EC 202 Autumn 2007 Week 2:
Costs and Cost Functions

Total cost of production:

Total cost of production The total cost of production of a firm is simply:
C = w1I1 + w2I2 + … + wNIN
,where Ii is the amount of input i that is used
and wi is the “price” paid for each unit of this
input.
More generally, the amount paid per unit of
input might depend on the quantities used:
C = w1(I1)I1 + w2(I2)I2 + …wN(IN)IN

Economic cost = Opportunity cost:

Economic cost = Opportunity cost The wis reflect the opportunity cost of the inputs used, i.e. their value in their best alternative use.
This does not always correspond to an actual expenditure, for two main reasons:
1. No actual payment was made (use of own
time, own house, own car,…)
2. The payment is unavoidable. What matter is the
price to be paid for the right to use the input as
opposed to not using it. If a payment does not
depend on whether or not the input is used then
it is not an economic cost of using the input (e.g.
contracts where payment is due regardless of use:
“take or pay”)

Example: Investing in Education:

Example: Investing in Education
David is currently working as a elementary school teacher
in Colchester, earning £21,000 a year. He rents a flat for
£480 a month. He has a two year lease that includes a
“no sublet” clause. He also owns a studio in Sitges, near
Barcelona. He uses this studio for his own vacations but
also rents it to tourists, making on average £ 3,500 a year. David gets a job offer in Sitges. When deciding whether or not to move, what kind of housing costs should he consider?
● The first two years of rent on the Colchester flat are
irrelevant: this cost is sunk and therefore does not
represent an economic cost.
● By using his own studio in Sitges full-time, David also
incurs a loss in revenues equal to £ 3,500 a year. This
is an economic cost.
● If you live in Sitges, where do you take your vacations?

Economic Cost Versus Accounting Cost:

Economic Cost Versus Accounting Cost Accounting costs keep track of the money actually paid.
The accounting cost of using an input can be higher or lower than the corresponding economic cost

Example 1:

Example 1 A firm signed a “take or pay” contract for 100 tons of coal: it must pay £100 a ton, whether or not it actually uses the coal.
Suppose that the firm actually uses 80 tons of coal. The corresponding economic cost is zero because the firm would have had to pay £10,000 anyway.
The accounting cost is equal to the actual expense, i.e. to £10,000.
Hence, in this example, economic cost is smaller than accounting cost.

Example 2:

Example 2 You start your own trucking company, purchasing 10 trucks at a cost of £60,000 per truck. Trucks last an average of 10 years and involve yearly maintenance work worth £2,000. In order to pay for the trucks you withdraw money from a money market account where you earn 5% per year.
In any given year, trucks can be resold at their sales value minus any depreciation.
What is the truck-related cost of running your trucking company during the first year?

Slide8:

Accounting costs are equal to the depreciation, i.e. one tenth of £600,000 = £60,000 plus maintenance costs of £2000.
Economic costs would also include the fact that by tying £600,000 into your truck you miss out on interest payments worth 5% or £30,000 a year.
Hence economic cost is higher than pure accounting cost.

Short run and long run costs:

Short run and long run costs In the long run, the quantities of all factors can be chosen freely.
In the short run, the amounts of some (Morgan-Katz-Rosen: all but one) factors of production are fixed. This means that one cannot use more than the fixed amount and that using less than the fixed amount does not decrease the total amount paid for this factor of production.

Variable costs, fixed costs and set-up costs:

Variable costs, fixed costs and set-up costs Variable costs are the costs that depend on the level of output chosen by the firm.
Fixed costs are the costs that are independent of the level of output chosen by the firm, i.e. they have to be paid for irrespective of how much the firm produces, or indeed of whether or not the firm produces at all. Hence fixed costs are sunk.
Set-up costs are independent of the level of output chosen by the firm, as long as this output is positive. When output is zero, set-up costs are also zero.

Short run and long run costs again:

Short run and long run costs again All short run economic costs are variable costs or short-term set-up costs.
Short run accounting costs would also include the fixed cost of paying the fixed inputs.

