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Unformatted text preview: Chapter 7 Firm Supply 7.1 Introduction In Chapter xxxx we studied the ﬁrms proﬁt maximization problem. We
obtained factor demands as the solution to the ﬁrm’s proﬁt maximization
problem. The proﬁt function told us the maximum attainable level of prof—
its for given factor prices and output price. Hotelling’s lemma told us that
the derivative of the proﬁt function with respect to the otput price was
equal to the ﬁrms optimal scale decision. The second order condition guar—
anteed that the scale decision was maximal and requred that the technolgy
be decreasing returns to scale. However, that the ﬁrm maximizes proﬁts
doesnot guarantee that these proﬁts are non—negative. If proﬁts were nega—
tive, then the ﬁrm would be better off not entering the industry. The ﬁrms
entry decision then hinges crucially on whether it obtains positive proﬁts
or not. In this chapter we will study the ﬁrm’s entry decision and we will derive
the ﬁrm’s supply function. We will see that the ﬁrm’s supply function is
its marginal cost curve above average cost. We will also study properties
of the supply function 7.2 The Firm’s Short Run Supply Function We have studied the ﬁrm’s short run cost minimization problem and we
have seen that it decides to set marginal revenue, which under competi—
tion is constant and equal to price, equal to marginal cost. The resulting 57 58 CHAPTER 7. FIRM SUPPLY short run supply function tells us the ﬁrm’s optimal quantity given cost
conditions in the factor market and the ﬁrm’s technology. Consider, then, the following short run proﬁt maximization problem
for the ﬁrm. The ﬁrm has already paid TE in ﬁxed capital costs which
we assume are sunk. For example, in the short run the ﬁrm has already
invested TE dollars and built a plant. The ﬁrm can not recover these costs
regardless of the amount of output it may choose to supply. Therefore, if
the ﬁrm chooses to supply zero units of output it will incur a loss equal
to the sunk cost fin this case, TE dollars. It seems reasonable to believe
that TE is the most the ﬁrm could lose since it always has the option of
producing zero units. Consider, then, the ﬁrm’s short run scale decision: max py — wlsr (E, y) — TE
y where we recall that the ﬁrm’s short run cost function is given by C' (11), T, E, y) 2 TE + M“ (E, y) . Recall that the ﬁrm is a price taker in goods markets, so
its supply decision will be heavily inﬂuenced by the unit price of the good.
In fact, we know that the ﬁrm’s optimality condition requires that it price
equal to marginal cost: _ dC' (w,T,E,y)
p — '
The ﬁrm will select a positive level of output as long as price exceeds
variable cost: _
M“ k,
p > #7
y since, under these conditions, producing a positive level of output will not
increase the ﬁrms losses beyond its sunk capital costs of TE dollars. In fact,
if the price the ﬁrm receives for a unit of output exceeds average variable
cost, then the ﬁrm will redeuce its losses or perhaps earn positive proﬁts.
since we are studying the short run and we assume that the number of
ﬁrms is ﬁxed, the ﬁrm keeps these proﬁts (under these conditions, in the
long run entry of new ﬁrms will drive proﬁts to zero). Fixed costs, therefore,
are important to the ﬁrm when it is decing whether or not to produce. Once
the ﬁrm determines that price covers average variable costs, ﬁxed costs do
not inﬂuence the ﬁrm’s optimal output choice. In the short run, then, the
ﬁrm may ﬁnd it optimal to produce a positive level of output despite the 7. 3. PROPERTIES OF THE SUPPLY FUNCTION 59 fact that it may receive a negative proﬁt. The reason is that by doing so,
the ﬁrm may be able to partially offset its sunk capital costs. If we look at Figure XXXX, we know that the marginal cost crosses aver—
age variable cost at its minimum point.The ﬁrms supply curve consists of
two portions. When price is below the shutdown price, p0 in the diagram,
the ﬁrm produces zero units of output and the supply function is a verti—
cal segment along the y—aXis. Once the price reaches p0, the ﬁrm ﬁnds it
proﬁtable to produce a positive amount of output and chooses its quantity
by equating marginal cost to price. The ﬁrm’s supply curve is then the
marginal cost curve above minimum average cost. Insert Figure XXXX. From Figure XXXX, we can obtain the ﬁrms total revenue and variable
costs. Insert Figure XXXX. To be completed. 7.3 Properties of the Supply Function Let us now look at the properties of the supply function.
Property I: Homogeneous of degree 0 in w, r, and p. if (01127073019) = if (w, 7“, p) This essentially follows from the homogeneity of degree 0 of the factor
demand functions. Property II: Supply functions must have a non—negative slope. dy* (mm?) > 0
dp _ We can decompose the supply function into a scale effect. To be completed. 60 CHAPTER 7. FIRM SUPPLY 7.4 The Firm’s Long Run Supply Function As we have previously discussed, we know that in the long run the ﬁrm is
able to vary its inputs. For the proﬁt maximizing competitive ﬁrm, this
means that capital costs are no longer sunk since in the long run the ﬁrm
can select the size of its plant. The ﬁrm’s long run proﬁt maximization
problem is: maXpy— CL(w,r,y)
y and the optimality condition requires that the ﬁrm equate marginal revenue
*price, in this case* to long run marginal cost: _ dCL (waray)
_ dy In the long run, the ﬁrm has no sunk costs and, therefore, will choose a
non—negative level of output only if it is guaranteed non—negative proﬁts.
The ﬁrm is guaranteed non—negative proﬁts if price exceed long run average
total costs: 17 W CL<w.r.y)
y The ﬁrm’s long run supply curve is, then, its marginal cost curve above long
run average cost. The ﬁrms long run supply curve is displayed in Figure
XXXX. Insert Figure XXXX. Consider Figure XXXX. At a price such as p* the ﬁrm is able to cover its
average costs and therefore chooses to supply a positive quantity. However,
there are two quantities of y that satisfy the price equal marginal cost
condition. They are labelled y1 and y2. we have asserted in the preceding
that the ﬁrm’s supply curve is its marginal cost curve above average costs.
That implicitly selects y2 as the optimal quantity and not yl. If we recall
the second order condition requires that marginal costs be increases at the
optimum. Quantity y2 satisﬁes this requirement. At quantity y1 the ﬁrm
is producing where marginal costs are decreasing and therefore violates the
second order condition. Quantity yl minimizes the ﬁrm’s proﬁt and so the
ﬁrm would never ﬁnd it optimal to produce yl units of output. To be completed. 7. 5. APPLICATIONS 7.5 Applications To be completed. 7.6 Exercises 61 ...
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This note was uploaded on 01/11/2012 for the course ECON 200 taught by Professor Riley during the Fall '10 term at UCLA.
 Fall '10
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