costs of production. in the previous section, we looked at production. in this section, we look at...
TRANSCRIPT
Costs of Production
In the previous section, we looked at production.
In this section, we look at the cost of production & determining the optimal level of output.
We’ll start with an interesting cost example, & then focus on determining the optimal level of output in general.
Optimal Level of Law Enforcement & Crime Prevention.
Consider the total cost of crime (CT), including both the cost of the criminal act itself (CA) & the cost of law enforcement & crime prevention (CP).
We’d like to know the optimal level of law enforcement & crime prevention (L) that will minimize the total cost CT = CP + CA .
Remember from calculus that to minimize a function, we take the first derivative and set it equal to zero.
So we will have dCT/dL = dCP/dL + dCA/dL = 0
or dCP/dL = – dCA/dL .
This means that the optimal enforcement level is where the cost of preventing an additional crime is equal to the cost of an additional criminal act.
To finish solving for the optimal level (L), we’d need to know the specific form of the cost functions involved.
Isocost Curve
The set of combinations of inputs that cost the same amount
Equation of an isocost
Suppose you have 2 inputs, capital K & labor L.The price of a unit of capital is PK.The price of a unit of labor is PL.Let a particular outlay amount be R.
Then all combinations of K & L such that PLL+ PKK = R lie on the isocost curve associated with that outlay.
If we rewrite the equation as K = R/PK – (PL/PK)L ,we see that the slope of the isocost is – (PL/PK) & the vertical intercept is R/PK .
Graph of an isocost
K
L
slope = – PL/PK = – 0.1
R/PK= 100
R/PL = 1000
For example, suppose you’re interested in the outlay amount $10,000. Suppose also that Labor cost $10 per unit & capital cost $100 per unit.
Then the slope of the isocost is – PL/PK = – 10/100 = – 0.1 .
The vertical intercept would be 10,000/100 = 100 & the horizontal intercept is 10,000/10 = 1,000.
Maximizing output for a given cost level
K
L
isocost
isoquants
At points A & B, we’re spending the outlay associated with this isocost, but we’re not producing as much as we can. We’re only making Q1 units of output.
We can’t produce Q3 or Q4 with this outlay. Those output levels would cost more.
At point E, we’re producing the most for the money, where the isocost is tangent to an isoquant.
A
B
E
Q1
Q2
Q3
Q4
At the tangency of the isocost & isoquant, the slopes of those curves are equal.
We found previously that the slope of the isoquant is – MPL/MPK , & the slope of the isocost is – PL/PK .
So at the tangency, – MPL/MPK = – PL/PK
or, multiplying by -1, MPL/MPK = PL/PK .
This expression is equivalent to MPL/PL = MPK/PK .
MPL/PL = MPK/PK
This condition means that to get the most output for your money, you should employ inputs such that the marginal product per dollar is equal for all inputs.
(Notice the similarity to the utility maximization condition that the marginal utility per dollar is equal for all goods.)
Minimizing cost for a given output level
K
Lisocosts C1 C2 C3
isoquant
At points A & B, we’re producing the desired quantity, but we’re not using the cheapest combination of inputs, so we’re spending more than necessary.
We can’t produce the desired output level at cost level C1. We need more money.
At point E, we’re producing the desired output at the lowest cost, where the isoquant is tangent to an isocost.
A
B
E
Q1
So whether we’re maximizing output for a given cost level, or minimizing cost for a given output level, the condition is the same:
MPL/PL = MPK/PK
The marginal product per dollar is equal for all inputs.
Short Run Costs of Production
Total Fixed Cost (TFC)
Total fixed cost is the cost associated with the fixed input.
Since TFC is constant, its graph is a horizontal line.
$
Quantity
TFC
Average Fixed Cost (AFC)
AFC = TFC/Q
AFC is the fixed cost per unit of output.
The AFC curve slopes downward & gets closer & closer to the horizontal axis.
$
Quantity
AFC
Total Variable Cost (TVC)
Total variable cost is the cost associated with the variable input.
The TVC curve is upward sloping.
It is often drawn like a flipped over S, first getting flatter & flatter, & then steeper & steeper.
This shape reflects the increasing & then decreasing marginal returns we discussed in the section on production.
$
Quantity
TVC
Average Variable Cost (AVC)
AVC = TVC/Q
AVC is the variable cost per unit of output.
We can determine the shape of the AVC curve based on the shape of the average product curve (AP).
Suppose X is the amount of variable input & PX is its price.
