • AC = C/q = 18 • MC = C/q = 18

  41. Shifting the Cobb-Douglas Cost Function • Suppose the production function is • we are assuming that technical change takes an exponential form and the rate of technical change is 3 percent per year

  42. Shifting the Cobb-Douglas Cost Function • The cost function is then • if input prices remain the same, costs fall at the rate of technical improvement

  43. Short-Run, Long-Run Distinction • In the short run, economic actors have only limited flexibility in their actions • Assume that the capital input is held constant at k1 and the firm is free to vary only its labor input • The production function becomes q = f(k1,l)

  44. Short-Run Total Costs • Short-run total cost for the firm is SC = vk1 + wl • There are two types of short-run costs: • short-run fixed costs are costs associated with fixed inputs (vk1) • short-run variable costs are costs associated with variable inputs (wl)

  45. Short-Run Total Costs • Short-run costs are not minimal costs for producing the various output levels • 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

  46. Because capital is fixed at k1, the firm cannot equate RTS with the ratio of input prices k1 l1 l2 l3 Short-Run Total Costs k per period q2 q1 q0 l per period

  47. Cost Functions • Cost Function: - the value of the conditional factor demands - the minimum cost of producing y unit of output • Short-run cost function: - the factors of production are fixed at predetermined levels - the price vectors and the variable vectors are composed of : FIXED AND VARIABLE FACTORS Fixed Cost SVC

  48. Total, Average, and Marginal Costs • Short-Run Total Cost = STC = • Short-Run Average Cost = SAC = • Short-run average variable cost= SAVC = • Short-run average fixed cost = SAFC = • Short-run marginal cost = SMC =

  49. Long Run Cost • Long run cost: • Long-run average cost: • Long-run marginal cost: • Note: Long-run fixed cost are zero

  50. SC (k2) SC (k1) C SC (k0) q0 q1 q2 Short-Run and Long-Run Costs Total costs The long-run C curve can be derived by varying the level of k Output

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