(i) Maximisation of output with given cost, that is

Maximise, Q = f(L, K) (Objective function)

Subject to C = w. L + r.K (Constraint)

(ii) Minimisation of cost with given quantity, that is,

Minimise, C = w.L + r.K (Objective function)

Subject to Q = f[L, K) (Constraint)

The condition for the purpose is the same whichever the case. For instance, suppose the producer aims at maximisation of Q subject to the cost constraint. The condition of equilibrium may be spelled out as

MPL/w = MPk/r

Or MPL/MPK = w/r

The right hand side of Equation 4.14 gives the price ration of the two factors. Its left hand side gives the MRTSL, K.

For diagrammatic representation of the producer’s equilibrium, we require the plot of the producer’s budget line, popularly called the ‘isocost curve’ and the ‘isoquant map’. The latter consists of a number of isoquants, one above the other, representing distinct levels of output possible for the producer. Figure 4.4 shows the isoquant map exactly on the same lines as the indifference map.

Fig. 4.4: Isoquant Map: Producer’s isoquant map comprises isoquants Q, Q’, Q” & Q’”. Thus. Q’” > Q” > Q’> Q. The one closest to the origin (Q) represents the lowest level of output available and the one farthest away from the origin represents the highest level of it.

The former, the isocost curve, is the locus of possible combinations of quantities of two factors that can be bought out of a given budget (C) at the respective prices, w and r.

Fig. 4.5: An isocost line represents the locus of possible factor combinations that can be employed by the producer at respective prices of w and r out of the fixed budget C. Slope of the isocost, AB, is (-w/r).

Figure 4.5 shows the isocost curve of the producer. Price of K remaining the same, a fall in price of factor L, decreases the price ratio (w/r) causing an outward pivotal shift in the isocost curve, and a rise in it increases it causing an inward pivotal shift in it as shown in Figure 4.6.

Fig. 4.6: Pivotal shifts in isocost line: Price of K remaining the same, a fall in price of L pivots the isocost BA upwards to BA’ and a rise in it pivots it downwards to BA”. If C = 200, w = 10 and r = 10, the isocost BA is 200 = 10L + 10K. If w falls to 5, it pivots to BA’ (200 = 5L + 10K) and if it rises to 20, it pivots to BA” (200 = 20L + 10K).

If the producer’s budget increases, or if prices of both the factors fall in the same proportion, the isocost line undergoes a parallel shift upwards (Figure 4.7a). The opposite is the effect of a decrease in producer’s budget or a simultaneous increase in prices of the two factors by the same proportion (Figure 4.7b).

Producer’s equilibrium can be shown diagrammatically as in Figure 4.8. Producer’s budget line (Figure 4.5) and his isoquant map (Figure 4.4) are both superimposed in Figure 4.8. The isocost line AB touches the isoquant Q” at point e. Thus, the highest attainable isoquant curve is Q”. The equilibrium level of employment of the two factors is (Le, Ke).

Illustration 4.1

Given the isoquant

Q = 1000 L2/3 K1/3

and the isocost

3000 = 100L + 50K determine the equilibrium level of employment of factors by the producer.

Fig. 4.7: If producer’s budget (C) increases, or if prices of both the factors fall in the same proportion, the isocost line undergoes a parallel shift upwards (panel a). The opposite is the effect of a decrease in producer’s budget, or a simultaneous increase in prices of the two factors by the same proportion (panel b).

For example, if producer’s budget doubles from 200 to 400 when prices of both the factors remain unchanged at 20 and 10 respectively, or when prices of the two factors fall to half as much each, i.e., to 10 and 5 respectively when producer’s budget remains unchanged at 200, the isocost line shifts upwards to right parallel to itself as shown in panel (a).

The opposite is the effect when producer’s budget falls to half as much, i.e., to 100, with factor-prices remaining the same, or when prices of the two factors increase to twice as much i.e., to 40 and 20 respectively while producer’s budget remains unchanged at 200 (panel b).

Solution

Differentiating the isoquant function partially with respect to L and K, we have

Fig. 4.8: Producer’s equilibrium level of employment: Producer’s isocost line AB (C = wL +

rK) touches the isoquant Q” at e. At this point, slope of the isoquant is equal to the slope of the isocost. In other words, MRTSLK = w/r. Equilibrium level of employment of factors by the producer is (L e, Ke).

Illustration 4.2

How will your answer to the problem in Illustration 4.1 change if the isoquants were

(a) Q = 100L + 50K

(b) Q = 50L + 100K

(c) Q = 200L + 50K

= 2 [MPL/MPK = w/r]

That is, MRTSL,K = w/r for all combinations represented by the isoquant or by the isocost. This also indicates that the isocost line and the isoquant function must both coincide (note that the isoquant function is linear, indicating that the two factors are substitutes). There exist infinitely many solutions, that is, any combination on the isocost line is as good as any other on it.

(This time, IQ is steeper than the isocost line) Again, isoquant function being linear, the factors are substitutes.

Factors being substitutes, (30, 0) would optimise producer’s output as it falls on the isocost as also on the highest IQ.

Discussions conducted so far in respect of producer’s equilibrium draw a perfect parallel with the consumer’s equilibrium be it mathematical treatment, or be it diagrammatic representation, or be it conceptual explanation. Continuing the analogy, let us work out the expansion path to see that the two treatments—indifference curve analysis for consumer’s equilibrium and the isoquant analysis for the producer’s equilibrium — are perfectly parallel concepts.

The only point of difference lies in labelling of the curves and the axes. It was with this objective that the same figures have been used even in the numerical analysis in both, the consumer’s equilibrium and the producer’s equilibrium.