Then, AVC = TVC/Q = (PXX)/Q
= PX(X/Q)
= PX [1/(Q/X)]
= PX [1/AP].
So since AP had an inverted U-shape, AVC must have a U-shape.
Average Variable Cost
$
Quantity
AVC
Total Cost
Quantity
TC = TFC + TVC
The TC curve looks like the TVC curve, but it is shifted up, by the amount of TFC.
$TC
TFC
Average Total Cost
$
Quantity
ATC
Like AVC, ATC is U-shaped, but it reaches its minimum after AVC reaches its minimum.
This is because ATC = AVC +AFC & AFC continues to fall & pulls down ATC.
AVC
Marginal Cost (MC)
MC is the additional cost associated with an additional unit of output.
MC = ΔTC/ ΔQ
Alternatively, MC = dTC/dQ .
MC is the first derivative of the TC curve or the slope of the TC curve.
We can determine the shape of the MC curve based on the shape of the marginal product curve (MP).
Suppose the firm takes the prices of inputs as given.
Then,MC = TC/Q
= PX X/ Q
= PX [1/(Q/X)]
= PX [1/MP].
So since MP had an inverted U-shape, MC must have a U-shape.
While MC is U-shaped, it is often drawn so it extends up higher on the right side.
$
Quantity
MC
Important Graphing Note: The MC must intersect the ATC at its minimum &
the AVC curve at its minimum.
$
Quantity
MC ATC
AVC
We have a similar graphical interpretation of ATC to the one we had for AP.
Q
Since ATC = TC/Q, the ATC of a particular value of Q1 can be interpreted as the slope of the line from the origin to the corresponding point on the curve.
TC
Q1 →
TC1
0
TC
We also have similar graphical interpretation of MC to the ones we had for MP.
The continuous MC is the slope of the total cost curve at a particular point.
The discrete MC is the slope of the line segment connecting 2 points on the total cost curve.
TC
Q
TC
Breaking Even
Recall that TR = PQ. If the price of output is fixed for the firm (as for a perfectly competitive firm), then TR is a straight line with slope P.
When the TR curve is above the TC curve, the firm will have positive economic profits.
When the TC curve is above the TR curve, the firm will have economic losses.
The firm will break even (have zero economic profits) where TR=TC.
TC
Q
TC
TR
Maximizing Profit
The firm will have the maximum profits where the vertical distance between TR & TC is the largest (& TR is above TC).
This is also where MR = MC (which you should recall from Micro Principles is the profit maximizing condition).
That means that the slope of the TR line equals the slope of the TC curve.
So the TR line will be parallel to a tangent to the TC line at the point where profits are maximized.
TC
Q
TC
TR
Profit-maximizing output level
Minimum Profit
TC
Q
TC
TR
Profit-minimizing output level
Notice that the TR line is also parallel to a tangent to the TC line here.
TR – TC reaches a minimum here, not a maximum.
Consider a function y = f(x) as shown. Notice that it has a minimum value at x1.
Notice also that the slope of the function (which is the same as the slope of the line tangent to the curve at that point) is zero. That is, f (x1) = 0.
x1 x
y
We’re going to digress a little to review from Calculus how to use first and second derivatives to determine minima and maxima.
Just to the left of x1, the curve slopes downward; it has a negative slope.
To the right of x1, the curve slopes upward; it has a positive slope.
x1 x
y
f (x) < 0 f (x) > 0
So as we move from left to right in the vicinity of x1, the slope is going from negative to zero to positive. It is increasing.
Recall that if a function is increasing, its derivative is positive.
In this case, the function itself is the slope or first derivative.
So its derivative is the second derivative.
Then, because the first derivative is increasing, the second derivative must be positive: f (x1) > 0.
To put all this together: At a minimum x1 ,the first derivative f (x1) = 0 and the second derivative f (x1) > 0 .
x1 x
y
f (x) < 0 f (x) > 0
f (x1) = 0
x1 x
y
f (x) < 0f (x) > 0
f (x1) = 0
Consider instead this function.At x1, we have a maximum.The derivative f (x1) = 0.
Here, as we move from left to right in the vicinity of x1, the slope is going from positive to zero to negative. The slope is decreasing.If a function is decreasing, its derivative is negative. Again here, the function is the slope or first derivative. So its derivative is the second derivative. Then, because the first derivative is decreasing, the second derivative must be negative: f (x1) < 0.To put all this together: At a maximum x1 ,the first derivative f (x1) = 0 and the second derivative f (x1) < 0 .
