Download bYTEBoss 13. Competitive markets 1

Competitive Markets 1
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1. Aggregate demand, aggregate supply, and competitive equilibrium
Holding each individual’s income and the price of other goods constant, we
can suppress these variables and write each individual’s demand for a
good as a function of the price of that good alone. If p is the price of the
good, then we can write the first consumer’s demand as d1(p), the
second consumer’s demand as d2 (p), the third consumer’s demand as
d3(p), etc. At each price, p, the aggregate demand for the good is the
sum of all the individual demands,
d1(p) + d2(p) + d3(p) + ... = D(p).
Under certain circumstances, aggregate supply is derived in exactly the
same manner. Suppose we are in the short-run, with N active firms,
labeled i = 1, 2, 3, ..., N. If the prices of all the inputs are constant, then
we can determine the short-run supply function for each firm (given the
level of its fixed factors) as in the previous section. Let si (p) be the
supply function for firm i, i = 1, 2, 3, ..., N. Then aggregate short-run
supply is
s1 (p) + s2 (p) + ... + sN (p) = S(p).
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If D(p) and S(p) are the aggregate short-run demand and supply functions,
then p* is a short-run competitive equilibrium price if
D(p*)= S(p*),
i.e., demand equals supply at price p*.
The equilibrium amount traded is
Q* = D(p*) =S(p*).
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In the long-run, the derivation of aggregate supply must also take into
account the possibility of entry and exit by the firms (i.e., the number of
active firms, N, is not fixed).
If AC(y) is the long-run average cost function for a firm, then the firm would
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want to enter the market whenever it expected the market price to
exceed the minimum of long-run average cost for an extended period
(since the firm could make more than the normal rate of return),
• want to exit the market whenever it expected the market price to be less
than the minimum of long-run average cost for an extended period
(since whatever its production level, the firm would make less than the
normal rate of return),
• be indifferent about being in the market whenever it expected the market
price to match the minimum of long-run average cost for an extended
period (since producing at an output that achieved the minimum of longrun average cost, the firm could make exactly the normal rate of return).
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If all firms have access to the same technology, then their entry/exit
incentives would be identical.
At any price above the minimum of long-run average cost, all firms would
want to enter, but since there is no limit to the number of potential firms,
aggregate supply would be “unbounded.” This will not be a feasible
market outcome.
At any price below the minimum of long-run average cost, all firms would
want to exit, and aggregate supply would be zero. This would be a longrun equilibrium if long-run aggregate demand were also zero at such a
price.
When the market price equals the minimum of long-run average cost, all
firms are indifferent about being in the market, and aggregate supply can
take any desired value by adjusting the number of active firms in the
market. This is the only possible long-run equilibrium price at which the
output could be non-zero. In this case the equilibrium price is
determined entirely by the supply side. It is the minimum of long-run
average cost. The quantity traded is determined by the demand side. It
is the aggregate demand at a price equal to the minimum of long-run
average cost.
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One additional complication might arise. As the aggregate production level
increases by large amounts, the increased usage of inputs might lead to
a change in the price of some inputs. We will discuss the case in which
the price of one input increases as its usage increases. [We do not
mean that an individual firm’s changes in use of the input affect its price
but rather that large changes in the aggregate usage of the input affect
its price. Individual firms still act as price takers.] Now the discussion of
the previous paragraph remains correct except that we must be sure we
are using the correct price for the input.
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Each level of aggregate usage of the input, Z, determines a price for the
input, w(Z). Given the price for the input, along with the prices of all
other inputs, we have a long-run cost minimization problem for the firm.
Solving this problem, we determine the long-run cost function for a firm, and
thus the long-run average cost function. The minimum of this long-run
average cost function depends on the input prices, in particular w(Z),
and thus on Z.
Call the minimum long-run average cost p(Z), let y(Z) be the output level
that achieves the minimum long-run average cost for a firm, and let z
(lower case Z) denote the amount of the input used by a firm to produce
the minimum-average-cost output level.
