The marginal physical product ( MPP ) of a factor is the additional output obtained by employing one more unit of that factor, while holding other factors constant. The MPP of either factor in the above Cobb–Douglas production function can be found by differentiating the function with respect to that factor (see pages A:10 – 13 for the rules of partial differentiation). Thus
The isoquant shows the whole range of alternative ways of producing a given output. Thus Figure 5.5 shows not only points a to e from the table, but all the intermediate points too.
Like an indiff erence curve, an isoquant is rather like a contour on a map. As with contours and indiff erence curves, a whole series of isoquants can be drawn, each one representing a diff erent level of output ( TPP ). The higher the output, the further out to the right will the isoquant be. Thus in Figure 5.6 , isoquant I 5 represents a higher level of output than I 4 , and I 4 a higher output than I 3 , and so on.
1.
?
Could isoquants ever cross?2. Could they ever slope upwards to the right?
Explain your answers.
KI 16 p 133
An isoquant map Figure 5.6
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The shape of the isoquant. Why is the isoquant ‘bowed in’
towards the origin? This illustrates a diminishing marginal rate of factor substitution ( MRS ). This, as we shall see very soon, is due to the law of diminishing returns.
The MRS 1 is the amount of one factor (e.g. K ) that can be replaced by a 1 unit increase in the other factor (e.g. L ), if output is to be held constant. So if 2 units of capital (Δ K = 2) could be replaced by 1 unit of labour (Δ L = 1) the MRS would be 2. Thus
MRS =ΔK ΔL=2
1= 2
The MRS between two points on the isoquant will equal the slope of the line joining those two points.
Thus in Figure 5.7 , the MRS between points g and h is 2 (Δ K /Δ L = 2/1). But this is merely the slope of the line joining points g and h (ignoring the negative sign).
When the isoquant is bowed in towards the origin, the slope of the isoquant will diminish as one moves down the curve, and so too, therefore, will the MRS diminish. Referring again to Figure 5.7 , between points g and h the MRS = 2. Lower down the curve between points j and k , it has fallen to 1.
?
Calculate the MRS moving up the curve in Figure 5.5 between each pair of points: e–d, d–c, c–b and b–a.Does the MRS diminish moving in this direction?
The relationship between MRS and MPP. As one moves down the isoquant, total output, by defi nition, will remain the same. Thus the loss in output due to less capital being used (i.e. MPP K × Δ K ) must be exactly off set by the gain in output due to more labour being used (i.e. MPP L × Δ L ). Thus
MPP L × Δ L = MPP K × Δ K KI 17
p 135
1 Note that we use the same letters MRS to refer to the marginal rate of factor substitution as we did in the previous chapter to refer to the marginal rate of substitution in consumption. Sometimes we use the same words too – just
‘marginal rate of substitution’ rather than the longer title. In this case we must rely on the context in order to tell which is being referred to.
Diminishing marginal rate of factor substitution
Figure 5.7
This equation can be rearranged as follows:
MPPL MPPK = ΔK
ΔL (= MRS)
Thus the MRS is equal to the inverse of the marginal produc- tivity ratios of the two factors.
Diminishing MRS and the law of diminishing returns. The principle of diminishing MRS is related to the law of diminishing returns. As one moves down the isoquant, increasing amounts of labour are being used relative to capital. This, given diminishing returns, would lead the MPP of labour to fall relative to the MPP of capital. But since MRS = MPP L / MPP K , if MPP L / MPP K diminishes, then, by defi nition, so must MRS . The less substitutable factors are for each other, the faster MRS will diminish, and therefore the more bowed in will be the isoquant.
Isocosts
We have seen how factors combine to produce diff erent levels of output, but how do we choose the level of output?
This will involve taking costs into account.
Assume that factor prices are fi xed. A table can be con- structed showing the various combinations of factors that a fi rm can use for a particular sum of money.
For example, assuming that P K is £20 000 per unit per year and P L is £10 000 per worker per year, Table 5.6 shows various combinations of capital and labour that would cost the fi rm £300 000 per year.
These fi gures are plotted in Figure 5.8 . The line joining the points is called an isocost . It shows all the combinations of labour and capital that cost £300 000.
As with isoquants, a series of isocosts can be drawn. Each one represents a particular cost to the fi rm. The higher the cost, the further out to the right will the isocost be.
1.
?
What will happen to an isocost if the prices of both factors rise by the same percentage?2. What will happen to the isocost in Figure 5.8 if the wage rate rises to £15 000?
