Download 1.2 Perfect Competition in an Industry

ESTOLA: GOODS’ MARKETS
7
Perfect
competition
Monopolistic Oligopoly Monopoly
competition
Number of sellers
many
many
a few
one
Barriers of entry
no
no
some
much
Differences
in goods
no
some
little
only one
good
some
much
banks
railway
Increasing returns no
no
to scale
An example
raw materials cars
1.2
1.2.1
Perfect Competition in an Industry
Firm in a Perfectly Competed Industry
A firm in a perfectly competed industry — like all other firms — aims to
operate as profitably as possible. A special feature in perfect competition
is that firms cannot affect the price of their product that adjusts in the
market according the demand of all consumers consuming, and the supply of
all producers producing the good. Price adjusts with time according to the
deviation between the aggregate production and consumption at the level
the production of the industry gets sold. This occurs because in the long-run
firms do not produce more than they can sell.
If the weekly production of a firm is greater than its weekly sales, it is
rational for the firm to decrease the price of its product in the case price
exceeds the firm’s marginal costs. If price is less than marginal costs, it
is rational for the firm to decrease its flow of production. In a perfectly
competed industry, firms’ products are almost perfect substitutes. Thus if
one firm sells at a lower price than others, customers buy from this firm. To
keep their customers, other firms must decrease their price accordingly.
If the aggregate weekly production of an industry is smaller than gets sold
at current price, those firms facing excess demand can raise their product
prices. In an excess demand situation, consumers must compete about who
can buy the scarce goods, and then some are ready to pay more. For this
reason, every firm can increase its product price because consumers buy from
the firms having goods left. Thus in a perfectly competed industry, firms
cannot decide the price of their product independently, but it is determined
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ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS
on the basis of the demand of all consumers consuming, and the supply of
all producers producing the good. In the following we study this process.
Next we derive the supply relation for a firm in perfect competition. The weekly profit of a firm producing homogeneous good k in a
perfectly competed industry is
Πk (t) = Rk qk (t) − Ck qk (t) = pk (t)qk (t) − Ck qk (t) ,
where the flow of production of the firm is denoted by qk (kg/week) and the
price of good k by pk ( /kg); the profit Πk , revenues Rk and costs Ck all
have unit /week. The dependence of the flow of production and price on
time t is assumed because later we analyze their adjustment with time. The
time derivative of Πk (Appendix A, Section 7.2) is
0
∂Πk 0
∂Πk 0
0
0
0
p (t) +
q (t) = qk (t)pk (t) + pk (t) − Ck qk (t) qk (t),
Πk (t) =
∂pk k
∂qk k
where marginal revenues equal pk and Ck0 qk (t) are marginal costs. Because
a firm in a perfectly competed industry cannot affect the price of its product,
the only variable by which the firm can affect its profit is qk . A profit-seeking
0
firm adjusts its flow of production as: qk0 (t) > 0 when
p
(t)
−
C
q
(t)
> 0,
k
k
k
0
0
0
and vice versa, and qk (t) = 0 when pk (t) = Ck qk (t) where qk (t) (kg/week 2 )
is the acceleration of production of the firm.
These adjustment rules can be explained as follows. Earlier on we explained that the price of good k adjusts with time at the level the production
of the industry gets sold (including the production of this firm). If price is
greater than the marginal costs of this firm, this firm can increase its profit
by increasing its flow of production. The firm knows, however, that if it
increases its flow of production, the aggregate flow of production of the industry increases that has a decreasing effect on price pk . Thus the firm has
to take account that if it increases its flow of production, this may require
it to decrease the price of its product to get its production
sold. In spite of
0
this remark, we identify the quantity pk (t) − Ck qk (t) as the force acting
upon the flow of production of good k of the firm.
The defined force measures the firm’s marginal profitability at current
price and the prevailing flow of production. The argument for the force
interpretation is as before; the more profitable producing one kilogram is,
the more eager a profit-seeking firm is to increase its flow of production. The
force consists of the benefits and costs in the firm’s decision-making, and the
profit maximizing situation corresponds to zero-force:
∂Πk
=0
∂qk
⇔
pk (t) = Ck0 qk (t) .
