Download Calculus Supplement: Chapter 10

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CHAPTER 10
The Price-Taking Firm
10.1 Conditional Factor Demands in the Long Run
In the previous chapter of this supplement, we considered the general problem of how a firm
that is a price taker in the factor markets chooses its cost-minimizing method of production.
Although we did not mention it at the time, the quantities of the factors used in the
cost-minimizing method of production are the “conditional” factor demands for those factors.
This idea is most easily illustrated through an example. From the previous chapter, recall
Dutch Dynamism, which uses capital, K, and labour, L, to produce output. Dutch
Dynamism’s production function is L ⁄ K ⁄ . Dutch Dynamism is a price taker in the factor
markets: It pays w per unit of labour and r per unit of capital. In the last chapter, we derived
the combination of factors that minimized the cost of producing x amount of output. The
quantity of labour is
12
12
L(w, r, x) = w−1⁄2r1⁄2x,
and the quantity of capital is
K(r, w, x) = w1⁄2r−1⁄2x.
Note that we have written these conditional factor demands as functions of the factor
prices and the amount of output to be produced, x. We do this to emphasize that the
quantities of the factors demanded depend on the values of these variables. In particular, we
have related the amount of labour the firm wishes to employ to its price; and we have
related the amount of capital the firm wishes to employ to its price. A relation between the
amount demanded of something and its price is a demand curve; so the above expressions
are the conditional factor demand curves for labour and capital, respectively.
Conditional factor demand curves always slope downward.1 Differentiating our expression
for the quantity of labour demanded, we can demonstrate this result for the labour demand
curve:
∂L
∂w
1
= − w−3⁄2r1⁄2x < 0.
2
Since the derivative is negative, the quantity of labour demanded falls as the wage rises.
Similarly, differentiating our expression for the quantity of capital demanded, we can also
demonstrate this result for the capital demand curve:
∂K
1
= − w1⁄2r−3⁄2x < 0.
∂r
2
We use the term conditional factor demand to differentiate these demand schedules from
the factor demand schedules considered in the text (much like we differentiated between
compensated demand and (uncompensated) demand in our analysis of the consumer). The
difference arises, because with the conditional factor demand, there is no output effect—x is
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CHAPTER 10: THE PRICE-TAKING FIRM
fixed—and only the factor substitution effect is present. Recall, from the text, that the
(unconditional) factor demand is affected by both the output effect and the factor substitution
effect. We will explore this distinction further in the next section.
10.2 Factor Demand and Profit Maximization for the Price-Taking Firm
Consider a firm that employs N different factors and is a price taker in both the output
market and the factor market. Let yn denote the quantity of the nth factor and pn its price.
Let P denote the price the firm receives for its output. The firm’s output from employing the
factor combination (y1, …, yN) is given by the production function F(y1, …, yN). Our
objectives are (1) to find the profit-maximizing amount of each factor to employ and (2) to
show the relation between this and the factor demands we derived in the previous section.
If the firm employs the factor combination (y1, …, yN), its profit, will be the revenue from
the output produced minus the sum of the expenditures on the factors:
P × F(y1, . . . , yN) − p1y1 − . . . − pNyN .
Although this expression is a function of many variables (i.e., y1, …, yN), maximizing it is
similar to maximizing a function of one variable. In particular, the necessary conditions are
analogous, the principle difference being that, here, we set the derivatives with respect to
each of the factors equal to 0. Thus, the factor combination (y1, …, yN) that maximizes profit
is the solution to the N equations2
P×
∂F
∂y1
− p1 = 0
through(10.2.1)
P×
∂F
∂yN
− pN = 0.
Recalling that )F/)yn is more conventionally written as MPPn (marginal physical product of
the nth factor) and rearranging Equations (10.2.1), we have
P × MPP1 = p1
through(10.2.2)
For a price taker in the output market, the output price, P, equals its marginal revenue,
MR. For a price taker in the factor markets, the price of the factor (e.g., pn) equals that
factor’s marginal factor cost, MFC. So Equations (10.2.2) are consistent with the rule that a
factor, say the nth, is employed up to the point at which its marginal revenue product, MR ×
MPPn, equals its marginal factor cost.
Consider any two factors, say the ith and jth. From Equations (10.2.2), we have
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MATHEMATICAL SUPPLEMENT TO ACCOMPANY MICROECONOMICS
Pi
MPPi
=P=
pj
MPPj
.
Rearranging that expression, we have
MPPj
MPPi
=
pj
pi
.
But this is just the familiar tangency condition for cost minimization! So we have shown
that the profit-maximizing factor combination is also the cost-minimizing factor combination
for producing the profit-maximizing level of output.