If N = Z/z firms are active and produce the minimum-average-cost output
level, then the aggregate output is Y(Z) = Ny(Z) and the aggregate
usage of the input is Nz = Z, which is consistent with our starting value.
This process has determined one point on the long-run aggregate supply
curve: when price is p(Z), the corresponding aggregate supply is Y(Z).
Repeating this process for different Z values, we trace out the long-run
aggregate supply curve.
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When w(Z) is increasing in Z, this long-run aggregate supply, S(p), will be
upward sloping, not horizontal.
Why is this true?
Why can’t it be downward sloping or horizontal?
Think through an argument to convince yourself.
Here p* is an equilibrium price if D(p*) =S(p*). The equilibrium amount
traded is
Q* = D(p*) = S(p*).
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Example:
Consider firms with identical cost function c(q) = 90 + 20q +
(0.1)q2 for q > 0 and c(0) = 0, where q is the firm’s output. [Note this is
the same cost function as in Example 1 of Section 2c in the “chapter”
Supply by a Single Firm. The results there are used in parts a and b
below.] Suppose aggregate demand is D(p) = 800 – 10p.
a. What is the supply function for a single firm?
Supply is s(p) = 0 for p < 26; s(26) is 0 and 30; and
s(p) = 5p - 100 for p > 26.
b. For a general price, p, compute the value of producer’s surplus for a
single firm.
For p ≤ 26, 0 is an optimal output, and producer’s surplus is 0. For p > 26,
producer’s surplus is 910 -100p + (2.5)p2.
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c. If there are 10 firms, what are the competitive equilibrium price and
quantity? What is each firm’s producer’s surplus?
The aggregate supply function is the sum of the supply functions for the individual
firms. At price p = 26, each firm was indifferent between two different outputs, so
the sum must take account of all the different possible combinations of optimal
choices. Since the firms are identical, for prices other than p = 26, S(p) = 10s(p).
Thus aggregate supply is
S(p) 0 if p<26
S(p) = 0, or 30, or 60, or ..., or 270, or 300 if p = 26.
S(p) 50p-1000 if p> 26
At p < 26, S(p) = 0 while D(p) > 0, so the equilibrium price must be at least 26.
At p = 26, S(26) is no more than 300 while D(26) = 540> 300. Thus the equilibrium
price must exceed 26.
At p> 26, S(p) = 50p - 1000 = 800 - 10p = D(p) at equilibrium price p*= 30> 26. The
corresponding aggregate quantity is Q*= D(p*)= 800- 10(30) = 500. Each firm
produces s(30)= 50 units of output, and, using the results from (b), obtains
producer’s surplus 910- 100(30) +(2.5)(30)2 = 160. Note producer’s surplus is
the same as profit since there are no sunk costs.
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d. In the long run, firms are free to enter or leave the industry. What is the
long-run competitive equilibrium price and quantity? How many firms
operate in the industry, and what is each firm’s producer’s surplus?
At p < 26, s(p) = 0 for each firm, so with free entry and exit, aggregate supply is
S(p) = 0 while D(p) > 0, and the equilibrium price must be at least 26.
At p > 26, s(p) > 0 and each active firm obtains positive profit. Since firms in the
industry earn more than the normal rate of return, other investors will want to
open firms in this industry (since it pays a better return than their other possible
investments). Thus the aggregate output at price p > 26 would be unbounded
(i.e., for any Q, firms would continue to enter until output exceeded Q). In
particular, supply would exceed D(p) for any p > 26.
The only possible equilibrium price is p = 26, at which firms earn the normal rate of
return (0 economic profit), and are indifferent between outputs 0 and 30. Since
D(26) = 800- 10(26) = 540, there must be just the right number of firms, N,
producing output 30 so that 30N =540.
Thus in the long-run equilibrium, price is 26, aggregate quantity is 540, and
N = 540/30 = 18 firms each produce output 30.