(1.1)
ESTOLA: GOODS’ MARKETS
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The relationship between the flow of production and price defined in (1.1)
expresses those flows of production at different prices that correspond to
the equilibrium states of the firm. A profit-seeking firm changes its flow of
production with time so that eventually Eq. (1.1) holds.
If the form of the cost function Ck is known, the optimal flow of production
of the firm at moment t can be solved from Eq. (1.1) as
−1
pk (t) ,
(1.2)
qk∗ (t) = Ck0
where Ck0 −1 is the inverse function of Ck0 . To be able to find this solution,
function Ck0 must be at least partially monotonous (Appendix A, Section 5.3)
so that it has a unique inverse at every flow of production.
In a perfectly competed industry, every firm decides its flow of production
with the aim to sell this production at the price determined in the market.
Firms know the price when they make their production decision, but they
do not know the flows of production of other firms. If then the aggregate
flow of production is greater than that which was previously sold, consumers
may not buy the increased production at current price. Some firms have
then unsold goods and they can decrease their production to diminish the
aggregate flow of production at the level that gets sold at current price. On
the other hand, if price is greater than the marginal costs of these firms, the
firms can decrease their product price to get their production sold.
Now, a price decrease by one firm forces other firms to follow this because
otherwise their products would not get sold. In this way the price determination in perfect competition leads to the marginal cost pricing with time. At a
higher price than that — assuming that firms’ marginal costs are higher than
their unit costs — firms can increase their profit by decreasing their price and
producing and selling more; this increases the aggregate flow of production
in the industry. If one firm decreases the price of its product under that of
other firms, the firm knows that other firms will follow this. This decreases
firms’ interest to lower their product price in order to increase their profit
and market share. However, a higher price than the marginal costs of firms
may attract new firms in the industry. Thus in a perfectly competed industry, the aggregate flow of production will increases with time when price is
higher than firms’ marginal costs. An increase in supply decreases price, and
thus price adjusts with time at the level of marginal costs of firms.
In many textbooks of economics it is also claimed that competition between firms forces the equilibrium price in perfect competition at the level
where the marginal costs of firms equal their unit costs. This occurs because those firms, for which this condition does not hold, can adjust their
production so that they can decrease their product price at that level. This
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ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS
way they gain customers from other firms that forces other firms to follow
the price decrease and to produce at the flow of production that creates the
minimum possible unit costs. If a firm produces at its minimum unit costs,
the firm can sell its products at the lowest possible price, see Figure 6.1.
Figure 6.1. The supply relation of a firm in perfect competition
The supply relation of a firm in a perfectly competed industry can be
derived as is shown in Figure 6.1. The unit and marginal costs of the firm
are described as in Chapter 5. Suppose that the price is pk0 in Figure 6.1.
Then every produced kilogram till the flow of production qk0 increases the
weekly profit of the firm, and the firm is motivated to increase its flow of
production till qk0 . However, the firm knows that its production affects the
aggregate supply in the industry and this way the market price of good k. For
this reason, the production decisions of firms cannot be analyzed separately
from the price adjustment mechanism, and the market mechanism must be
analyzed as a complete system. This is done in Sections 6.3.1-2.
Figure 6.1 shows that the force by which the firm affects the aggregate
production of the industry is positive at price pk0 if qk < qk0 , and negative if
qk > qk0 . An increase in the flow of production from qk0 creates more costs
than revenues at price pk0 which decreases the profit of the firm. At price
pk0 , the weekly profit of the firm gets its maximum at the flow of production
qk0 ; this corresponds to the zero force situation.
If price increases from pk0 to pk1 in Figure 6.1, every sold kilogram till the
flow of production qk1 increases the profit of the firm. If the whole production
of the firm gets sold, the optimal flow of production at price pk1 is qk1 . We
can thus think that the optimal flow of production of the firm is derived by
its marginal costs. By the uniqueness of the marginal cost function above
the unit cost function, the marginal cost function defines the unique flow
of production at every price the firm is willing to produce in the case its
production gets sold. Thus it is profitable for the firm to adjusts its flow of
production with time at the level its marginal costs equal the price.
