Answer to Question #130984 in Microeconomics for yasir

Question

What is the slope of an iso-cost line equal to and why? Provide mathematical explanation

Answer 1

The slope of an iso-cost is equal to the ratio of factor prices, that is, $slope = \dfrac {w}{r}$ where w is the price of labour, wage rate, and r is the price of capital, rent. It hence represents the relative prices of factor inputs, and can be taken to represent the opportunity cost of labour.

Firstly, an iso-cost is a locus of points of factor inputs, capital (K) and labour (L), that cost the same amount. The equation of the iso-cost is therefore written, $C = rK + wL$ , where K and L are units of capital and labour respectively, and C is the total cost.

When no labour is hired, $L = 0 \space units, \space C =rK \\ and \space K = \dfrac {C}{r}$

Also, When no capital is hired,

$K = 0 \space units, \space C = wL \\ and \space L = \dfrac {C}{w}$

With capital (K) plotted on the vertical axis and labour on the horizontal axis,

$Slope = \dfrac {K}{L}$

$= \dfrac {\dfrac {C}{r}}{\dfrac {C}{w}}$

$= \dfrac {C}{r}×\dfrac{w}{C}$

$= \dfrac {w}{r}$

$= \dfrac {price \space of \space labour}{price \space of \space capital}$

$= \dfrac {wages}{rent}$

The gradient is always negative.

Alternatively, since the iso-cost is a straight line, it must fit the general form of the equation of a straight line

$y = mx + c$

Now, reducing $C = rK + wL$ into the general form by making K the subject of the formula gives:

$C - wL = rK$

dividing by r throughout gives:

$K = \dfrac {C}{r} - \dfrac {wL}{r}$

$K = -\dfrac {w}{r}L + \dfrac {C}{r}$ , where the gradient

is $- \dfrac {w}{r}$ as shown above.

Thus, the gradient of the iso-cost is the wage-rent ratio, that is, it is the ratio of the relative prices of factor inputs, labour and capital.