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Each firm’s producer’s surplus is 0. This can be found using the results of
(b), from which producer’s surplus is 910- 100(26) + (2.5)(26)2 =0.
The result could also have been determined by a general principle: since
there is an unlimited number of identical potential firms and only 18
produce in equilibrium, those that produce must have the same surplus
as those that do not produce. Those that do not produce get 0 surplus
(the normal rate of return), so the producers must get 0 surplus as well.
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e. Suppose the “90” in each firm’s cost function is the salary that must be
paid to get a competent manager to work for the firm and run its
production process. [Each of the many potential managers has alternative
job opportunities that pay salary 90 per period.]
Five new potential managers just graduated from a special engineering and
management joint program in which they learned how to reorganize the
production process for a firm in this industry. Any firm with one of these
managers will have “14q + (0.1)q2” instead of “20q + (0.1)q2” in its cost
function, and all the firms know this.
Unfortunately the information is industry specific, so the managers have the
same alternative job opportunities that pay salary 90 per period as all the
other potential managers. Taking account of the competition to hire
managers, what is the new long-run equilibrium in the industry? Do these
new managers work in this industry, and so, what are their salaries and
how do their firms’ equilibrium outputs and levels of producer’s surplus
compare to those of other firms in the industry?
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If these new managers bring any advantage to a firm, then the firms will
compete to hire the managers, driving the salary for the special
managers up until the firms that hire them are back to the normal rate of
return, or producer’s surplus 0.
[What is special is the managerial ability of these 5 managers, not any
particular firm. Thus the return to the firms that hire a special manager
must be no better than for a firm that does not hire one of the special
managers. The 5 new managers are identical to each other, but are not
identical to typical managers. Thus in equilibrium the 5 will all get the
same salary, but that salary can be higher than the 90 per period paid to
a typical manager.]
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To determine the advantage one of these managers brings to a firm,
suppose one of the special managers could be hired for 90 per period.
Then the firm would have cost function
90 + 14q + (0.1)q2, and its supply function would be
0
s*(p)= {
if p<20
0,or30 if p = 20.
5p-70 if p> 20
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At p = 26, such a firm would produce s*(26) = 60 and obtain profit
(26)(60) - [90 + (14)(60) + (0.1)(60)2] =270.
Note that at price 26, 5 such firms produce output 300 < D(26), so all 5
managers can work in this industry in equilibrium. As the firms compete
to hire these special managers, the special managers’ salaries are bid
up to 90 + 270 = 360. A firm with one of the special managers at salary
360 per period has cost function
360 + 14q + (0.1)q2,
and its supply function would be
0 if p<26
s**(p)= { 0, or 60 if p = 26.
5p-70 if p>26
The increased salary for the special manager has raised the firm’s shutdown price back up to 26, the same as for a firm with a regular manager
at salary 90 per period.
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The equilibrium price remains 26, with aggregate quantity
D(26) = 540 as originally.
At price 26, since the firms with special managers produce 60 rather than
the 30 produced by firms with regular managers, in the new equilibrium
there will be only 13 firms, 5 with special managers and 8 with regular
managers.
All firms obtain producer’s surplus 0. Even though firms are indifferent
between hiring regular and special managers [the reduced production
cost is exactly counterbalanced by the increased salary], all the special
managers must be employed in this industry since they strictly prefer it.
Special managers get salary 360 per period in this industry as opposed to
only 90 per period at their alternative jobs. Regular managers still get
salary 90 per period.
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2. Equilibrium total surplus and the efficiency of equilibrium
[Throughout this section, assume all consumers have quasi-linear utility
functions (zero income effects), with the good we are considering here
the non-linear one.]
Given aggregate demand, D(p), aggregate supply, S(p), and the equilibrium
price, p*, and quantity, Q*, we can determine the aggregate consumers’
surplus and producers’ surplus. As long as there are no externalities, no
other individuals or firms are affected (outside market effects).
