Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

122 THEORY OF PRODUCTION [CHAP. 6
6.5 THE MARGINAL RATE OF TECHNICAL SUBSTITUTION
The marginal rate of technical substitution of L for K (MRTS LK ) refers to the amount of K that a firm can give
up by increasing the amount of L used by one unit and still remain on the same isoquant. The MRTS LK is also
equal to MP L /MP K . As the firm moves down a isoquant, the MRTS LK diminishes.
EXAMPLE 6. In moving from point B to point C on isoquant I in Fig. 6-3, the firm gives up 3 units of K for one additional
unit of L. Thus, the MRTS LK ¼ 3. Similarly, from point C to D on isoquant I, the MRTS LK ¼ 2. Thus, the MRTS LK
diminishes as the firm moves down an isoquant. This is so because the less K and the more L the firm is using (i.e., the
lower the point on the isoquant), the more difficult it becomes for the firm to substitute L for K in production.
EXAMPLE 7.
Table 6.2.
Table 6.3 gives the MRTS LK between the various points on the negatively sloped portion of the isoquants in
Table 6.3
Isoquant I Isoquant II Isoquant III
L K MRTS LK L K MRTS LK L K MRTS LK
2 11 4 13 6 15
1 8 3 10 5 12
2 5 3.0 4 7 3.0 6 9 3.0
3 3 2.0 5 5 2.0 7 7 2.0
4 2.3 .7 6 4.2 .8 8 6.2 .8
5 1.8 .5 7 3.5 .7 9 5.5 .7
6 1.6 .2 8 3.2 .3 10 5.3 .2
7 1.8 9 3.5 11 5.5
Note that the MRTS LK between two points on the same isoquant is given by the absolute (or positive value of the) slope of
the chord between the two points, while the MRTS LK at a point on the isoquant is given by the absolute slope of the isoquant
at that point. The MRTS LK is also equal to the MP L /MP K . For example, if the MP K is 1 2
at a particular point on an isoquant
while the MP L is 2, this means that one unit of L is 4 times more productive than one additional unit of K at this point. Thus,
the firm can give up four units of K by using one additional unit of L and still produce the same level of output (remain on the
same isoquant). Therefore, the MRTS LK ¼ MP L /MP K ¼ 2/(1/2) ¼ 4 at the given point.
6.6 CHARACTERISTICS OF ISOQUANTS
Isoquants have the same characteristics as indifference curves: (1) in the relevant range isoquants are
negatively sloped, (2) isoquants are convex to the origin and (3) isoquants never cross.
EXAMPLE 8. The relevant portion of an isoquant is negatively sloped. This means that if the firm wants to use less K, it
must use more L to produce the same level of output (i.e., remain on the same isoquant). The firm will not operate on the
positively sloped range of an isoquant because it could produce the same level of output by using less of both L and K. For
example, point A on isoquant I in Fig. 6-4 involves both more L and more K than at point B (also on isoquant I). If we draw
lines separating the relevant (i.e., the negatively sloped) from the irrelevant (i.e., the positively sloped) portions of the isoquants
in Fig. 6-3, we get “ridge lines” OY and OX of Fig. 6-4. The range of isoquants between the ridge lines corresponds to
stage II of production for L and K (see Problems 6.13 and 6.14).
In the relevant range, isoquants are not only negatively sloped but also convex to the origin because of diminishing
MRTS LK . In addition, isoquants cannot cross. If two isoquants crossed, the point of intersection would imply that the
firm could produce two different levels of output with the same combination of L and K. This is impossible if we
assume, as we do, that the firm uses the most efficient production techniques at all times.