Special case: 2 inputs:

Special case: 2 inputs Q = F(K,L) In the long run, both L and K can be chosen freely.
In the short run, K is fixed at Ko. L can be chosen freely
The unit cost of L is w, the unit (flow) cost of K is r C = wL + rK

Isocost curves in the Short run:

Isocost curves in the Short run An isocost is the set of all combinations of inputs that result in the same total cost
In the short run, capital is fixed at Ko, hence C = wL. This means that all combinations of factors resulting in the same value C must also correspond to the same amount of labour, L.
Hence, in the short run, the isocosts are vertical lines

Cost minimisation in the short run:

Cost minimisation in the short run 0 L K K0 Q0 Q1 L*(Qo) L*(Q1) L*(Q) is the cost-minimising amount of L required to produce Q
Hence C = wL*(Q) is the short run cost function. Higher cost ● ● Output Expansion
Path

Short run economic cost function:

Short run economic cost function The output expansion path is the set of least-cost input combinations corresponding to different levels of output
The cost function of a firm indicates the lowest possible cost of producing any given level of output.
The cost function can be obtained from the output expansion path, as shown on the following graph.

Slide16:

0 Q C wL*(Qo) wL*(Q1) Qo Q1 L K Ko L*(Qo) L*(Q1) Qo Q1 C(Q)

Short run economic cost function (2):

Short run economic cost function (2) A short run cost function C(Q) is defined for a given technology (i.e. a given production function), given product characteristics, given the prices of variable inputs and given amounts of the factors that are fixed in the short run, i.e. C = C(Q,w,Ko), where the shape of C() depends on the shape of the underlying production function Q = F(K,L)
Hence a change in w, Ko or technology would shift the cost function.

Shifts in the cost function:

Shifts in the cost function 0 Q C(Q) Increase in w or
decrease in Ko Decrease in w, increase in
Ko, technological
progress

Properties of short run cost functions:

Properties of short run cost functions Q = F(Ko,L) yields the corresponding short-run cost
function C = C(Q,w,Ko).
Short run marginal cost is : Hence, marginal cost is positive. If MPL is decreasing in
L (i.e. the “law” of decreasing marginal returns) then
short run marginal cost is increasing in Q.

Slide20:

Short run average cost is ACSR(Q) = CSR(Q)/Q
When all short run economic costs are variable costs (i.e. no set up costs), short run average economic costs are equal to short run average variable costs: ACSR(Q) = AVCSR(Q)

Average and marginal…again:

Average and marginal…again If L is the only variable input in the short run, then there is also a precise relationship between the MPL and ACSR:
● If MPL is increasing (decreasing) in L (and hence
in Q) at all levels, then MPL must also be higher
(lower) than the average product of labour
defined as APL = Q/L.
● If a marginal magnitude is higher than the
corresponding average magnitude, then the
average magnitude increases. Hence if MPL
is increasing then APL must also be increasing.
● But average cost is just wL/Q = w/APL.
Hence increasing (decreasing) APL implies
decreasing (increasing) AC. So, If MPL is
decreasing (increasing) at all levels of Q then
AC is increasing (decreasing) at all levels of
output.

Average and marginal cost:

Average and marginal cost Since Average Cost increases or decreases as long as
marginal cost is higher or lower than average cost, if the
two curves intersect then MC must go through the
minimum of the AC (or AVC) curve. AC
MC 0 Q MC AC MC AC 0 Q 0 Q AC MC

Isocost curves in the long run:

Isocost curves in the long run In the long run, both K and L are variable
Hence total costs are C = wL + rK
For a given level of total costs, Co, the isocost line is K = (Co/r)-(w/r)L
This is the equation of a line with vertical intercept Co/r and slope
–(w/r)

Long run isocost lines:

Long run isocost lines

Cost minimisation in the long run:

Cost minimisation in the long run

Conditions for Cost-minimisation:

Conditions for Cost-minimisation For an interior solution, we must have slope isocost = slope isoquant, i.e. MRTS = w/r
Since MRTS = MPL/MPK, the same condition can be written as
w/MPL = r/MPK
i.e. the cost per additional unit of production must be the same regardless of which input is used to increase output.

Output expansion path in the long run:

Output expansion path in the long run

Long run cost function:

Long run cost function A long run cost function C(Q) is defined for a given technology (i.e. a given production function), given product characteristics and given input prices, i.e. C = C(Q,w,r), where the shape of C() depends on the shape of the underlying production function Q = F(K,L)
Hence a change in w,r or technology would shift the long run cost function.

The relationship between long run and short run cost functions:

The relationship between long run and short run cost functions On the one hand, the cost of factors that are fixed in the short run becomes an economic cost in the long run. This tends to make LR economic costs higher than SR economic costs.
On the other hand, in the LR, the firm can choose more efficient combinations of inputs. This tends to make LR economic costs lower than SR economic costs.
Only the second force applies to accounting costs, hence SR accounting costs are never lower than LR accounting costs.

