Chapter 10 Cost Functions.

Chapter 10 Cost Functions

Explicit Costs and Implicit Costs
Explicit Costs – Costs that involve a direct monetary outlay. Implicit Costs – Costs that do not involve outlays of cash.

Opportunity Cost The relevant concept of cost is opportunity cost: the value of a resource in its best alternative use. The only alternative we consider is the best alternative

Economic Costs and Accounting Costs
Economic Costs – Sum of a firm’s explicit costs and implicit Costs. Accounting Costs – Total of a firm’s explicit costs.

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 Sunk Cost Expenditure that has been made and cannot be recovered Because a sunk cost cannot be recovered, it should not influence the firm’s decisions. E.g: Consider the purchase of specialized equipment for a plant. Suppose the equipment can be used to do only what it was originally designed for and cannot be converted for alternative use. The expenditure on this equipment is a sunk cost. Because it has no alternative use, its opportunity cost is zero.

Two Simplifying Assumptions
Inputs are hired in perfectly competitive markets firms are price takers in input markets There are only two inputs homogeneous labor (l), measured in labor-hours homogeneous capital (k), measured in machine-hours

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

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 We will assume that the firm has already chosen its output level (q0) and wants to minimize its costs

Cost-Minimizing Input Choices
We seek to minimize total costs given q = f(k,l) = q0 Setting up the Lagrangian: ℒ = wl + vk + [q0 - f(k,l)] FOCs are ℒ /l = w - (f/l) = 0 ℒ /k = v - (f/k) = 0 ℒ / = q0 - f(k,l) = 0

Dividing the first two conditions we get The cost-minimizing firm should equate the MRTS for the two inputs to the ratio of their prices

Solution to cost minimization:
Slope of isoquant = slope of isocost line (or) Ratio of marginal products = ratio of input prices

At point E This implies the firm could spend an additional dollar on labor and save more than a dollar by reducing its employment of capital and keep output constant

At point F This implies the firm could spend an additional dollar on capital and save more than a dollar by reducing its employment of labor and keep output constant

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

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

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

Interior Solution Suppose: Q = 50L1/2K1/2
Suppose that the wage rate of the laborer is given as $5, the rental price is given as $20 and the firm produces an output of 1000. What is the cost minimizing level of output?

Corner Solution The cost-minimizing input combination for producing Q0 units of output occurs at point A where the firms uses no capital. At this corner point the isocost line is flatter than the isoquant.

Contingent Demand for Inputs
In Chapter 4, we considered an individual’s expenditure-minimization problem to develop the compensated demand for a good Can we develop a firm’s demand for an input in the same way?

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

In the present case, cost minimization leads to a demand for capital and labor that is contingent on the level of output being produced The demand for an input is a derived demand it is based on the level of the firm’s output

Some Key Definitions An increase in Q0 moves the isoquant Northeast.
Expansion Path: A line that connects the cost-minimizing input combinations as the quantity of output, Q, varies, holding input prices constant. Normal Inputs: An input whose cost-minimizing quantity increases as the firm produces more output. Inferior Input: An input whose cost-minimizing quantity decreases as the firm produces more output.

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

An Expansion Path As output increases, the cost minimization path moves from point A to B to C when inputs are normal

q00 The expansion path is the locus of cost-minimizing tangencies q0 q1 k per period The curve shows how inputs increase as output increases E l per period

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

An Expansion Path As output increases, the cost minimization path moves from point A to B to C when labor is an inferior input

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 )

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

Cost Minimization Dividing the first equation by the second gives us
The MRTS depends only on the ratio of the two inputs 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 C = C(v,w,q) As output (q) increases, total costs increases

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
With constant returns to scale, total costs are proportional to output Total costs AC = MC Both AC and MC will be constant Output

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)

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

Total costs rise dramatically as output increases after diminishing returns set in 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 Output

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
1) Suppose we have a fixed proportions technology such that q = f(k,l) = min(ak,bl) Production will occur at the vertex of the L-shaped isoquants (q = ak = bl) C(w,v,q) = vk + wl = v(q/a) + w(q/b)

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

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

Properties of Cost functions
1) Homogeneity 2) Total cost functions are nondecreasing in q, v, and w. 3) Total cost functions are concave in input prices.

Properties of Cost Functions
Homogeneity cost functions are all homogeneous of degree one in the input prices a doubling of all input prices will not change the levels of inputs purchased

Nondecreasing in q, v, and w cost functions are derived from a cost-minimization process any decline in costs from an increase in one of the function’s arguments would lead to a contradiction

Concave in input prices costs will be lower when a firm faces input prices that fluctuate around a given level than when they remain constant at that level the firm can adapt its input mix to take advantage of such fluctuations

Concavity of Cost Function
At w1, the firm’s costs are C(v,w1,q1) C(v,w1,q1) w1 Cpseudo If the firm continues to buy the same input mix as w changes, its cost function would be Cpseudo Costs C(v,w,q1) Since the firm’s input mix will likely change, actual costs will be less than Cpseudo such as C(v,w,q1) w

Some of these properties carry over to average and marginal costs homogeneity effects of v, w, and q are ambiguous

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 (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

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 C0 = C0(q,v,w) = qC0(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 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)

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:

If v = 3 and w = 12, the relationship is C = 480 to produce q =40 AC = C/q = 12 MC = C/q = 12

