Everything you need to know
about isocosts and isoquants to
prove HO, Stolper-Samuelson,
and Rybczynski theorems.
The behavior of the firm
• Firms are assumed to attempt to maximize profits.
• First, the firm must identify the profit maximizing
level of output – the quantity where marginal
revenue equals marginal cost.
• Next, the firm must minimize the cost of producing
that level of output.
• It must choose the appropriate technology and
apply it correctly. As a part of this, it must
combine resources according to the least-cost
recipe.
2
Learning Objectives
• Calculate and graph a firm’s isocost line
• Work out how the isocost line changes when
resource prices or total cost change
• Make a map of production recipes (technology)
using isoquants
• Explain the choices that firms make
• Prove three theorems relating to the HO model
3
A Cost Function: Two Resources
• Assume that there are two resources, Labor (L)
and Capital (K).
• The money payments to these resources are
Wages (W) and Rent (R). An isocost line is similar
to the budget line. It’s a set of points with the
same cost, C. Let’s plot K on the y axis and L on
the x axis.
WL + RK = C; solve for K by first subtracting WL from both
sides.
RK = C - WL; next divide both sides by R.
K = C/R – (W/R)L; note that C/R is the y intercept and W/R is
the slope.
5
An isocost line
K (machines rented)
C/R
Absolute value of slope equals
The relative price of Labor, W/R.
C/W Labor hours used in
production
6
A Numerical Example
Bundles of:
Labor
Machine rental
with C = $30 ($6 per labor hour)
($3 per machine hour)
a
0
10
b
1
8
c
2
6
d
3
4
e
4
2
f
5
0
Points a through f lie on the isocost line for C = $30/hour.
7
Capital, K (machines rented)
The Isocost Line
10
a
b
8
c
6
d
4
e
2
f
0
1
2
3
4
5
6
7
8
9
10
Labor, L (worker-hours employed)
8
The Isocost Line
• Wage-rental ratio
– With K on the y axis and L on the x axis, the slope
of any isocost line equals W/R, the wage-rental
ratio. It is also the relative price of labor.
• The y-intercept shows the number of units of K
that could be rented for $C.
• The x-intercept shows the number of units of L
that could be hired for $C.
11
Capital, K (machines rented)
Changes in One Resource Price
10
Cost = $30; R = $3/machine
The money wage, W = ...
a
8
6
A Change
in W
4
2
0
…$6
…$10
1
h
2
3
f
4
5
6
7
8
9
10
Labor, L (worker-hours employed)
12
Capital, K (machines rented)
Changes in Cost
10
A Change
in Cost; every point
between g and h costs $18.
8
6
g
4
2
W = $6; R = $3;C = $30
h
0
1
2
3
4
5
6
7
8
9
10
Labor, L (worker-hours employed)
14
Each point on a given
isoquant represents different
recipes for producing the
same level of output.
Capital, K (machines rented)
An Isoquant
12
10
8
i
6
Quantity of Soybeans = 1 (kg./hour)
4
2
j
0
1
2
3
4
5
6
7
8
9
10
Labor, L (worker-hours employed)
17
6 7 8 9 10
m
4
5
Quantity of Soybeans = 2 (kg./hour)
3
k
2
j
1
Quantity of Soybeans = 1 (kg./hour)
0
Capital, K (machines rented)
An Isoquant Map
Different isoquants represents
different levels of output.
0
1
2
3
4
5
6
7
8
9 10
Labor, L (worker-hours employed) 18
Capital, K (machines rented)
Cost Minimization
Choose the recipe where the
desired isoquant is tangent to
the lowest isocost.
12
10
a
8
C = $36
6
W = $6; R = $3;C = $30
4
equ.
2
C = $18
0
1
2
3
4
5
6
7
8
9
10
Labor, L (worker-hours employed)
21
Conclusion: Buy resources such
that the last dollar spent on K adds
the same amount to output as the
last dollar spent on L.
• The |slope| of the isocost line = W/R.
• The |slope| of the isoquant = MPL/MPK
– This will be demonstrated on the board.
22
Proof of the HO theorem (price definition).
24
Preparing to Prove the Heckscher-Ohlin theorem.
