ch 10: COSTS 1

Cost function

Derivation of the cost function mathematically and graphically

Some more concepts associated with costs◦ Total fixed cost and total variable cost

◦ Average fixed cost and average variable cost

◦ Marginal cost

Short run vs long run costs

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What does the cost of production dependson?◦ Level of output◦ The factor prices (prices of inputs)◦ The type of production technology◦ Efficiency of productionNaturally a cost function is one that somehow relates all of

the above.

Definition:◦ It is a function C(q, wK, wL) that relates the minimum costs

C to a given output amount q.◦ Note “minimum costs C to a given output”, implying an

efficient relation between C and q.

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Mathematically◦ We obtain the cost function by solving the cost

minimization problem:

such that

◦ In words: choose the inputs so as to minimize the costs toproduce an output q at given prices of labor and capital.

Again, use Lagrange multipliers to solve◦ Let K*(q) and L*(q) be the solutions to the cost minimization problem for

output level q

◦ The cost function is C(q)= wKK* (q) + wLL* (q)

◦ This is the so-called long-run cost function

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min,

LKfq

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Graphically solving the cost minimizationproblem.◦ Move to lowest possible isocost line that still has a point

common with the relevant isoquant.

◦ If it helps “isocost lines and budget lines are the same”thing”

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Solving the minimization problem graphically

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Mathematical steps, Scheme below works if isoquants are convex„enough‟

Step (a) In optimum:◦ Slope isoquant = slope isocost curve

this implies

Meaning:“In the optimum an extra dollar invested in labor mustraise production by just as much as an extra dollar invested incapital”

Step (b) In optimum we also have q=F(K,L)

Step (c) Combine conditions of steps (a) and (b) Two equations &two unknowns Find K* and L

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Cost minimization problem Let q=2K1/2L1/2 (Production function)

let wL=2 and wK=4 (capital and labor costs)

such that

Step (a):

Step (b):

Step (c):

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So the cost function will be:

Example 2: the Leontief technology q = min(1/6L,K)

Solution cost minimization problem is now obtained by common

sense, not by mathematical recipe

L and K must be on the efficient locus: 1/6L=K (why?)

So q= 1/6 & L=K and it follows that: L*(q) = 6q and K*(q)=q

Cost function

C(q) = WLL*(q)+ wLK*(q) = wL6q +wKq = (wL6 +wK)q

For example, if wL=wK=1, then C(q)=7q

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Fixed Cost:◦ the cost that does not vary with the level of output in

the short run.

◦ FC = rK0

Variable Cost:◦ the cost that varies with the level of output in the

short run.

◦ VCq1= wL1

Total Cost:◦ the cost of all the factors of production

employed.

◦ TCq1= FC + VCq1= rK0+ wL1

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Short-run and long-run cost functions In the long-run (LR) the producer can vary every

input

In the short-run (SR) some inputs are assumed tobe fixed

Usual assumption in case of two inputs: q = F(K,L).K is

assumed fixed and L variable

Practical implication #1: Costs must be higher inthe short-run than in the long-run.

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Practical implication #2 SR cost function has fixed costs as well as variable costs

LR cost function: only variable costs

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Other concepts of short run costs. Average fixed cost (AFC) :

◦ is fixed cost divided by the quantity of output. The ray from theorigin to a point on the FC curve.

◦ AFCq1= FC/Q1= rK0/Q1

Average variable cost (AVC):◦ is variable cost divided by the quantity of output. The ray from the

origin to a point on the VC curve.◦ AVCq1= VC/Q1= wL1/Q1

Average total cost (ATC): is total cost divided by the quantity of output. The ray

from the origin to a point on the TC curve.◦ ATCq1= AFCq1+ AVCq1

Marginal cost (MC):◦ is the change in total cost that results from producing an

additional unit of output.◦ MC=dc(q)/dq

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The most important cost in production decision is the

marginal cost.

Similar to marginal product (C.9), when MC is less than the

average cost (either ATC or AVC), the average cost curve must

be decreasing with output; and when MC is greater than

average cost, average cost must be increasing with output.

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Allocating production between 2 productionprocesses.

If have two distinct production processes,allocating inputs so that MC of the two productionprocesses are equal.◦ MCA= MCB

Link between Production and Cost.◦ MC = w/MPL

◦ AVC = w/APL

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The Isocost curve:◦ similar to the budget line in consumer problem, it is

the locus of all possible input bundles that can bepurchased for a given level of total expenditure C. Theslope is –w/r if K and L are the only two inputs with Kon the vertical axis.

rK+ wL= C or K = C/r –(w/r)L

The minimum cost for a given level of output is thebundle at which the isocost curve is tangent to theisoquant curve (C9).

MRTS = MPL/MPK = w/r

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Graph of long run total cost, long run average cost and long run marginal cost. Fixed cost?

Next slide

ch 10: COSTS 19

ch 10: COSTS 20

Explaining practical implication #1 above better:

◦ Costs must be higher in the short-run than in the long-run

Long-run costs cannot exceed short-run costs, because you

have more flexibility in choosing inputs in the right way

Deriving the LR cost function : Deriving the SR cost

function :

Result: LR cost function is lower envelope of SR cost functions

ch 10: COSTS 21

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ch 10: COSTS 22