If v = 3 and w = 27, the relationship is C = 720 to produce q =40 AC = C/q = 18 MC = C/q = 18

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

The cost function is then if input prices remain the same, costs fall at the rate of technical improvement

Contingent demand for inputs

Contingent demand functions for all of the firms inputs can be derived from the cost function Shephard’s lemma 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

Shepherd’s lemma is one result of the envelope theorem the change in the optimal value in a constrained optimization problem with respect to one of the parameters can be found by differentiating the Lagrangian with respect to the changing parameter

Suppose we have a fixed proportions technology The cost function is

For this cost function, contingent demand functions are quite simple:

Suppose we have a Cobb-Douglas technology The cost function is

For this cost function, the derivation is messier:

The contingent demands for inputs depend on both inputs’ prices

Suppose we have a CES technology The cost function is

The contingent demand function for capital is

The contingent demand function for labor is

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
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)

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 MRTS will not be equal to the ratio of input prices

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

Long run-all variables are variable and the expansion path is from A – B – C
Short run-some variables are fixed (capital)-the expansion path is from D –B –E

Short run: One input is fixed, capital
Short run: One input is fixed, capital Firm can vary the other input, labor. SO demand for labor will be independent of price of K Short run demand for labor will also depend on quantity produced. As quantity increased, labor used increases holding capital fixed.

Short-Run Marginal and Average Costs
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

Long and Short Run Total Cost Functions
STC(Q,K0) Total Cost ($/yr) TC(Q) The long-run C curve can be derived by varying the level of k K0 is the LR cost-minimising quantity of K for Q0 Q0 Q1 Q (units/yr)

• Long and Short Run Total Cost Functions
STC(Q,K0) Total Cost ($/yr) TC(Q) The long-run C curve can be derived by varying the level of k A • TC0 K0 is the LR cost-minimising quantity of K for Q0 Q0 Q1 Q (units/yr)

• • Long and Short Run Total Cost Functions
STC(Q,K0) Total Cost ($/yr) TC(Q) • The long-run C curve can be derived by varying the level of k TC1 C A • TC0 K0 is the LR cost-minimising quantity of K for Q0 Q0 Q1 Q (units/yr)

• • • Long and Short Run Total Cost Functions
STC(Q,K0) Total Cost ($/yr) • TC(Q) TC2 B • The long-run C curve can be derived by varying the level of k TC1 C A • TC0 K0 is the LR cost-minimising quantity of K for Q0 Q0 Q1 Q (units/yr)

Short-Run and Long-Run Costs
The geometric relationship between short- run and long-run AC and MC can also be shown q0 q1 AC MC SAC (k0) SMC (k0) SAC (k1) SMC (k1) Output

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 LAC 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

Returns to Scale & Economies of Scale
1) When the production function exhibits increasing returns to scale, the long run average cost function exhibits economies of scale so that AC(Q) decreases with Q, all else equal. 2) When the production function exhibits decreasing returns to scale, the long run average cost function exhibits diseconomies of scale so that AC(Q) increases with Q, all else equal. When the production function exhibits constant returns to scale, the long run average cost function is flat: it neither increases nor decreases with output.

LONG-RUN COST WITH ECONOMIES AND DISECONOMIES OF SCALE
As an Envelope Curve The long-run average cost curve LAC is the envelope of the short-run average cost curves SAC1, SAC2, and SAC3. With economies and diseconomies of scale, the minimum points of the short-run average cost curves do not lie on the long-run average cost curve.

Economies of Scope If a firm produces multiple goods, the cost of one good may depend on the output level of the other. Outputs are linked if a single input is used to produce both. There are economies of scope if it is cheaper to produce goods jointly than separately. Economies of Scope – a production characteristic in which the total cost of producing given quantities of two goods in the same firm is less than the total cost of producing those quantities in two single-product firms.

Economies of Experience
Economies of Experience – cost advantages that result from accumulated experience, or, learning-by-doing. Experience Curve (or Learning Curve)– a relationship between average variable cost and cumulative production volume – typical relationship is AVC(N) = ANB, where N – cumulative production volume, A > 0 – constant representing AVC of first unit produced, -1 < B < 0 – experience elasticity (% change in AVC for every 1% increase in cumulative volume – slope of the experience curve tells us how much AVC goes down (as a % of initial level), when cumulative output doubles

Graphing the Learning Curve
A firm’s production cost may fall over time as managers and workers become more experienced and more effective at using the available plant and equipment. The learning curve shows the extent to which hours of labor needed per unit of output fall as the cumulative output increases. N is the cumulative units of output produced and L the labor input per unit of output. A, B, and b are constants, with A and B positive, and β between 0 and 1. When N is equal to 1, L is equal to A + B, so that A + B measures the labor input required to produce the first unit of output. When β equals 0, labor input per unit of output remains the same as the cumulative level of output increases; there is no learning. When β is positive and N gets larger and larger, L becomes arbitrarily close to A. A, therefore, represents the minimum labor input per unit of output after all learning has taken place. The larger β is, the more important the learning effect.

ECONOMIES OF SCALE VERSUS LEARNING
Economies of Scale: A firm’s average cost of production can decline over time because of growth of sales when increasing returns are present (a move from A to B on curve AC1), Learning Curve: A firm’s average cost of production can decline because there is a learning curve (a move from A on curve AC1 to C on curve AC2).

Chapter 10 Cost Functions.

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Chapter 10 Cost Functions.

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