To illustrate these ideas, suppose widgets are made with labour and capital according to
the production function F(L,K) = L1/2 + K1/2. Suppose the price of labour is €1, the price of
capital is €2, and the price of a widget is €8. Equations (10.2.2) for this problem are
8×
1
2
× L−1⁄2 = 1
and
8×
1
2
× K−1⁄2 = 2.
Solving, we have L = 16 and K = 4. The profit-maximizing level of output is, therefore,
161/2 + 41/2 = 6 widgets. The firm’s profit is (€8 × 6) – (€1 × 16) – (€2 × 4) = €24.
We can also use this example to explore the difference between conditional factor
demand and factor demand, thereby illustrating the importance of the output effect. Suppose,
now, that the price of labour is w, the price of capital is r, and the price of a widget is P.
Equations (10.2.2) for this problem are now
P×
1
2
× L−1⁄2 = w
and
P×
1
2
× K−1⁄2 = r.
Solving, we have
L=
S D
P
2w
2
and K =
SD
P
2
(10.2.3)
2r
Note that both factor demands are downward sloping, as we knew they must be (recall
footnote 2 on p. 328 of the text).
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CHAPTER 10: THE PRICE-TAKING FIRM
To see the difference that the output effect makes, we will now calculate the conditional
factor demands. We will do so using the Lagrange Method from Chapter 8 of this
supplement. The Lagrangean is
L = wL + rK + μ[x − (L1⁄2 + K1⁄2)] ,
where, recall, the Lagrange multiplier, m, is marginal cost. Differentiating the Lagrangean
with respect to L, K, and m and setting the derivatives equal to 0 yields
1
× L−1⁄2 = 0,
2
1
∂L ⁄ ∂K = r − μ × × K−1⁄2 = 0,
2
∂L ⁄ ∂L = w − μ ×
and
∂L ⁄ ∂μ = x − L−1⁄2 − K1⁄2 = 0.
Solving these equations for the unknowns (L, K, and m), we obtain
μ=
L=
2wrx
w + r1
rx
S D
w+r
2
(10.2.4)
,
and
K=
S D
wx
w+r
2
.
The last two equations are the conditional factor demands for labour and capital,
respectively.
At the profit-maximizing factor combination, the factor demands and the conditional factor
demands are equal. To see this, remember that a price taker produces up to the level where
price, P, equals marginal cost. From the Lagrange Method, marginal cost equals m. So P =
m, or, from Equations (10.2.4).
P=
2wrx
(10.2.5)
w+r
Substituting that expression for P into Equation (10.2.3), we get
L=
S D
rx
2
w+r
and K =
S D
wx
2
w+r
.
These are the conditional factor demands found above.
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MATHEMATICAL SUPPLEMENT TO ACCOMPANY MICROECONOMICS
Although the factor demands and the conditional factor demands are equal at the
profit-maximizing factor combination, their slopes are different at that combination. To see
this, differentiate the factor demand for labour with respect to the price of labour, w:
dL
dw
=
P2
2w
=−
3
2r2x2
w(w + r)2
The second equality follows substituting Equation (10.2.5) for P. Differentiating the
conditional factor demand for labour with respect to w,
dLc
dw
=
2r2x2
(w + r)3
,
Where the superscript c indicates that this is conditional factor demand. Since w < w + r,
it follows that
dLc
dw
>
dL
dw
.
That is, the factor demand curve is more steeply downward sloping than the conditional
factor demand curve. Here, there is a negative output effect: When the price of labour rises,
the firm reduces the amount of labour it employs, both because it is substituting out of
labour—the factor substitution effect, dLc/dw—and because it is reducing the amount of
output it produces—the negative output effect.
10.3 Exercises
10.3.1 Verify for the widget manufacturer considered in Section 10.2 that L = 16 and K = 4
is really the cost-minimizing factor combination for producing 6 widgets. That is, solve the
cost minimization problem directly and check that you obtain L = 16 and K = 4.
10.3.2 Consider Dutch Dynamism again. What happens to its conditional factor demand
for labour as the price of capital rises, all else being equal? Briefly explain why you should
have expected this answer given that only two factors are used in production.
10.3.3 Consider the widget manufacturer again. What happens to its conditional factor
demand for labour as the price of capital rises, all else being equal? What happens to its
factor demand for labour as the price of capital rises, all else being equal? Explain the
differences that you find.
1
You can see that this must be true by the fact that cost minimization subject to obtaining
an output target is really the same problem as expenditure minimization subject to
obtaining a utility target (which was dealt with in Chapter 4 of this supplement). The
consumption bundle that solves this latter problem consists of the compensated demands,
and compensated demand curves, recall, always slope downward.
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CHAPTER 10: THE PRICE-TAKING FIRM
2
To be precise, we have only shown these equations to be a necessary condition. The
mathematics required to show that these equations are also a sufficient condition lie
outside the scope of this supplement.
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