To summarize our conclusions on first and second derivatives and maxima and minima:
At a minimum x1,the first derivative f (x1) = 0 and the second derivative f (x1) > 0 .
At a maximum x1,the first derivative f (x1) = 0 and the second derivative f (x1) < 0 .
Memory Aid
If the second derivative is positive, we have two happy twinkly eyes and a smiling mouth which has a minimum.
If the second derivative is negative, we have two sad eyes and a sad mouth which has a maximum.
Let’s return to maximizing profit and see how we use our Calculus in this context.
Example: Suppose the price of a product is $10. The cost of production is TC = Q3 – 21Q2 + 49Q+100.What is the profit maximizing output level?
We need to determine the profit function , take its 1st derivative, set that equal to zero, & solve for Q. = TR –TC
= PQ – TC
= 10Q – (Q3 – 21Q2 + 49Q+100)
= 10Q – Q3 + 21Q2 – 49Q – 100
= – Q3 + 21Q2 – 39Q – 100
d/dQ = – 3Q2 + 42Q – 39.
Setting the 1st derivative equal to zero we have – 3Q2 + 42Q – 39 = 0
This equation can be solved either by the quadratic formula or factoring.
1. Quadratic formula:
)3(2
)39)(3(44242
2
4 22
a
acbbQ
6
468176442
6
129642
6
3642
)6(7 13or 1
2. Factoring:
– 3 (Q2 – 14Q +13) = 0
– 3 (Q – 1)(Q – 13) = 0
So either Q -1 = 0 or Q -13 = 0 ,
& Q = 1 or Q = 13,
which is what we found by the quadratic formula.
Are these both relative maxima, minima, or one of each?
We need to look at the 2nd derivative of our profit function.
– 3Q2 + 42Q – 39 = 0
We had = – Q3 + 21Q2 – 39Q – 100
d/dQ = – 3Q2 + 42Q – 39
The 2nd derivative is – 6Q + 42
To determine whether profit is maximized or minimized at our values of 1 and 13, we need to know if the second derivative is positive or negative at each of those values.
When Q = 1, – 6Q + 42 = 36 > 0
which means that is a minimum when Q =1 .
When Q = 13, – 6Q + 42 = – 36 < 0
which means that is a maximum when Q =13 .
What are our maximum & minimum profit values?
= – Q3 + 21Q2 – 39Q – 100
Our maximum , which is when Q = 13, is:
= – (13)3 + 21(13)2 – 39(13) – 100 = 745
Our minimum , which is when Q = 1, is:
= – (1)3 + 21(1)2 – 39(1) – 100 = – 119
Graph of the Profit Function = – Q3 + 21Q2 – 39Q – 100 Q Profit-5 745-4 456-3 233-2 70-1 -390 -1001 -1192 -1023 -554 165 1056 2067 3138 4209 521
10 61011 68112 72813 74514 72615 66516 55617 39318 170
19 -119 Notice: minimum profit (or greatest loss) of -119 occurs when Q = 1, 20 -480 and the maximum profit of 745 occurs when Q = 13.
As we approach Q = 1 from the left, the slope of the profit curve goes from negative to zero (at Q = 1) to positive to the right of Q =1. The slope is increasing and the curve is convex.
As we approach Q = 13 from the left, the slope of the profit curve goes from positive to zero (at Q = 13) to negative to the right of Q =13. The slope is decreasing and the curve is concave.
Recall that the first derivative f' tells us how the function f changes as our independent variable (X or Q) increases. If f' > 0 , f is increasing; if f' < 0 , f is decreasing.
Similarly, the second derivative f" tells us how f' is changing as our variable (X or Q) increases. If f" > 0 , f' is increasing (and f is convex); if f" < 0 , f' is decreasing (and f is concave).
-600
-400
-200
0
200
400
600
800
1000
-10 -5 0 5 10 15 20 25
Profit
Quantity (Q)
Long Run Costs of Production
The Long Run ATC Curve(or the planning curve)
shows the least per unit cost at which any output can be produced after the firm has had time to make all appropriate adjustments in its plant size.
Cost
Quantity of output
SRATC1
At a relatively low output level, in the short run, the firm might have SRATC1 curve as its short run average cost curve.
Cost
Quantity of output
SRATC2
At a slightly higher output level, in the short run, the firm might have SRATC2 curve as its short run average cost curve.
