2 Important Ways in which the Problem of Constrained Maximum can be Solved Mathematically

Mathematically, the problem of ‘constrained maximum’ (producer equilibrium) can be stated and solved in two different ways:

Case 1: Maximisation of Output Subject to a Cost Constraint:

The rational producer seeks to maximise his output, given his total-outlay and the prices of the factors (Fig. 7.10 (a)). The problem can be stated as follows:

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

Subject to C = w L + r K (cost constraint)

Where ‘w’ and ‘r’ are factor prices for labour and capital respectively.

This problem of constrained maximum can be solved by using Lagrangian multipliers. Since there is only one constraint in this problem, we will use one Lagrangian multiplier, say λ.

The augmented objective function can be written as

Z = Q + λ (C –wL – rK)

The maximisation of the ‘Z’ function implies maximisation of the output. The first order condition require

Solving the first two equations for X, we obtain

Thus, the producer is in equilibrium, when it equates the ratio of the marginal productivities of factors (MRTS) to the ratio of their prices.

The second order condition for equilibrium requires that the marginal product curves of the two factors should have a negative slope.

These conditions are sufficient for establishing convexity of the isoquants.

Case 2: Minimisation of Cost for a Given Level of Output:

In this case, the producer wants to produce a given output with the minimum cost outlay. Here, we have a single isoquant depicting the given level of output and a set of iso-cost lines (Fig. 7.10 (b)). Lines closer to the origin show lower cost outlay. Because of the assumption of constant factor prices, these lines are parallel. The problem can be stated as follows:

Minimise C = f (Q) = wL + rK (production function)

Subject to Q = f (L, K) (output constraint)

Using Lagrangian multiplier, the augmented objective function can be written as

Z = C + X [Q – f (L, K)]

Or, Z = (w L + r K) + X [(Q – f (L, K)]

The first order condition requires

Dividing the two expressions, we find

The condition for producer’s equilibrium in this case is precisely the same as in case 1 just stated before.

The second order condition concerning the convexity of the isoquants is again fulfilled by the assumption of negative slopes of the marginal product of factors.

In the two cases just stated above, E₁ and E₂ are the points of equilibrium, as shown in the Fig. 7.9 (a) and 7.9 (b) respectively. Higher levels of output (to the right of E₁) are desirable, but, not attainable due to cost constraint, while lower levels of output (to the left of E₁) are not desirable. Similarly, points above E₂ involve higher costs, while lower points are not attainable due to output constraint.