Chapter 5
Chapter 6
Chapter 8
Chapter 21
Mathematical
Postscripts
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The Factor-Proportions Model
Factor Prices and Costs
• The unit isoquant in Figure 5P-1 shows
combinations of capital and labor that can be used
to produce one unit of the good.
– It’s increasingly difficult to substitute capital for labor as
the capital-labor ratio increases, and vice-versa.
• Producers choose the mix of capital and labor that
minimizes their cost.
– Such as point E, the point at which the unit isoquant is
tangent to a line whose slope equals –w/r.
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Fig. 5P-1: Efficient Production
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1-3
The Factor-Proportions Model
(cont.)
• The cost of production equals the sum of the cost
of capital and labor inputs, where the input
coefficients (aK and aL) have been chosen to
minimize costs.
c = aK r + aL w
• Because the mix of factors was chosen to
minimize cost, an infinitesimal change in the
capital-labor ratio must have no effect on cost.
0 = r daK + w daL
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1-4
The Factor-Proportions Model
(cont.)
• A change in the factor prices has two effects:
– It will change the choice of factor mix (aK and aL) , and it
will change the cost of production c.
• The cost-minimizing labor-capital ratio depends on
the ratio of the price of labor to that of capital.
• For small changes in factor prices, the change in
production cost is the following (the last two terms
sum to zero):
dc = aKdr + aLdw + rdaK + wdaL = aKdr + aLdw
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1-5
The Factor-Proportions Model
(cont.)
dc  aK r  dr   aLw  dw 

   


c  c  r   c  w 
cˆ  K rˆ  Lwˆ
K  L  1
aK r
K 
c
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is share of capital in total
production costs
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The Factor-Proportions Model
(cont.)
• The price of each good
must equal production
cost:
PF  aKF r  aLFw
PC  aKC r  aLCw
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• Equations for the rate
of change for factor
prices:
PˆF  KF rˆ  LFwˆ
PˆC  KC rˆ  LCwˆ
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The Factor-Proportions Model
(cont.)
• The economy’s factors
must be fully
employed:
aKFQF  aKCQC  K
aLFQF  aLCQC  L
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• For given prices,
equations for rate of
change for outputs:
KFQˆ F  KCQˆC  Kˆ
LFQˆ F  LCQˆC  Lˆ
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The Factor-Proportions Model
(cont.)
Goods Prices and Factor Prices
• If the price of food rises relative to the price of
cloth, then the real price of capital rises in terms of
both goods, while the real price of labor falls in
terms of both goods.
rˆ  PˆF  PˆC  wˆ
• In particular, if the price of F were to rise with no
change in the price of C, the wage rate would fall.
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1-9
The Factor-Proportions Model
(cont.)
Factor Supplies and Output
• If the prices of the goods stay constant, while the
supply of capital rises relative to the supply of
labor, then the output of food grows relative to the
supply of capital and the output of cloth shrinks
relative to the supply of labor.
Qˆ F  Kˆ  Lˆ  QˆC
• In particular, if the supply of K were to rise with no
change in the supply of L, the output of cloth would
fall.
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1-10
Fig. 6P-1: Consumption Effects of a Price
Change
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1-11
The Monopolistic Competition Model
• Consider the effects of changes in the size of the
market on equilibrium in a monopolistically
competitive industry.
• Each firm has the total cost C = F + cX, where c is
marginal cost, F a fixed cost, and X the firm’s
output.
• This implies an average cost AC = C/X = F/X + c.
• Each firm faces a demand curve X = S [1/n – b (P
– P)], where S is total industry sales (taken as
given), n is the number of firms, and P is the
average price charged by other firms (taken as
given).
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1-12
The Monopolistic Competition Model
(cont.)
• Profits
p = PX – C = (PS – c) [1/n – b (P – P)] – F
• Each firm chooses its price to maximize
profts by setting
dp /dP = X – SbP + Sbc = 0
• Since all firms are symmetric, P = P and X
= S/n so P = 1/bn + c and AC = Fn/S + c
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1-13
The Monopolistic Competition Model
(cont.)
• In zero-profit equilibrium, the price
charged by a typical firm must equal its
average cost P = AC.
1/bn + c = Fn/S + c
• Solving implies n = sqrt (S/bF).
• An increase in the size of the market, S,
will lead to an increase in the number of
firms, n, but not in proportion.
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1-14
The Monopolistic Competition Model
(cont.)
• The price charged by the representative
firm is
P = 1/bn + c = c + sqrt (S/bF).
• An increase in the size of the market leads
to lower prices.
• The sales per firm equal
X = S/n = sqrt (S/bF).
• The scale of each individual firm also
increases with the size of the market.
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Fig. 21P-1: Indifference Curves for
Uncertain Consumption Costs
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1-16
Fig. 21P-2: Maximizing Expected Utility
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1-17
Fig. 21P-3: Nondiversified Portfolios
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1-18
Fig. 21P-4: Effects of a Rise in H1 on
Consumption
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1-19
Fig. 21P-5: Effects of a Rise in H1 on
Portfolio Shares
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