Cost
Quantity of output
SRATC3
At a still higher output level, in the short run, the firm might have SRATC3 curve as its short run average cost curve.
Cost
Quantity of output
LRATC
SRATC1
SRATC2
SRATC3
SRATC4
SRATC5
In the long run, the firm can pick any appropriate plant size. At each output level, the firm picks the plant that has the SRATC curve with the lowest value.
Cost
Quantity of output
LRATC
SRATC1
SRATC2
SRATC3
SRATC4
SRATC5
So, the LRATC curve is made up of segments of the SRATC curves.
In many industries, the number of possible plant sizes is virtually unlimited.
Then the long-run ATC curve is made up of points of tangency of the theoretically unlimited number of short-run ATC curves.
Then the long run ATC curve is smooth.
Cost
Quantity of output
LRATCSRATC1
SRATC2
SRATC3
SRATC4
SRATC5
The downward-sloping section of the Long Run ATC curve reflects
Economies of Scale.
Economies of Scale: As plant size increases, there are factors which
lead to lower average costs of production.
Labor Specialization: Jobs can be subdivided and workers performing very specialized tasks can become very efficient at their jobs.
Managerial Specialization: Management can also specialize in a larger firm (in areas such as marketing, personnel, or finance).
Equipment that is technologically efficient but only effectively utilized with a large volume of production can be used.
The upward-sloping section of the Long Run ATC curve reflects
Diseconomies of Scale.
Diseconomies of Scale: As plant size increases, there are factors which
lead to higher average costs of production.
Expansion of the management hierarchy leads to problems of communication, coordination, and bureaucratic red tape, and the possibility that decisions will fail to mesh. (“The left hand doesn’t seem to know what the right hand is doing.”) The result is reduced efficiency.
In large facilities, workers may feel alienated and may shirk (not work as much as they should). Then additional supervision may be required and that adds to costs.
Sometimes there is a segment of the LR ATC curve which is horizontal.
In that section, the LR ATC is constant, & there are
Constant Returns to Scale.
Once we have the LR ATC, we can determine the LR total cost TC.
Remember that ATC = TC/Q.
So TC = (ATC) Q.
From the LR TC curve, we get the LR MC, either from
MC = ΔTC/ΔQ
or MC = dTC/dQ
As in the case of the short run MC & ATC, it is also true for the long run curves that
MC < ATC when ATC is decreasing,
MC > ATC when ATC is increasing, &
MC = ATC when ATC is at its minimum.
Furthermore, when the firm has built the optimal scale of plant for producing a given level of output,
long run MC & short run marginal cost will be equal at that output.
That is, the LR MC & SR MC will intersect at that output.
We can also determine the LR TC curve from the expansion path.
L
K
E3E2
E1
K3
K2
K1
O L1 L2 L3
The expansion path shows how the quantities of inputs change as output increases, but the prices of inputs remain fixed.
In particular, suppose that the price of labor is $10. The TC of producing output 50 at E1 is the same as the cost of any of the point on that isocost line.
L
K
E1
40
30
20
O 25 37 45
In particular, at point H1, where only labor is used, the cost is the price of labor times the amount of labor or (10)(25) = 250.
So (50, 250) will be one point on the LR TC curve.
50
100
150
H1
Similarly, the LR TC of output 100 is (10)(37) = 370.
L
K
E3
E2
E1
40
30
20
O 25 37 45
So, (100, 370) is another point on the LR TC curve.
The LR TC of output 150 is (10)(45) = 450
So, (150, 450) is a third point on our LR TC curve.
50
100
150
So our LR TC curve might look like this:
LR TC
Q
LR TC
Q LR TC
50 250
100 370
150 450
O 50 100 150
450370250
We’ve discussed economies & diseconomies of scale.When a firm produces more than one product, it may also experience economies or diseconomies of scope.
Economies of scope exist when a single firm producing multiple products jointly can produce them more cheaply than if each product was produced by a separate firm.
Economies of scope may occur because
1. Production of different products use common facilities or inputs.
Example: Automobile & truck production may use the same factory assembly line and raw materials.
2. Production of one product produces by-products that the producer can sell.
Example: A cattle producer raises cattle to sell for beef, but can also sell the hides.
A measure of economies of scope is
)QTC(Q
)QTC(Q– )TC(Q )TC(Q
21
2121
where TC(Q1) is the total cost of producing Q1 units of product 1 only, TC(Q2) is the total cost of producing Q2 units of product 2 only, & TC(Q1+Q2) is the total cost of producing them jointly.
