## Long-Run Optimality for a Profit-Maximising Firm (Numerical Problem)

To explain this point, let us consider three isoquants I, II and III, out of which isoquant II is tangent to the iso-cost AB. Isoquant III is located right to it and isoquant I is located left of isoquant II.

Isoquant III (or any isoquant higher than isoquant II) yields more output than isoquant II and hence is preferred to isoquant II, but it violates cost constraint since it lies at the right side of the iso-cost line. So isoquant III is unattainable.

Isoquant I (or any isoquant lower than isoquant II) satisfies the cost constraint but yields less production than isoquant II. Thus, isoquant II which is tangent to isocost is the only one that satisfies both the conditions.

Mathematically, maximisation of production within a cost constraint requires one condition to be fulfilled – the slope of isoquant should be equal to the slope of isocost curve.

In figure 8.17, movement from point A to B indicates ‘loss’ of output arising from withdrawal of capital by ?K which is just compensated by ‘gain’ in output by increasing labour input by ?L, since A and B yield same quantity of output. Thus, loss in total productive capacity of ?K amount of capital, i.e., ?K ? MPK must be equal to gain in total productive capacity of ?L quantity of labour i.e., ?L x MPL.

Again, MRTS measures the slope of AB segment of the given isoquant. If distance between A and B reduces, ultimately it will converge to a point. So, MRTS also measures slope of isoquent at a particular point.

We have already mentioned that the point at which production of a given quantity of output is possible at least cost, must ensure equality between slope of isocost line (i.e., – w/r) and the slope of isoquant (i.e, MRTS) at that point.

The above condition ensures maximisation of output within the cost constraint.

#### Worked out Numerical Problem:

The production function is given as Q = K2 + L2 where w = Rs. 4 and r = Rs. 6. If the total cost of the firm is Rs. 780, calculate the maximum number of units that can be produced within the cost constraint.