The supply relation of a firm shows the flows of production at different
prices that correspond to the optimal states of the firm. If price decreases,
the optimal flow of production of a firm decreases according to its marginal
cost function. However, if price pk decreases below unit costs of a firm, this
firm makes losses. In this situation, the firm can stop its operation either
permanently or temporarily, or continue by making losses. The last option
is rational if the firm can cover its variable costs, and the firms’ managers
believe that the firm can reduce its costs, or that the price will increase, in
the future. On this basis, we can define the marginal cost function as
the supply relation of a firm. Marginal costs show the optimal flow of
ESTOLA: GOODS’ MARKETS
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production at every price greater than the unit costs of the firm.
Example. Let the cost function of a firm in perfect competition be
Ck (qk ) = C0 + g(qk )qk , where the flow of production qk has unit kg/week,
g(qk ) = c1 + c2 qk with unit /kg is the variable unit cost function and
C0 , c1 , c2 are positive constants with units /week, /kg and ( ×week)/kg 2 ,
respectively. The weekly profit of the firm is then
Πk = pk qk − Ck (qk ) = pk qk − C0 − c1 qk − c2 qk2 ,
(1.3)
and the supply relation of the firm can be defined as
dΠk
=0
dqk
⇔
⇔
pk = c1 + 2c2 qk
qk∗ =
p k − c1
.
2c2
(1.4)
The marginal revenues of the firm equal with price pk , and marginal costs
c1 + 2c2 qk linearly increase with the flow of production. The supply relation
of the firm is the last form of Eq. (1.4). The optimal flow of production
increases with price and decreases when constants c1 , c2 increase. If pk < c1 ,
the firm makes losses and it is optimal to stop the production. 1.2.2
Aggregate Production of an Industry
In the previous section we showed that a firm in a perfectly competed industry reacts to a price change by changing its flow of production. Because every
firm in the industry faces the same price change, firms change their production flows according to the difference between the price and their marginal
costs. As an example of this, we study a situation where the profit of firm i
is of the form (1.3) and its flow of production is denoted by qki . Thus
2
Πki = pk qki − Cki (qki ) = pk qki − C0i − c1i qki − c2i qki
,
and the supply relation of the firm is
dΠki
=0
dqki
⇔
⇔
pk = c1i + 2c2i qki
∗
qki
=
pk − c1i
.
2c2i
(1.5)
Next we assume n firms in the industry and derive the optimal aggregate flow
of production of these firms. For simplicity, the cost functions are assumed
of identical form for every firm, but the constants in them may vary. The
optimal aggregate flow of production is then
qkA =
∗
qk1
+
∗
qk2
+ ··· +
∗
qkn
=
n
X
i=1
= pk
∗
qki
=
n
X
pk − c1i
i=1
n
n
X
X
1
c1i
−
= pk A − B,
2c
2c
2i
2i
i=1
i=1
2c2i
(1.6)
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ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS
P
P
where A = ni=1 2c12i , B = ni=1 2cc1i2i are positive constants. The aggregate
supply function thus positively depends on price pk ,
qkA = Apk − B,
(1.7)
and qkA ≥ 0 if pk ≥ B/A, where B/A defines the minimum price at which
production occurs. The assumption that firms cost functions are of identical form was made to simplify the aggregation. In the real world, varying
cost functions of firms make the aggregation more complicated. In the next
section we show that in the case of varying cost functions, we can approximate the supply relations of firms by linear ones in the neighborhood of the
equilibrium states and in this way solve the aggregation problem.
To describe the behavior of a perfectly competed industry, we need to
model the aggregate production of the firms in the industry. It is possible
that this can be done by using a relatively simple average cost function for
the firms. Above we proposed one such candidate, but much work is needed
before this question is solved in an empirically satisfactory way.
1.2.3
Equilibrium in Perfect Competition
In this section we omit subindex k referring to the good to simplify the
notation, and we separate supply and demand by subscripts s, d. Thus pk =
p, qksi = qsi , qkdj = qdj , Cki = Ci and hkj = hj . According to Sections 4.10
and 6.1.3 when every firm and consumer have adjusted optimally, we have:
p = Ci0 (qsi ), i = 1, . . . n and p = hj qdj , Tj , p, pG , j = 1, . . . , m. (1.8)
Adding the n and m equations in (1.8), separately, and dividing the results
by n and m, respectively, we get
n
p=
m
1 X
1X 0
Ci (qsi ) =
hj qdj , Tj , p, pG ;
n i=1
m j=1
(1.9)
the middle term is the P
average of firms’ marginal
P costs at the aggregate
flow of production qs = ni=1 qsi , and term (1/m) m
j=1 hj is the consumers’
average willingness-to-pay
for one kilogram of good k at the aggregate flow
P
of consumption qd = m
q
j=1 dj . In the equilibrium, price equals the average of
firms’ marginal costs and consumers’ willingness-to-pay, and no agent likes
to change his behavior. This corresponds to the neoclassical equilibrium.
