Economics(assignment)
jackson muchoki
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Lecture 09: Cost Curves Perlo¤ Chapter 7
Vladimir Petkov
VUW
06 April 2016
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 1 / 24
Short Run Costs
Remember that in the short run some inputs are �xed.
To produce a given level of output, a �rm will incur costs for both its �xed and variable inputs.
A �xed cost (F) is the expense that cannot be adjusted in the short run, since the input that generates this cost is �xed.
A variable cost (VC) is the expense that changes with the amount of output produced.
Note: in the long run, all costs are variable.
A �rm�s total cost (C) is the sum of the variable cost and the �xed cost:
C = VC + F.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 2 / 24
Short Run Costs: Cobb-Douglas Example
Suppose that the �rm�s production function is given by
f (K, L) = Kα L1�α.
In the short run, capital is �xed at K = K̄. Therefore, the labor needed to produce q̄ units of output satis�es
q̄ = K̄α L1�α.
Solving for L yields
L = �
q̄ K̄α
� 1 1�α .
Thus, the short-run total cost is
C = wL + rK̄ = w �
q̄ K̄α
� 1 1�α + rK̄.
The �xed cost is rK̄ and the variable cost is w �
q̄ K̄α
� 1 1�α .
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 3 / 24
The Short Run Expansion Path
The short-run decision about factor proportions is not done by the tangency optimization routine. If there are only two inputs and one of them is �xed, the other is uniquely determined by the target output level. Input prices are irrelevant for the short-run choice of L.
L
K
K
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 4 / 24
Obtaining The Short Run Cost Curve
We can drop the isocost line and just map the cost of hiring labor. If MPL is diminishing, the graph is as follows:
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 5 / 24
Cost Measures
Marginal Cost. A �rm�s marginal cost (MC) is the amount by which the total cost changes if the �rm produces one more unit of output:
MC = dC(q)
dq .
Remember that C(q) = F + VC(q). Since F does not change with q, it is also true that MC = dVC(q)/dq. Average cost. A �rm�s average cost (AC) is the total cost per unit of output:
AC = C(q)
q .
Note that AC = C(q)q = F+VC(q)
q = F q +
VC(q) q = AFC + AVC.
AFC = F/q is the average �xed cost; AVC = VC(q)/q is the average variable cost.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 6 / 24
Cost Measures: Example
Suppose that a �rm has a variable cost and a �xed cost given by
VC(q) = 100q � 4q2 + 0.2q3, F = 450, respectively. The marginal cost of this �rm is
MC = VC0(q) = 100 � 8q + 0.6q2. The average �xed cost is
AFC = F/q = 450/q.
The average variable cost is
AVC = VC(q)
q =
100q � 4q2 + 0.2q3 q
= 100 � 4q + 0.2q2.
The average total cost is
AC = AVC + AFC = 100 � 4q + 0.2q2 + 450/q. Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 7 / 24
Short-Run Cost Curves
Typical short-run total cost and the variable cost curves are shown in the �gure below. The vertical distance between the two curves is the �xed cost.
C = F + VC
q
C, dollars per year
F
C
VC
Kr
=F
=F
=F
=F
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 8 / 24
Short-Run Cost Curves (Continued)
Now let us draw the average cost and marginal cost curves.
Graphically, the average cost at a given output level is the slope of the ray from the origin that goes through the C curve. The marginal cost is given by the slope of the C curve. The AVC, the AC and the MC curves are usually U-shaped. The AF curve is monotonically decreasing: the �xed cost is spread over more and more units of output.
The vertical distance between the AC and the AVC curves is the average �xed cost. So these two curves are getting closer and closer.
Note that the MC curve intersects the AC and the AVC curves at their minimum!
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 9 / 24
Short-Run Cost Curves (Continued)
q
q
Cost
Cost
VC
MC
AVC
q1 q2
F
AFC
AC
C
A
A
B
B
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 10 / 24
Short-Run Cost Curves (Continued)
Now we investigate why the MC curve and the AC curves are U-shaped. The short-run cost is C(q) = wL(q)+ rK̄. Remember that MPL = dq/dL. Thus, dL/dq = 1/MPL. Thus, the marginal cost is
dC(q) dq
= w dL dq =
w MPL
.