This measure indicates the savings of joint production compared to separate production, as a percentage of joint production.
)QTC(Q
)QTC(Q– )TC(Q )TC(Q
21
2121
Example 1: The total cost of producing Q1 units of product 1 only is 50,000. The total cost of producing Q2 units of product 2 only is 90,000. The total cost of producing them jointly is 120,000. Determine if there are economies or diseconomies of scope, and measure them.
000,120
000,120000,90000,50 167.0
There are economies of scope, since joint production is less costly than the sum of the separate productions.
000,120
000,20
)QTC(Q
)QTC(Q– )TC(Q )TC(Q
21
2121
Example 2: The total cost of producing Q1 units of product 1 only is 50,000. The total cost of producing Q2 units of product 2 only is 90,000. The total cost of producing them jointly is 150,000. Determine if there are economies or diseconomies of scope, and measure them.
000,150
000,150000,90000,50 067.0
There are diseconomies of scope, since joint production is more costly than the sum of the separate productions.
000,150
000,10
)QTC(Q
)QTC(Q– )TC(Q )TC(Q
21
2121
Example 3: The total cost of producing Q1 units of product 1 only is 50,000. The total cost of producing Q2 units of product 2 only is 90,000. The total cost of producing them jointly is 140,000. Determine if there are economies or diseconomies of scope, and measure them.
000,140
000,140000,90000,50 0
There are neither economies nor diseconomies of scope, since joint production costs the same amount as the sum of the separate productions.
000,140
0
How do you determine the profit-maximizing output levels for a multi-product firm?
Set MR equal to MC for each product.
Two-Product Firm Example
A firm produces Q1 units of item 1 & Q2 units of item 2.TC = 30 Q1 + 30 Q2 – 4 Q1 Q2
MC1 = dTC/dQ1 = 30 – 4Q2
MC2 = dTC/dQ2 = 30 – 4Q1
Demand for product 1: P1 = 26 – 2Q1
MR1 = dTR1/dQ1 = d(P1Q1)/dQ1
= d(26Q1 – 2Q12)/dQ1
= 26 – 4Q1
Demand for product 2: P2 = 42 – 4Q2
MR2 = dTR2/dQ2 = d(P2Q2)/dQ2
= d(42Q2 – 4Q22)/dQ2
= 42 – 8Q2
Equate MR1 = 26 – 4Q1 to MC1 = 30 – 4Q2
& MR2 = 42 – 8Q2 to MC2 = 30 – 4Q1 .
26 – 4Q1 = 30 – 4Q2
4Q2 – 4Q1 = 4
Q2 – Q1 = 1
Q2 – 1 = Q1
42 – 8Q2 = 30 – 4Q1
12 = 8Q2 – 4Q1
3 = 2Q2 – Q1
Q1 = 2Q2 – 3
Setting the Q1 expressions equal to each other,
Q2 – 1 = 2Q2 – 3
2 = Q2
Q1 = Q2 – 1 = 2 – 1 = 1
So the profit-maximizing output levels are Q1 = 1 and Q2 = 2
From the demand functions, the prices are
P1 = 26 – 2Q1 = 26 – 2(1) = 24, and
P2 = 42 – 4Q2 = 42 – 4(2) = 34
TR = TR1 + TR2 = P1Q1 + P2Q2 = 24(1) + 34(2) = 24 + 68 = 92
TC = 30 Q1 + 30 Q2 – 4 Q1 Q2 = 30(1) + 30(2) – 4(1)(2) = 82
= TR – TC = 92 – 82 = 10
You probably recall from Microeconomic Principles that accounting profit and
economic profit differ.
The difference results from the fact that the accountant only includes explicit costs in TC, while the economist includes both explicit & implicit costs.
Implicit costs do not leave a paper trail. They are opportunity costs such as the foregone earnings of the owner, and foregone interest on money invested in the firm.
Because of the differences in the cost definitions, zero accounting profit & zero economic profit mean different things.
Zero accounting profit means that revenue is just sufficient to cover explicit costs.
Zero economic profit means that a business is doing no better or worse than the typical business.
It is making a normal accounting profit.
Firms may have objectives in addition to profit-maximization. These may include
• maintaining or increasing market share,• achieving better social conditions in the
community,• protecting the ecological environment, & • establishing an image as a good employer
and a valuable part of the community.
Often these additional goals contribute to long term profit maximization.
For example, a better image makes it possible to attract more productive employees and more customers.