Question

Explain why cost minimization for each level of output implies that the firm hires that basket of inputs for which a). the marginal rate of technical substitution equals the input price ratio. b). the marginal products per dollar of the two inputs are equal. If you use a diagram in your answer, make the diagram is large and label all curves, axes, and points.

Answer 1

IC represents the Isocost line. IC0 and IC1 are two different Isocost lines and where r represents the rental rate of capital , w represents the wage rate for labor.

Slope of isoquant line is the ratio of input prices = -w/r.

Isoquant is shown by Q0.

Now, we can see at point G, it is not on isoquant . Although this input combination can produce Q0 but it is technically inefficient because firm would be wasting inputs. And input combination A can produce the same level of output Q0 by using fewer units of labor and capital.

Points E and F are technically efficient because lie on the isoquant but they are not cost-minimising . Because these points lies on the higher iso cost line.By moving from point E to A or F to A , the firm can produce the same amount of output, but at a lower total cost (represented by IC0).

At point A ,the slope of the isoquant is equal to the slope of the iso cost line.

Negative of the slope of the isoquant is the marginal rate of technical substitution i.e MRTS . MRTS = MPL/MPK.

It implies that slope of isoquant is -MRTS.

Slope of isocost line = -w/r

As stated earlier that cost minimising condition occurs:

-MRTS = -w/r

MRTS =w/r

It implies that at cost minimising level , marginal rate of technical substitution equals the input price ratio.

MPL/MPK =w/r

MPL/w = MPK/r

This condition tells that at a cost minimsing input combination , the additional output per dollar spent on labor equal the additional output per dollar spend on capital. That is marginal products per dollar of the two inputs are equal at cost minimising input combination.

Suppose take point E, At point E, slope of the isoquant is more than the slope of the isocost line. Therefore , MPL/MPK < w/r or MPL/w> MPK/r.

This implies that a firm operating at E could spend an additonal dollar on labor and save more than one dollar by reducing its capital amount in a manner that keeps output constant. This is not cost minimising input combination.