Answer to Question #130984 in Microeconomics for yasir

Question #130984
What is the slope of an iso-cost line equal to and why? Provide mathematical explanation
1
Expert's answer
2020-09-02T11:38:10-0400

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}\u00d7\\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.


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