In our example in class and the handouts, we demonstrated the theorem
by assuming that Country A was relatively capital abundant, giving it
comparative advantage in the capital-intensive good, S. Before we prove
the theorem, study the figure above. Two isoquants are shown. Each
represents all the technically efficient combinations of resources that
could be used to produce one unit of a particular product. The isoquant
for S lies closer to the K axis because S is the K-intensive good. The
isoquant for T lies closer to the L axis because T is the L-intensive good.
Only one isoquant is drawn for each good. However, our assumption of
constant returns to scale means that the isoquant for two units of a good
will require twice as much of each input. Thus the map of isoquants is
regularly spaced. If we can prove a theorem for one output level, then it
will be valid for all output levels. The assumption of constant returns to
scale (and no fixed costs) also implies that average cost and marginal
cost are constant and equal to each other for all levels of production.
Note also that the assumption that both countries have access to the
same set of technologies means that their isoquant maps are identical.
25
Proof of the Heckscher-Ohlin theorem.
To prove the theorem from the price definition of factor abundance, we
must show that a higher wage-rental ratio in A implies that B will have a
lower autarkic relative price of T (and that A will have a lower autarkic
relative price of S). Suppose that the autarkic relative price of S (PS/PT)
in A equals 1. Then the line segment GH is the pre-trade isocost line
facing A’s firms. Why? Given PS = PT and that P = MC = AC, the cost of
producing one unit of S must equal the cost of producing one unit of T.
Point G represents the following ratios:
(MCSA = PSA = MCTA = PTA)/RA.
By the same reasoning, point H represents the following ratios:
(MCSA = PSA = MCTA = PTA)/WA .
Thus the slope of the line connecting points G and H equals the pretrade wage-rental ratio in A, WA/RA.
26
Proof of the Heckscher-Ohlin theorem (continued)
Now consider country B, which represents the labor-abundant country.
Country B’s greater relative supply of labor means that it will have a
lower autarkic wage-rental ratio. Two separate, parallel isocost lines are
required to represent B’s optimal input choices for one unit of each good.
These choices are represented by points X and Y. Note that the isocost
line CX lies above the isocost line EY. Thus, the marginal cost of
producing one unit of S in B is greater than the marginal cost of
producing one unit of T in B. That is, MCSB > MCTB. Since price equals
marginal cost, it follows that PSB > PTB. This is exactly what we were
seeking to prove. That is, if the relative price of S in A equals one, then
the relative price of S in B is greater than one. Thus A has a comparative
advantage in S and B has a comparative advantage in T.
27
Proof of the Stolper-Samuelson theorem.
Home is Labor abundant, T is
Labor intensive. The initial
isocost line is tangent to the S
and T isoquants at F and D. $1
worth of each good costs $1 to
produce.
With trade PT/PS rises.
(Assume only PT changes.)
Note that W/R must rise
proportionately more.
28
Proof of the Stolper-Samuelson theorem.
When PT rises, Home will sell fewer units of T for $1. The new isoquant
is T’. The new isocost line is tangent to the S isoquant at F’ and tangent
to the T’ isoquant at D’.
R’ < R. The money return to capital falls. PS is unchanged, but PT has
risen. Thus the real return to capital falls.
W’ > W. The money wage rises. Workers can now buy more S, as its
price is unchanged. But can they buy more T?
The increase in the wage rate is shown by the proportion (1/W)/(1/W’).
This is greater than the proportionate increase in the price of T, which is
the ratio of line segment 0D to 0R, 0D / 0R. The real wage in terms of T
has also risen. Thus the real return to Labor rises.
29
Proof of the Stolper-Samuelson theorem (more words)
The Stolper-Samuelson theorem states that the factor that is used
intensively in a product whose relative price has risen gains, while the
other factor loses. In the context of the HO model, this means that the
abundant factor gains from trade while the scarce factor loses. Consider
the figure.
As before, we illustrate two isoquants representing $1 of output of S and
T, respectively. Given the initial prices, wage and rental rates, the
optimal combination of inputs are shown by points F and D.
When trade is opened, the country with comparative advantage in good
T will experience an increase in the relative price of T. At a higher price
for T, $1 worth of this product would lie on a lower isoquant (remember
that isoquants refer to physical units). Thus the T=$1 isoquant would
become the isoquant labeled T’. If some of both goods are still to be
produced, the $1 isocost line must rotate to maintain tangency with the
two isoquants S and T’. How can this be accomplished? Wages and
rents must change. The new isocost line has intercepts $1/R’ and $1/W’.