In the appendix of this section we show that we can approximate the
average of firms’ marginal costs as
n
a0 a1
1X
Ci0 qsi ≈
+ 2 qs ,
C 0 qs ≡
n i=1
n
n
qs =
n
X
i=1
qsi ,
(1.10)
ESTOLA: GOODS’ MARKETS
13
where constants a0 > 0, a1 have units /kg, ( × week)/kg 2 , respectively;
a1 < 0 implies increasing returns to scale in the industry and vice versa.
In the appendix of this section we show that we can approximate the
average willingness-to-pay for one kilogram of good k of the m consumers as
m
b0
b1
b3
1 X
b2
b4
hj ≈
+ 2 qd + p + 2 T + pG ,
h(qd , p, T, pG ) ≡
m j=1
m m
m
m
m
where P
the aggregate flow of consumption of the m consumers
Pm is denoted by
m
qd =
j=1 qdj , the aggregate budgeted funds by T =
j=1 Tj , and constants b0 ≥ 0, b1 < 0, b3 have units /kg, ( × week)/kg 2 and week/kg,
respectively; b2 < 0, b4 are dimensionless. In the following in this section
we assume that pG = pG0 , Tj = Tj0 ; this way we can omit these quantities
from the analysis; see the appendix of this section. An approximate average
of the consumers’ willingness-to-pay for one kilogram of good k is then
m
h(qd , p) ≡
b1
b2
b0
1 X
+ 2 qd + p.
hj (qdj , p) ≈
m j=1
m m
m
(1.11)
The equilibrium state in the industry can then be approximated as
a0 a1
+ 2 qs ,
n
n
b1
b2
b0
+ 2 qd + 2 p.
Aggregate demand relation: p =
m m
m
Aggregate supply relation: p =
(1.12)
(1.13)
The aggregate supply and demand relations can also be presented as
na0 n2
+ p,
a1
a1
2
mb0
m − b2
= −
+
p,
b1
b1
qs = −
(1.14)
qd
(1.15)
where a0 , a1 , b0 , n, m > 0 and b1 , b2 < 0.
Remark! The aggregate demand relation (1.15) equals with that derived
in Section 4.12 with a = mb0 /(m2 − b2 ) > 0, b = b1 /(b2 − m2 ) > 0. Setting qd = qs in system (1.14) — (1.15) we can solve the equilibrium
price p∗ . The equilibrium flows of production and consumption can then be
solved by substituting p∗ in (1.14) and (1.15). These give
p∗ =
a0 b 1 n − a1 b 0 m
> 0,
b1 n2 + a1 (b2 − m2 )
qd∗ = qs∗ =
n[a0 (m2 − b2 ) − b0 mn]
.
a1 (b2 − m2 ) + b1 n2
(1.16)
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ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS
The reader can check with the measurement units of the constants that the
solutions are dimensionally well-defined. The condition for qd∗ = qs∗ > 0 is
that p1 = a0 /n < b0 m/(m2 − b2 ) = p2 , where p1 is the price at which the
aggregate supply, and p2 the price at which the aggregate demand is zero.
The equilibrium state is displayed in Figure 6.2.