On the other hand, the average variable cost is
AVC = VC
q =
wL q =
w APL
.
If the law of diminishing returns holds, MPL and APL will eventually start declining. Thus, MC and AVC will start increasing. Since AFC becomes smaller, this means that AC will start increasing.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 11 / 24
Short-Run Cost Curves (Continued)
We now show that the MC curve intersects the AC curve at the minimum point of AC. The output that minimizes average cost solves
dAC dq
= d �
C(q) q
� dq
= 0.
Di¤erentiate using the quotient rule:� C(q)
q
�0 =
C0(q)q � C(q) q2
= 0.
This condition can be rewritten as
C0(q) = C(q)
q .
In other words, at the output level which minimizes AC, we will have AC = MC.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 12 / 24
E¤ects of Taxes On The Cost Curves
Suppose that the government imposes a per-unit tax of $10.
Then the AC and the MC curves will shift up by 10.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 13 / 24
E¤ects of Taxes On The Cost Curves (Continued)
Now consider a lump-sum tax: a one-o¤ payment to the government. Lump-sum taxes are like �xed costs. They will not a¤ect the AVC and MC curves.
q
q
Cost
Cost
VC
MC
AVC
q1 q2
F
AFC
AC
C
A
A
B
B AFC’
AC’
F’
C’
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 14 / 24
Changing Input Prices
An increase in the cost of capital r increases F. So we get the same picture as on the previous slide.
If the labor cost goes up, then the MC and the AVC curves shift up.
q
Cost MC
AVC
AVC’
MC’
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 15 / 24
Long-Run Cost Curves
Now suppose that all inputs are variable. Last time we discussed the long-run expansion path.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 16 / 24
Long-Run Cost Curves (Continued)
From this expansion path, we derive the long-run total cost curve. No �xed costs in the long run: the C curve starts from the origin!
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 17 / 24
Long-Run Marginal and Average Cost Curves
Just as with short-run costs, we can also de�ne long-run marginal cost (LRMC) and long-run average cost (LRAC).
LRMS = dC(q)
dq , LRAC =
C(q) q
Q
Cost, $ per unit
LRAC(Q)
LRMC(Q)
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 18 / 24
The Shape of The Long-Run Cost Curves
Usually LRMC and LRAC are also U-shaped. But this is due to the properties of the production functions, not to the �xed cost.
A cost function exhibits economies of scale if AC falls with output. If AC rises with output, then we have diseconomies of scale. Increasing returns to scale in the production function are a su¢ cient but not necessary condition for economies of scale.
Usually up to some output level we have economies of scale, but after that the average cost begins to increase.
Hence the U-shaped curves.
The output which minimizes LRAC is called the minimum e¢ cient scale.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 19 / 24
Changing Input Prices
If the cost of an input increases, the total cost curve rotates outward. Thus, the LRAC and the LRMC will shift up.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 20 / 24
Relationship Between Long-Run and Short-Run Costs
Suppose that in the short run K̄ is �xed at the long-run optimum K�. Then CSR = CLR. For all other K̄, we will have CSR > CLR.
L
Q
K
C
A B K2 K1
L1 L2
Q=1 million
Q=2 million
1m
A
LRTC(Q)
C
L3
STC(Q) when K=K1
1rK
2m
B
C
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 21 / 24
Relationship Between Long-Run and Short-Run Costs (Continued)
The long-run cost curves are the envelope curves of the short-run cost curves. Note that these tangencies are typically not at min SRAC. The points of tangency are where K̄ = K�.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 22 / 24
Relationship Between Long-Run and Short-Run Costs (Continued)
If we have CRS, then the long-run cost function is linear. So AC = MC = const.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 23 / 24
Learning By Doing
Learning by doing: average cost decreases with past output. LBD is like dynamic economies of scale.
Vladimir Petkov (VUW) Lecture 09: Cost Curves 06 April 2016 24 / 24