Consumers’ surplus is the area above the equilibrium price, p*, below
aggregate demand, D(p), to the right of the vertical axis and out to the
equilibrium quantity, Q*.
Producers’ surplus is the area below the equilibrium price, p*, above
aggregate supply, S(p), to the right of the vertical axis and out to the
equilibrium quantity, Q*.
The total surplus is the area between the demand and supply curves, to the
right of the vertical axis and out to the quantity traded, Q*.
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Note that if fewer units had been traded, the aggregate surplus would have
been smaller (consumers have a higher value for the units out to Q*
than the cost to produce them).
Similarly, if more units had been traded, the aggregate surplus would have
been smaller (consumers have a lower value for the units beyond Q*
than the cost to produce them).
Thus the competitive equilibrium output is efficient in the sense that there is
no way to change the amount produced, who gets the units produced,
and who gets the money, so that everyone is better off. If someone is
made better off, someone else must be made worse off.
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The figures above indicate the total surplus and its distribution between
consumers and firms for the upward sloping supply and the horizontal
supply cases. With horizontal supply, there is no producers’ surplus.
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What happens to surplus in the long-run when the industry has increasing
costs? The figure below shows the situation. It looks just like the first
figure except that what was labeled producers’ surplus there has been
labeled ??? here.
In long-run equilibrium with identical firms, all firms must be getting the
normal rate of return (otherwise firms would want to enter or leave the
market). Thus this area cannot represent surplus for the producers in
this market. What is it?
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With a little work it can be shown that this area represents extra surplus for
the suppliers of the input whose price increases. For example, that input
might be a particular type of skilled labor. In order to attract more of this
labor, the corresponding wage must be increased. When the wage is
increased, a worker who would have been willing to work for $12 per
hour might receive $15 per hour. The increased wage was necessary to
attract other workers, but this worker is made better off by the wage
increase ($3 per hour) times the number of hours worked (say 40 hours
per week).
Thus total surplus is still represented by the area between the demand and
supply curves, to the right of the vertical axis and out to the quantity
traded, Q*. The “mystery” triangle is not producers’ surplus for the firms
in this industry, but it is surplus for factors on the production side of the
market. [See the example in the previous section, where the special
managers obtain a surplus of 270 per period in addition to the normal
return of 90 per period.]
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3. Long-run versus short-run equilibrium
A short-run equilibrium need not be a long-run equilibrium since the
number of firms and some inputs are fixed in the short-run. In the longrun, firms may want to change the levels of the previously fixed factors
or to exit, or other firms may want to enter the market.
A long-run equilibrium must be a short-run equilibrium. Given the long-run
equilibrium price, every firm must have made an appropriate entry/exit
decision, and each firm must have chosen an appropriate “plant size”
(i.e., level of all factors to be fixed in the short-run). Because of free
entry, at a long-run equilibrium with identical firms, each active firm has
no other choice than to produce an output that minimizes its long-run
average cost. The “plant size” it builds must be optimal for producing
the minimum-long-run-average-cost output, so the firm’s short-run
average total cost must also be minimized. Thus it produces where
short-run average total cost equals short-run marginal cost equals
long-run average cost equals long-run marginal cost. The firm does not
“want to be in that position,” it is forced to be.
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Example: All firms have the same cost function, with an avoidable fixed
cost. For zero output, c(0) = 0, c(q) = 9 + 14q + q2 for q > 0, where q is
the firm’s output. This cost function applies in both the short and long
run. [Note this is the same cost function as in Problem 2 of Section 3 in
the “chapter” Supply by a Single Firm. The results there are used in part
a below.]
a. What is the supply function for a single firm?
Supply is
s(p) = 0 for p < 20;
s(20) is 0 and 3; and
s(p) = (p/2) - 7 for p > 20.
