BASIC PRINCIPLES AND EXTENSIONS Chapter 12 COSTS MICROECONOMIC THEORY BASIC PRINCIPLES AND EXTENSIONS EIGHTH EDITION WALTER NICHOLSON Copyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.

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

Definitions of Costs Labor Costs to accountants, expenditures on labor are current expenses and hence costs of production to economists, labor is an explicit cost labor services are contracted at some hourly wage (w) and it is assumed that this is also what the labor could earn in alternative employment

Definitions of Costs Capital Costs accountants use the historical price of the capital and apply some depreciation rule to determine current costs economists refer to the capital’s original price as a “sunk cost” and instead regard the implicit cost of the capital to be what someone else would be willing to pay for its use we will use v to denote the rental rate for capital

Definitions of Costs Costs of Entrepreneurial Services To an accountant, the owner of a firm is entitled to all profits, which are the 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 these services are inputs and some cost should be imputed to them part of accounting profits would be considered as entrepreneurial costs by economists

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

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 = TC = 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

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 “derived demand” theory of labor and capital inputs (see Chapter 21) But for now we will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

Cost-Minimizing Input Choices To minimize the cost of producing a given level of output, a firm should choose a point on the isoquant at which the RTS is equal to the ratio w/v it should equate the rate at which K can be traded for L in the productive process to the rate at which they can be traded in the marketplace

Cost-Minimizing Input Choices Mathematically, we seek to minimize total costs given q = f(K,L) = q0 Setting up the Lagrangian L = wL + vK + [q0 - f(K,L)] First order conditions are L/L = w - (f/L) = 0 L/K = v - (f/K) = 0 L/ = q0 - f(K,L) = 0

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

Cost-Minimizing Input Choices q0 Given output q0, we wish to find the least costly point on the isoquant TC1 TC2 TC3 Costs are represented by parallel lines with a slope of -w/v K per period TC1 < TC2 < TC3 L per period

Cost-Minimizing Input Choices The minimum cost of producing q0 is TC2 This occurs at the tangency between the isoquant and the total cost curve K per period TC1 TC3 TC2 K* L* The optimal choice is L*, K* q0 L per period

L = f(K,L) + D(TC1 - wL - vK) Output Maximization The dual formulation of the firm’s cost minimization problem is to maximize output for a given level of cost The Lagrangian is L = f(K,L) + D(TC1 - wL - vK) The first-order conditions are identical to those for the primal problem

Output Maximization q00 The maximum output attainable with total cost TC2 is q0 q0 q1 This occurs at the tangency between the total cost curve and isoquant q0 K per period TC2 = wL + vK K* L* The optimal choice is L*, K* L per period

Derived Demand In Chapter 5, we considered how the utility-maximizing choice is affected by the change in the price of a good we used this technique to develop the demand curve for a good Can we develop a firm’s demand for an input in the same way?

Derived Demand To analyze what happens to K* if v changes, we must know what happens to the output level chosen by the firm The demand for K is a derived demand it is based on the level of the firm’s output We cannot answer questions about K* without looking at the interaction of supply and demand in the output market

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 the firm may demand, we can trace the locus of cost-minimizing choices called the firm’s expansion path

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

The Firm’s Expansion Path The expansion path does not have to be a straight line 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

Minimizing Costs for a Cobb-Douglas Function Suppose that the production function for hamburgers is q = 10K 0.5 L 0.5 Total costs are given by TC = vK + wL Suppose that the firm wishes to minimize the cost of producing 40 hamburgers

Minimizing Costs for a Cobb-Douglas Function The Lagrangian expression is L = vK + wL + (40 - 10K 0.5 L 0.5) The first-order conditions are L/K = v - 5(L/K)0.5 = 0 L/L = w - 5(K/L)0.5 = 0 L/ = 40 - 10K 0.5 L 0.5 = 0

Minimizing Costs for a Cobb-Douglas Function Dividing the first equation by the second gives us This production function exhibits constant returns to scale so the expansion path is a straight line

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 TC = TC(v,w,q) As output 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 K1 units of capital and L1 units of labor input are required to produce one unit of output TC(q=1) = vK1 + wL1 To produce m units of output TC(q=m) = vmK1 + wmL1 = m(vK1 + wL1) TC(q=m) = m  TC(q=1) TC is proportional to q