Since the numerators of these fractions are the same as before, we can
deduce what has happened to rents and wages by simply comparing
1/R with 1/R’ and 1/W with 1/W’. (continued on next slide)
30
Proof of the Stolper-Samuelson theorem (continued)
In the first case, the fraction has risen. This could occur only if R’ < R,
that is , if rents have fallen. On the other hand, a comparison of
horizontal intercepts shows that W has risen. These changes in W and R
are nominal changes. What has happened to the purchasing power of
capitalists and laborers? For capitalists, the answer is straightforward.
We have assumed that the price of S has stayed fixed while the price of
T has risen. A fall in R, therefore, means that capitalists have lost
purchasing power in terms of either product – they are definitely worse
off when the price of the labor-intensive good T rises relative to the price
of the capital-intensive good S. What about labor? The rise in W
definitely means that labor can afford to purchase more S, because its
price has been assumed to remain constant. However, the price of T has
risen. Can labor buy more or less of this product? The answer is more.
How do we know? Graphically, the increase in wages can be found by
comparing the proportions (1/W)/(1/W’). This increase is greater than the
proportionate increase in the price of T, which can be found by the ratio
of the line segments 0D/0R. (Given CRS, the isoquants are evenly
spaced.) This proves the theorem.
31
Notes on Stolper Samuelson
Why does the ratio 0D/OR represent the proportionate
increase in the price of T? This is a bit hard to see at first.
Suppose that R and W (and C) rose by the same proportion as
PT. Then the $1 isocost line would shift in (parallel) to be
tangent at R. This lower quantity of T costs $1 to produce, and
would sell for $1. Since price equals marginal cost equals
average cost, the increase in cost at D (which now costs more
than $1) will equal the increase in price. Thus 0D/0R
represents the proportionate increase in price.
32
A more formal explanation:
Why does the ratio 0D/OR represent the proportionate
increase in the price of T?
We know that PTT = P’TT’ (= $1). Divide both sides by PTT’ to
get :
T/T’ = P’T/ PT
0D = T and 0R = T’. T
Thus 0D/0R = P’T/ PT, the proportionate increase in price.
Looking at the graph reveals that W’/W > 0D/OR.
Therefore, W’/W > P’T/ PT.
Thus workers’ real wage has risen in terms of both T and S
because the money wage has increased by more than the price
of either good.
33
Proof of the Rybczynski theorem
34
Proof of the Rybczynski theorem
The Rybczynski theorem states that if a country experiences an increase
in its endowment of any one factor (say, labor), then, holding all other
things constant (including factor and product prices), the output of the
good that uses the factor intensively will rise, and the output of the other
good will fall. To prove this theorem, refer to the isoquant map shown in
the figure. Each of the two isoquants shown represents the output of $1
worth of one good, S or T. Suppose that the relative price of S is equal to
1. As discussed in the proof of the HO theorem, this must imply that
there is an isocost line that is just tangent to the two isoquants, just as
drawn. Furthermore, we know that the vertical and horizontal intercepts
of this isocost line must equal $1/R and $1/W, respectively. The
tangency points F and D determine the optimal input combinations to
produce $1 of S output and $1 of T output. If wages and rental rates are
held fixed, the assumptions of constant returns to scale guarantees that
the slope of the rays from the origin passing through point F and D
determines the optimal capital/labor ratios for the two industries, given
those factor prices.
(continued on next slide)
35
Proof of the Rybczynski theorem (continued)
How does the economy divide its output between the two products? This
depends upon the overall supply of available factors of production.
Suppose that the economy is initially endowed with a set of factors
defined by point E. To find the optimal production of S and T, complete
the parallelogram from point E to the two rays emanating from the origin.
This defines points G and H on the two rays. These points represent
optimal production levels of S and T, given the prices prevailing in the
economy. How do we know that this is true? First, we know that output
must occur on the rays. Second, we want to use all available resources.
If we add the factor combination represented by the line 0G to the point
H, we reach the total endowment level E. Similarly, if we add 0H to G,
we also reach point E.
Now we are in position to prove the theorem. Suppose that the country’s
endowment of labor rises, but capital and prices do not change. This
pulls the country’s endowment point horizontally away form E to , say, E’.
By completing the parallelogram with points E’ and 0 at the corners, we
see that the optimal production levels of S and T have changed from
their old levels. In particular, the output of S has fallen (to G’), while that
of T has risen (to H’). This proves the theorem.
36