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Example: All firms have the same cost function, with an avoidable fixed
cost. For zero output, c(0) = 0, c(q) = 9 + 14q + q2 for q > 0, where q is
the firm’s output. This cost function applies in both the short and long
run. [Note this is the same cost function as in Problem 2 of Section 3 in
the “chapter” Supply by a Single Firm. The results there are used in part
a below.]
b. Suppose there are 10 identical firms. What is the aggregate supply
function?
The aggregate supply function is the sum of the supply functions for the
individual firms. At price p = 20, each firm was indifferent between two
different outputs, so the sum must take account of all the different
possible combinations of optimal choices. Since the firms are identical,
for prices other than p = 20, S(p) = 10s(p). Thus aggregate supply is
0 if p<20
S(p)={ 0, or 3, or 6, or 9, or..., or 27, or 30 if p = 20.
5p-70 if p>20
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Example: All firms have the same cost function, with an avoidable fixed
cost. For zero output, c(0) = 0, c(q) = 9 + 14q + q2 for q > 0, where q is
the firm’s output. This cost function applies in both the short and long
run. [Note this is the same cost function as in Problem 2 of Section 3 in
the “chapter” Supply by a Single Firm. The results there are used in
part a below.]
c. If aggregate demand is D(p) = 112 - 2p, what are the competitive
equilibrium price, aggregate output, output per firm, and profit per firm?
At p < 20, S(p) = 0 while D(p) > 0, so the equilibrium price must be at least 20.
At p = 20, S(20) is no more than 30 while D(20) = 72 > 30. Thus the
equilibrium price must exceed 20.
At p > 20, S(p) = 5p - 70 = 112 - 2p = D(p) at equilibrium price p*=26 > 20.
The corresponding aggregate quantity is Q*= D(p*)=112- 2(26) = 60.
Each firm produces s(26) = 6 units of output, and obtains profit
p*s(p*) - c(s(p*))= (26)(6)- [9 + (14)(6) + (6)2]= 156 - 129 = 27.
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d. If there is a shift in aggregate demand to D(p) = 118 -5p, what are the
new short-run competitive equilibrium price, aggregate output, and profit
per firm?
At p < 20, 5(p) = 0 while D(p) > 0, so the equilibrium price must be at least
20.
If we use the formula for supply that applies when p > 20, and set
S(p) = D(p), we obtain 5p - 70 = 118- 5p, with solution p = 18.8. This is
not an equilibrium price! The formula we used for supply was correct for
p > 20, but 18.8 < 20. Thus the equilibrium price must be 20.
At p = 20, S(20) is any multiple of 3 between 0 and 30 while D(20) = 18,
which is a possible value for supply at price 20. The equilibrium price is
p*= 20, with corresponding aggregate quantity
Q*= D(p*) = 118- 5(20) = 18.
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Each firm produces an output from s(20), but both 0 and 3 are allowed.
To obtain the required aggregate output of 18, six of the ten firms must
produce output 3 while the other four firms produce output 0.
We have no way to determine which six firms will produce the output.
Each firm producing output 3 obtains profit
p*3- c(3) = (20)(3) - [9 + (14)(3) + (3)2]= 0,
while each firm producing output 0 obtains profit
p*0- c(0) = 0 - 0 = 0.
Since the firms start out identical, they must obtain the same profit level in
the competitive equilibrium, but they need not produce the same output.
[But all outputs chosen must be in s(p*).]
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Example: All firms have the same cost function, with an avoidable fixed
cost. For zero output, c(0) = 0, c(q) = 9 + 14q + q2 for q > 0, where q is
the firm’s output. This cost function applies in both the short and long
run.
e. If there is free entry and exit with aggregate demand D(p) = 112 - 2p,
then what are the long-run competitive equilibrium price, aggregate
quantity, quantity per firm, and number of firms?
Long-run aggregate supply is zero at prices less than 20 and is any multiple
of 3 at price 20.
Since D(20) = 72 > 0, the equilibrium price cannot be less than 20.
Thus p*= 20, aggregate quantity is D(p*) = 72, and since each firm
produces 3, there must be 72/3 = 24 firms in the long-run equilibrium.
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