*COST FUNCTIONS

*The Firms Expansion PathThe firm can determine the cost-minimizing combinations of k and l for every level of output: q= (l*, k*)If input prices remain constant for all amounts of k and l, we can trace the locus of cost-minimizing choicescalled the firms expansion path

*The Firms Expansion Pathl per periodk per periodThe curve shows how inputs increase as output increases

*The Firms Expansion PathThe expansion path does not have to be a straight line (k/l ratio may change)the use of some inputs may increase faster than others as output expandsdepends on the shape of the isoquantsThe expansion path does not have to be upward slopingif the use of an input falls as output expands, that input is an inferior input

*Homothetic Production FunctionsSuppose that the production function is Cobb-Douglas:q = k l The Lagrangian expression for cost minimization of producing q0 isL = vk + wl + (q0 - k l )

*The first-order conditions for a minimum areL/k = v - k -1l = 0L/l = w - k l -1 = 0L/ = q0 - k l = 0

Homothetic Production Functions

*Dividing the first equation by the second gives us

This production function is homotheticthe RTS depends on the ratio of the two inputs; RTS changes as k/l changesthe expansion path is a straight line

Homothetic Production Functions

*Suppose that the production function is CES:q = (k + l )/The Lagrangian function for cost minimization of producing q0 isL = vk + wl + [q0 - (k + l )/]

Homothetic Production Functions

*The first-order conditions for a minimum areL/k = v - (/)(k + l)(-)/()k-1 = 0L/l = w - (/)(k + l)(-)/()l-1 = 0L/ = q0 - (k + l )/ = 0

Homothetic Production Functions

*Dividing the first equation by the second gives us

This production function is also homotheticHomothetic Production Functions

*Total Cost FunctionThe total cost function shows the minimum costs that must be incurred to achieve any output level (at a given set of input prices) C = C(v,w,q)As output (q) increases, total costs increase

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

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

*Graphical Analysis ofTotal CostsSuppose that k1 units of capital and l1 units of labor input are required to produce one unit of outputC(q=1) = vk1 + wl1To 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)

*Graphical Analysis ofTotal CostsOutputTotalcostsAC = MCBoth AC andMC will beconstant

*Graphical Analysis ofTotal CostsSuppose instead that total costs start out as concave and then becomes convex as output increasesone possible explanation for this is that there is a third factor of production that is fixed as capital and labor usage expandstotal costs begin rising rapidly after diminishing returns set in

*Graphical Analysis ofTotal CostsOutputTotalcosts

*Graphical Analysis ofTotal CostsOutputAverage and marginalcosts

*Some Illustrative Cost FunctionsSuppose we have a fixed proportions PF such thatq = f(k,l) = min(ak,bl)Production will occur at the vertex of the L-shaped isoquants (q = ak = bl and k*/l*=a/b)C(w,v,q) = vk + wl = v(q/a) + w(q/b)

*Some Illustrative Cost FunctionsSuppose we have a Cobb-Douglas case such thatq = f(k,l) = k l Cost minimization requires that

*Some Illustrative Cost FunctionsIf we substitute into the production function and solve for l, we will getA similar method will yield

*Some Illustrative Cost FunctionsNow we can derive total costs as where which is a constant that involves only the parameters and

*Some Illustrative Cost FunctionsSuppose we have a CES PF such thatq = f(k,l) = (k + l )/To derive the total cost, we would use the same method and eventually get

*Properties of Cost FunctionsHomogeneitycost functions are all homogeneous of degree one in the input prices C(tw,tv,q) = t.C(w,v,q)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, but will double the CPure inflation will not change a firms input decisions but will shift the cost curves up

*Properties of Cost FunctionsNondecreasing in q, v, and wcost functions are derived from a cost-minimization processany decline in costs from an increase in one of the functions arguments would lead to a contradiction

*Properties of Cost FunctionsConcave in input pricescosts will be lower when a firm adjust input mix appropriately in response to changes in input prices than the costs in the situation when input mix remains constantConcavity of CF implies that the firm chooses the optimal input mixLinear CF implies that the firm keeps the input mix constant when input price changes

*Concavity of Cost FunctionwCosts

*Properties of Cost FunctionsSome of these properties carry over to average and marginal costshomogeneityeffects of v, w, and q are ambiguous

*Input SubstitutionA change in the price of an input will cause the firm to alter its input mixWe wish to see how k/l changes in response to a change in w/v, while holding q constant

*Input SubstitutionPutting this in proportional terms as gives an alternative definition of the elasticity of substitutionin the two-input case, s must be nonnegativelarge values of s indicate that firms change their input mix or k/l ratio significantly if relative prices of inputs change

*Shifts in Cost CurvesThe cost curves are drawn under the assumption that input prices and the level of technology are held constantany change in these factors will cause the cost curves to shift

