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 entrepreneurial costs are included in capital costs Inputs are hired in perfectly competitive markets –firms are price takers in input markets

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

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 (q 0 ) and wants to minimize its costs

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

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

Cost Minimization Suppose that the production function is Cobb-Douglas: q = k  l  The Lagrangian expression for cost minimization of producing q 0 is ℒ = vk + w l + (q 0 - k  l  )

Cost Minimization The FOCs for a minimum are  ℒ /  k = v -  k  -1 l  = 0  ℒ /  l = w -  k  l  -1 = 0  ℒ /  = q 0 - k  l  = 0

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

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

q0q0 Given output q 0, we wish to find the least costly point on the isoquant C1C1 C2C2 C3C3 Costs are represented by parallel lines with a slope of - w/v Cost-Minimizing Input Choices l per period k per period C 1 < C 2 < C 3

C1C1 C2C2 C3C3 q0q0 The minimum cost of producing q 0 is C 2 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

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

The Firm’s Expansion Path l per period k per period q 00 The expansion path is the locus of cost- minimizing tangencies q0q0 q1q1 E The curve shows how inputs increase as output increases

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

Cost Minimization Suppose that the production function is CES: q = (k  + l  )  /  The Lagrangian expression for cost minimization of producing q 0 is ℒ = vk + w l + [q 0 - (k  + l  )  /  ]

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 /  = q 0 - (k  + l  )  /  = 0

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

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

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

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

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)

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

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

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

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

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

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

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

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

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

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

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

Partial Elasticity of Substitution The partial elasticity of substitution between two inputs (x i and x j ) with prices w i and w j is given by S ij is a more flexible concept than  –it allows the firm to alter the usage of inputs other than x i and x j when input prices change

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

Technical Progress Improvements in technology also lower cost curves Suppose that total costs (with constant returns to scale) are C 0 = C 0 (q,v,w) = qC 0 (v,w,1)

Technical Progress Because the same inputs that produced one unit of output in period zero will produce A(t) units in period t C t (v,w,A(t)) = A(t)C t (v,w,1)= C 0 (v,w,1) Total costs are given by C t (v,w,q) = qC t (v,w,1) = qC 0 (v,w,1)/A(t) = C 0 (v,w,q)/A(t)

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:

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

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

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

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

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 k 1 and the firm is free to vary only its labor input The production function becomes q = f(k 1, l )

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 )

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

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

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 SVC Fixed Cost

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 =

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

Short-Run and Long-Run Costs Output Total costs SC (k 0 ) SC (k 1 ) SC (k 2 ) The long-run C curve can be derived by varying the level of k q0q0 q1q1 q2q2 C

Geometry of Costs (3) output AFC Output naik  AFC turun output AVC Output naik  AVC naik output AC Minimum Efficient scale

Short-Run and Long-Run Costs Output Costs The geometric relationship between short- run and long-run AC and MC can also be shown q0q0 q1q1 AC MC SAC (k 0 )SMC (k 0 ) SAC (k 1 ) SMC (k 1 )

Long-run and Short-run Cost Curve output AC y* AC(y*,z*)

Short-Run and Long-Run Costs At the minimum point of the AC curve: –the MC curve crosses the AC curve MC = AC at this point –the SAC curve is tangent to the AC curve SAC (for this level of k) is minimized at the same level of output as AC SMC intersects SAC also at this point AC = MC = SAC = SMC

Exercise (1) Suppose that a firm uses two inputs x 1 and x 2 with cobb- douglas technology If the firm is restricted to operate at level of k. Calculate: (a) Short-run cost (b) SAC (c) SAVC (d) SAFC (e) SMC (f) Long-run cost

Exercise (2) Production function for the book: Where: q = the number of pages in the finished book; S = the number of working hours spent by Smith, and J = the number of hours spent working by Jones. Smith values his labor as $ 3 per working hour. He has spent 900 hours preparing the first draft. Jones, whose labor is valued at $ 12 per working hour, will revise Smith’s draft to complete the book. a.How many hours will Jones have to spend to produce a finished book of 150 pages? b.What is the marginal cost of th 150 th page of the finished book?