Units of capital (at £20 000 per unit) 0 5 10 15 No. of workers (at a wage of £10 000) 30 20 10 0
Combinations of capital and labour costing the firm £300 000 per year Table 5.6
Definitions
Marginal rate of factor substitution The rate at which one factor can be substituted by another while holding the level of output constant:
MRS = Δ F 1 /Δ F 2 = MPP F2 / MPP F1
Isocost A line showing all the combinations of two factors that cost the same to employ.
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5.3 THE LONG-RUN THEORY OF PRODUCTION 151
The slope of the isocost equals PL
PK
This can be shown in the above example. The slope of the isocost in Figure 5.8 is 15/30 = ½. But this is P L / P K (i.e.
£10 000/£20 000).
Isoquants and isocosts can now be put on the same dia- gram. The diagram can be used to answer either of two ques- tions: (a) What is the least-cost way of producing a particular level of output? (b) What is the highest output that can be achieved for a given cost of production?
These two questions are examined in turn.
The least-cost combination of factors to produce a given level of output
First the isoquant is drawn for the level of output in ques- tion: for example, the 5000 unit isoquant in Figure 5.5 . This is reproduced in Figure 5.9 .
Then a series of isocosts are drawn representing diff erent levels of total cost. The higher the level of total cost, the further out will be the isocosts.
The least-cost combination of labour and capital is shown at point r , where TC = £400 000. This is where the isoquant just touches the lowest possible isocost. Any other point on the isoquant (e.g. s or t ) would be on a higher isocost.
Comparison with the marginal productivity approach. We showed earlier that the least-cost combination of labour and capital was where
KI 3 p 13
TC 8 p105
MPPL PL =MPPK
PK
In this section it has just been shown that the least-cost combination is where the isoquant is tangential to an iso- cost (i.e. point r in Figure 5.9 ). Thus their slope is the same.
The slope of the isoquant equals MRS , which equals MPP L / MPP K ; and the slope of the isocost equals P L / P K .
∴ MPPL MPPK=PL
PK
∴ MPPL PL =MPPK
PK
Thus, as one would expect, the two approaches yield the same result.
KI 14 p 107
An isocost Figure 5.8
The least-cost method of production Figure 5.9
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Highest output for a given cost of production
An isocost can be drawn for the particular level of total cost outlay in question. Then a series of isoquants can be drawn, representing diff erent levels of output ( TPP ). This is shown in Figure 5.10 . The higher the level of output, the further out will lie the corresponding isoquant. The point at which the isocost touches the highest isoquant will give the factor combination yielding the highest output for that level of cost. This will be at point h in Figure 5.10 .
Again this will be where the slopes of the isocost and isoquant are the same: where P L / P K = MRS .
If the prices of factors change, new isocosts will have to be drawn. Thus in Figure 5.10 , if the wage rate goes up, less labour can be used for a given sum of money. The isocost will swing inwards round point x . The isocost will get steeper.
Less labour will now be used relative to capital.
Maximising output for a given total cost Figure 5.10
*LOOKING AT THE MATHS
We can express the optimum production point algebraically. This can be done in either of two ways, corresponding to Figures 5.9 or 5.10 . (The method is similar to that used for finding the optimum consumption point that we examined
on page 113 .)
(a) Corresponding to Figure 5.9
The first way involves finding the least-cost method of producing a given output ( Q ). This can be expressed as
Min P K K + P L L (1) subject to the output constraint that
Q = Q ( K , L ) (2)
In other words, the objective is to find the lowest isocost ( equation 1 ) to produce on a given isoquant ( equation 2 ).
(b) Corresponding to Figure 5.10
The second involves finding the highest output that can be produced for a given cost. This can be expressed as
Max Q ( K , L ) (3)
subject to the cost constraint that
P K K + P L L = C (4) In other words, the objective is to find the highest isoquant ( equation 3 ) that can be reached along a given isocost ( equation 4 ).
There are two methods of solving (a) and (b) for any given value of P K , P L and either Q (in the case of (a)) or C (in the case of (b)).
The first involves substituting the constraint equation into the objective function (to express K in terms of L ) and then finding the value of L and then K that minimises the objective function in the case of (a) or maximises it in the case of (b). This involves differentiating the objective function and setting it equal to zero.
A worked example of this method is given in Maths Case 5.2 in MyEconLab.
The second method, which is slightly longer but is likely to involve simpler calculations, involves the use of ‘Lagrangian multipliers’. This method is explained, along with a worked example, in Maths Case 5.3 . It is the same method as we used in Maths Case 4.2 when finding the optimal level of consumption of two products.