*Size of Shifts in Costs CurvesThe increase in costs will be largely determined by the relative significance of the input (whose price increased) in the production processIf firms can easily substitute another input for the one whose price increased, there may be little increase in costs

*Technical ProgressImprovements in technology also reduce cost for each level outputSuppose that total costs (with constant returns to scale) without technical change areC0 = C0(q,v,w) = qC0(v,w,1)

*Technical ProgressAfter technical change the same inputs that produced one unit of output in period zero will produce A(t) units in period tCt(v,w,A(t)) = A(t)Ct(v,w,1)Ct(v,w,1) = C0(v,w,1)/A(t) Total costs are given byCt(v,w,q) = qCt(v,w,1) = qC0(v,w,1)/A(t) = C0(v,w,q)/A(t)

*Shifting the Cobb-Douglas Cost FunctionThe Cobb-Douglas cost function is whereIf we assume = = 0.5, the total cost curve is :

*Shifting the Cobb-Douglas Cost FunctionIf v = 3 and w = 12, the relationship isC = 480 to produce q =40AC = C/q = 12MC = C/q = 12

*Shifting the Cobb-Douglas Cost FunctionIf v = 3 and w = 27, the relationship isC = 720 to produce q =40AC = C/q = 18MC = C/q = 18

*Contingent Demand for InputsEarlier, we considered an individuals expenditure-minimization problemwe used this technique to develop the demand for a goodCan we develop a firms demand for an input in the same way?

*Contingent Demand for InputsThe demand for an input is a derived demandIn the present case, cost minimization leads to a demand for capital and labor that is contingent on the level of output being producedit is based on the level of the firms output

*Contingent Demand for InputsContingent demand functions for all of the firms inputs can be derived from the cost functionShephards lemmathe contingent demand function for any input is given by the partial derivative of the total-cost function with respect to that inputs price

*Contingent Demand for InputsSuppose we have a fixed proportions PF:

The cost function is

*Contingent Demand for InputsFor this cost function, demand functions for l and k are:

*Contingent Demand for InputsSuppose we have a Cobb-Douglas PFThe cost function is

*Contingent Demand for InputsFor this cost function, the demand function for input k is:

*Contingent Demand for InputsThe contingent demands for inputs depend on both inputs prices

*Short-Run, Long-Run DistinctionIn the short run, economic actors have only limited flexibility in their choiceAssume that the capital input is held constant at k1 and the firm is free to vary only its labor inputThe production function becomesq = f(k1,l)

*Short-Run Total CostsShort-run total cost for the firm isSC = vk1 + wlThere 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)

*Efficiency of Short-Run Costscosts given by short-run CF are not minimal costs for producing each output level: C(k1, l, q)the RTS will not be equal to the ratio of input prices at each output levelthe firm does not have the flexibility of choosing optimal input mix for each qto vary its output in the short run, the firm uses non-optimal input combinations

*Short-Run Total Costsl per periodk per periodq0q1q2

*Short-Run Marginal and Average CostsThe short-run average total cost (SAC) function isSAC = total costs/total output = SC/qThe short-run marginal cost (SMC) function isSMC = change in SC/change in output = SC/q

*Relationship between Short-Run and Long-Run CostsOutputTotal costsThe long-runC curve canbe derived byvarying the level of k

*Relationship between Short-Run and Long-Run CostsOutputCostsThe geometric relationshipbetween short-run and long-runAC and MC canalso be shown

*Relationship between Short-Run and Long-Run CostsAt the minimum point of the AC curve:the MC curve crosses the AC curveMC = AC at this pointthe SAC curve is tangent to the AC curveSAC (for this level of k) is minimized at the same level of output as ACSMC intersects SAC also at this pointAC = MC = SAC = SMC

*Important Points to Note:A firm that wishes to minimize the economic costs of producing a particular level of output should choose that input combination for which the rate of technical substitution (RTS) is equal to the ratio of the inputs rental prices

*Important Points to Note:Repeated application of this minimization procedure yields the firms expansion paththe expansion path shows how input usage expands with the level of outputit also shows the relationship between output level and total costthis relationship is summarized by the total cost function, C(v,w,q)

*Important Points to Note:The firms average cost (AC = C/q) and marginal cost (MC = C/q) can be derived directly from the total-cost functionif the total cost curve has a general cubic shape, the AC and MC curves will be u-shaped

*Important Points to Note:All cost curves are drawn on the assumption that the input prices are held constantwhen an input price changes, cost curves shift to new positionsthe size of the shifts will be determined by the overall importance of the input and the substitution abilities of the firmtechnical progress will also shift cost curves

*Important Points to Note:Input demand functions can be derived from the firms total-cost function through partial differentiationthese input demands will depend on the quantity of output the firm chooses to produceare called contingent demand functions

*Important Points to Note:In the short run, the firm may not be able to vary some inputsit can then alter its level of production only by changing the employment of its variable inputsit may have to use nonoptimal, higher-cost input combinations than it would choose if it were possible to vary all inputs