Graphical Analysis of Total Costs TC Total costs are proportional to output Total costs AC = MC Both AC and MC will be constant Output

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 another 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 TC Total costs rise dramatically as output rises after diminishing returns set in Output

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

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

Homogeneity The total cost function is homogeneous of degree one in input prices if all input prices were to increase in the same proportion (t), the total costs for producing any given output level would also be multiplied by t a simultaneous increase in input prices does not change the ratio of input prices cost-minimizing combination of K and L unchanged

Homogeneity The average and marginal cost functions will also be homogeneous of degree one in input prices In a “pure” inflationary period (one where the prices of all inputs rise at the same rate), there will be no incentive for firms to alter their input choices

Change in the Price of One Input If the price of only one input changes, the firm’s cost-minimizing choice of inputs will be affected a new expansion path must be derived

Change in the Price of One Input An increase in the price of one input must increase TC for any output level AC will also rise If the input is not inferior, MC will also rise

Change in the Price of One Input A change in the price of an input means that the firm must alter its cost-minimizing choice of inputs in the two-input case, an increase in w/v will be met by an increase in K/L Define the elasticity of substitution as s must be nonnegative

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 If firms can easily substitute another input for the one that has risen in price, there may be little increase in costs

Cobb-Douglas Cost Function Suppose that the production function is q = 10K 0.5L0.5 First-order conditions for cost minimization require that w/v = K/L Dividing by K yields q/k = 10(v/w)0.5

Cobb-Douglas Cost Function This means that K = (q/10)w 0.5v -0.5 Multiplying both sides by v, we get vK = (q/10)w 0.5v 0.5 A similar chain of substitutions yields wL = (q/10)w 0.5v 0.5

Cobb-Douglas Cost Function Because TC = vK + wL we have TC = (2/10)qw 0.5v 0.5 If w = v = $4, then TC = 0.8q

Cobb-Douglas Cost Function Because the production function exhibits constant returns to scale, AC and MC will be constant AC = TC/q = 0.8 MC = TC/q = 0.8 If v rises to $9, TC, AC, and MC rise TC = (2/10)qw 0.5v 0.5 = 1.2q AC = TC/q = 1.2 MC = TC/q = 1.2

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)

Short-Run Total Costs Short-run total cost for the firm is STC = vK1 + wL There are two types of short-run costs: short-run fixed costs (SFC) are costs associated with fixed inputs short-run variable costs (SVC) are costs associated with variable inputs

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 Because capital is fixed at K1, K per period Because capital is fixed at K1, the firm cannot equate RTS with the ratio of input prices K1 q2 q1 q0 L per period L1 L2 L3

Short-Run Marginal and Average Costs The short-run average total cost (SATC) function is SATC = total costs/total output = STC/q The short-run marginal cost (SMC) function is SMC = change in STC/change in output = STC/q

Short-Run Average Fixed and Variable Costs Short-run average fixed costs (SAFC) are SAFC = total fixed costs/total output = SFC/q Short-run average variable costs are SAVC = total variable costs/total output = SVC/q

Relationship between Short-Run and Long-Run Costs STC (K2) Total costs STC (K1) q0 q1 q2 TC The long-run TC curve can be derived by varying the level of K STC (K0) Output

Relationship between Short-Run and Long-Run Costs The geometric relationship between short- run and long-run AC and MC can also be shown Costs q0 q1 AC MC SATC (K0) SMC (K0) SATC (K1) SMC (K1) Output

Relationship between 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 SATC curve is tangent to the AC curve SATC (for this level of K) is minimized at the same level of output as AC SMC intersects SATC also at this point the following are all equal: AC = MC = SATC = 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 firm’s expansion path the expansion path shows how input usage expands with the level of output The relationship between output level and total cost is summarized by the total cost function [TC(q,v,w)] the firm’s average cost (AC = TC/q) and marginal cost (MC = TC/q) can be derived directly from the total cost function

Important Points to Note: All cost curves are drawn on the assumption that the input prices and technology are held constant when an input price changes, cost curves shift to new positions the size of the shifts will be determined by the overall importance of the input and by the ease with which the firm may substitute one input for another technical progress will also shift cost curves

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