Figure 6.2. The equilibrium state in a perfectly competed industry
Remark! In Figure 6.2 on the horizontal axis are the aggregate flows
of production and consumption of good k both measured in units kg/week,
and on the vertical axis is the price of good k, the average marginal costs
of firms, and the average willingness-to-pay of consumers for one kilogram;
these all measured in units /kg. 1.2.4
Mathematical Appendix
The first order Taylor series approximations of firms’ marginal cost functions
in the neighborhood of the equilibrium flows of production qi0 are:
Ci0 qsi = Ci0 qsi0 + Ci00 (qsi0 ) qsi − qsi0 + i , i = 1, . . . , n,
(1.17)
where i is the residual term. Assuming i = 0 and summing over i, we get1 :
n
n
X
X
0
0
Ci qsi ≈
Ci (qsi0 ) − Ci00 qsi0 qsi0 + Ci00 qsi0 qsi
i=1
i=1
a1
qs ,
n
P
where qs = ni=1 qsi and constants
≈ a0 +
a0 =
n
X
Ci0
(qsi0 ) −
Ci00
(qsi0 ) qsi0 ,
i=1
a1 =
n
X
Ci00 (qsi0 ) ,
i=1
2
have units /kg and ( × week)/kg , respectively. Because marginal costs
are positive at every qsi , then a0 > 0 (let qsi → qsi0 and i → 0 in (1.17) ∀i).
Increasing (decreasing) returns to scale in the industry correspond to a1 < 0
(a1 > 0).
We approximate the willingness-to-pay function of consumer j in the
neighborhood of his equilibrium point zj0 = (qdj0 , p0 , Tj0 , pG0 ) as
hj (qdj , p, pG , Tj ) = hj (zj0 ) +
+
∂hj
∂hj
(zj0 )(qdj − qdj0 ) +
(zj0 )(p − p0 )
∂qdj
∂p
∂hj
∂hj
(zj0 )(Tj − Tj0 ) +
(zj0 )(pG − pG0 ) + j , j = 1, . . . , m. (1.18)
∂Tj
∂pG
Pn
Pn
Pn
Pn
Because i=1 ci qi = c i=1 qsi + i=1 (ci − c)qsi where c = (1/n) i=1 ci , the approximation is the more accurate the less ci = Ci00 (qsi ) or qsi vary, i = 1, . . . , n.
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ESTOLA: GOODS’ MARKETS
15
Assuming j = 0 ∀j and summing over j, we get2 :
m
m X
X
∂hj
∂hj
hj (qdj , p, pG , Tj ) =
hj (zj0 ) −
(zj0 )qdj0 −
(zj0 )p0
∂q
∂p
dj
j=1
j=1
X
m
m
X
∂hj
∂hj
∂hj
∂hj
−
(zj0 )Tj0 −
(zj0 )pG0 +
(zj0 )qdj +
(zj0 )p
∂Tj
∂pG
∂q
∂p
dj
j=1
j=1
+
m
X
∂hj
j=1
m
X
∂hj
(zj0 )Tj +
(zj0 )pG
∂Tj
∂p
G
j=1
b1
b3
q d + b 2 p + T + b4 p G ,
m
m
wherePthe aggregate flow of consumption of the m consumers
is denoted by
Pm
qd = m
q
,
their
aggregate
budgeted
funds
by
T
=
T
j=1 dj
j=1 j , and
≈ b0 +
b0 =
m h
X
hj (zj0 ) −
j=1
∂hj
∂hj
∂hj
(zj0 )qdj0 −
(zj0 )p0 −
(zj0 )Tj0
∂qdj
∂p
∂Tj
m
m
i
X ∂hj
X ∂hj
∂hj
−
(zj0 )pG0 , b1 =
(zj0 ), b2 =
(zj0 ),
∂pG
∂qdj
∂p
j=1
j=1
b3 =
m
X
∂hj
j=1
m
X
∂hj
(zj0 ), b4 =
(zj0 ).
∂Tj
∂pG
j=1
The units of the constants are: b0 :
/kg, b1 : ( × week)/kg 2 , b3 :
week/kg, and b2 , b4 are dimensionless. Because the willingness-to-pay of
every consumer is non-negative at every qdj , Tj , p, pG , we can conclude that
b0 ≥ 0 (let qdj → qdj0 , Tj → Tj0 , p → p0 , pG → pG0 and j → 0 ∀j in (1.18)).
In the following we assume that pG = pG0 and Tj = Tj0 ∀j. An approximate average of the consumers’ willingness-to-pay is then:
m
h(qd , p) ≡
1 X
b0
b1
b2
hj (qdj , p) ≈
+ 2 qd + p,
m j=1
m m
m
(1.19)
where
b0 =
b2 =
m X
j=1
m
X
j=1
2
m
X
∂hj
∂hj
∂hj
hj (zj0 ) −
(zj0 )qdj0 −
(zj0 )p0 , b1 =
(zj0 ),
∂qdj
∂p
∂qdj
j=1
∂hj
(zj0 ).