Economies of scale and the cost function:

Economies of scale and the cost function There are increasing, decreasing or constant economies of scale if the long run average cost decreases, increases or stay constant as output increases
Economies of scale are a long-term concept: the use of all factors of production can be chosen without constraints

Economies of Scale and Returns to Scale:

Economies of Scale and Returns to Scale In the absence of set up costs and for given factor prices increasing, decreasing or constant economies of scale arise when the underlying production function exhibits increasing, decreasing or constant returns to scale.
With set-up costs there will be economies of scale even if the technology exhibits constant returns to scale: AC(Q) = AV(Q) + (set-up cost)/Q. CRS implies AV(Q) is constant, but (set-up cost)/Q decreases with Q.

Should we ever have decreasing economies of scale?:

Should we ever have decreasing economies of scale? The prices of some factors of production might increase a the firm’s demand for such inputs increases
Hence one can have decreasing economies of scale even if the underlying technology exhibits constant returns to scale

Comparative Statics: Input Demand Functions:

Comparative Statics: Input Demand Functions

Input demand functions:

Input demand functions Input demand functions give us the quantity of
inputs used as a function of the level of output to
be achieved and the prices of the factors of
production. The shape of these Conditional input
demand functions depends on the underlying
technology:
L = L(w,r,Q)
K = K(w,r,Q)

Properties of Conditional input demand functions:

Properties of Conditional input demand functions L(w,r,Q) is decreasing in w: as labour becomes more expensive relative to K, the firm uses more K and less L. 0 L K Lo Ko K1 L1

Properties of Conditional input demand functions (2):

Properties of Conditional input demand functions (2) With only two factors of production, the demand for an input is increased by an increase in the price of the other factor. With more than 2 factors, demand can increase or decrease with the price of other inputs.
L(w,r,Q) can be increasing or decreasing in Q. However, as Q increases (decreases) the demand for at least one input must increase (decrease).

Example 1: Leontief:

Example 1: Leontief Q = Min(K,L) For each level of output Q,
we need Q units of K and
Q units of L. Hence
C(Q) = (w+r)Q is the
long run cost function L K 0

Example 2: linear production function:

Example 2: linear production function If the production function is Q = L + K, what is the (long
(run) economic cost function of the firm? Isoquant
For Qo Isocost
lines If (w/r) > 1, then the firm
only uses K. Hence C(Q) = rQ
If (w/r) < 1, then the firm
only uses L. Hence C(Q) = wQ
If (W/r) = 1, then the firm is
indifferent between any mix
of L and K that adds up to Q,
hence C(Q) = wQ = rQ.
Putting all cases together, we
have C(Q) = Min(w,r)Q

Example 3: Cobb Douglas:

Example 3: Cobb Douglas Q = KaLb
Cost minimisation in the long run implies w/r = MRTS
MRTS = (b/a)(K/L) hence K = (a/b)(w/r)L so Q = (a/b)a(w/r)aLa+b , or L = Q1/(a+b)(b/a)a/(a+b)(w/r)-a/(a+b) and K = Q1/(a+b)(b/a)-b/(a+b)(w/r)b/(a+b)
C(Q) = wL + r K, hence

Cobb-Douglas:

Cobb-Douglas Hence, for example, with a = b = 0.5 we get C(q) = 2w1/2r1/2Q
We have increasing economies scale if a + b > 1.
C(q) is concave in both w and r
Alternatively we could have minimised the following Lagrangian: L= wL+rK+λ[Q-KaLb] with respect to K,L and λ.

Non-linear isocost curves:

Non-linear isocost curves So far we have assumed that the firm took w and r as given. But there are cases where w and/or r depend on the amount of L or K that are used.
In such cases, the isocost curves are not linear.

Example 1: Overtime:

Example 1: Overtime Given the number of qualified workers in the area, the
firm can hire up to 50 units of labour at a flat rate of wo.
If the firm wants to use more labour (more working
hours), it must pay a higher rate of w1 > wo to all units
of labour beyond 50. 0 L K Wo/r W1/r 50 Challenge: suppose Q=(KL)1/2
what would be the firm’s demand
function for labour ?

Example 2: quantity discounts:

Example 2: quantity discounts The firm uses two inputs, I and L.
The unit price of L is w.
The unit price of I is r(I), where r is a decreasing function of I (r’ < 0,
r’’ > 0).
0 L K Isocost curve

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