∂p
See the previous footnote.
16
ESTOLA: PRINCIPLES OF QUANTITATIVE MICROECONOMICS
An approximate average of the consumers’ willingness-to-pay for one kilogram of the good thus linearly depends on the aggregate flow of consumption
of the good and its price.
1.2.5
Adjustment by Price Mechanism
Traditionally, the dynamics in a perfectly competed industry has been assumed to take place by price adjustment via excess demand as:
p0k (t) = fp (Fp ), fp0 (Fp ) > 0, fp (0) = 0, Fp = qkd − qks ,
(1.20)
where qkd − qks is the excess demand (supply) when it is positive (negative).
This mechanism was suggested by Paul Samuelson (1941,1942). Assuming
the demand and supply relations as in the previous section, and denoting
2
0
0
the constants in them as A0 = na
> 0, A1 = na1 > 0, B0 = − mb
> 0,
a1
b1
2
m −b2
B1 = − b1 > 0, the excess demand becomes the following:
qkd − qks = B0 − B1 pk (t) + A0 − A1 pk (t) = B0 + A0 − (B1 + A1 )pk (t).
Taking the Taylor series approximation of function fp in (1.20) in the neighborhood of point Fp = 0 and assuming the error term zero, we get fp ≈
fp0 (0) × Fp and we can approximate (1.20) as
p0k (t) = fp0 (0) × [A0 + B0 − (A1 + B1 )pk (t)],
(1.21)
where fp0 (0) is a positive constant with unit /kg 2 . From (1.21) we see that
A0 +B0
the quantity in brackets is positive if pk < A
= p∗k , and vice versa.
1 +B1
Thus price increases (p0k (t) > 0) when it is below, and decreases when it is
above its equilibrium value. The equilibrium is thus stable and price adjusts
with time toward its equilibrium. According to (1.14) and (1.15), a price
raise increases supply and decreases demand and vice versa. Thus firms and
consumers react to price changes, and the demand and supply relations show
how both parties adjust their flows of production and consumption.
Equation (1.21) can be explained as follows. If the whole production of
the industry gets sold, and even more could have been sold at current price,
the lack of goods forces consumers to compete about who can buy the scarce
goods. This allows some firms to raise their product price, and other firms
can follow this because also their productions get sold. In the opposite case,
any firm can assure that its production gets sold by decreasing its product
price, even though the whole production of the industry does not get sold.
The price reduction of one firm forces other firms to follow this if they like
to sell their whole production. The difference qkd (t) − qks (t) can thus be
ESTOLA: GOODS’ MARKETS
17
interpreted as the force acting upon price pk , and every consumer and
producer affect this force by their decisions.
We could have added also static friction in the price to explain that in
the real world, a certain non-zero excess demand may be required before
price starts to react. For example, the existing goods in firms’ inventories
prohibit a price raise before these goods are sold. However, we assume, for
simplicity, that prices do not have static friction. A positive force thus acts
upon the price of good k when the aggregate flow of consumption exceeds
that of aggregate production at the prevailing price, and vice versa.
Remark! In Eq. (1.21) we gave up the Newtonian principle ‘force causes
acceleration’, because in Eq. (1.21), the force causes the velocity and not the
acceleration of the price. The reason for this is that when we model changes
in consumption and production, the adjusting variables are flow variables
while price is a ‘stock’ variable. However, Eq. (1.21) corresponds to the
principle of modelling in economics because in an excess demand situation,
it is rational for profit-seeking firms to charge a higher price and for utilityseeking consumers to pay more. 1.2.6
Time Path of Price*
The price adjustment mechanism described in the previous section yielded
differential equation (1.21) the solution of which is
pk (t) =
A0 + B0
0
+ C0 e−fp (0)(A1 +B1 )t ,
A1 + B1
where C0 ( /kg) is the constant of integration. Because A1 + B1 > 0,
with t → ∞, pk (t) adjusts toward the equilibrium price in the industry
A0 +B0
1 n−a1 b0 m
p∗k = A
= b1an02b+a
2 , see Section 6.3.2. The process is thus stable.
1 +B1
1 (b2 −m )