Question

What is meant by an expansion path? Illustrate expansion paths for a normal input and an inferior input.

Answer 1

An expansion path is a graph which shows how a firm's cost minimizing input mix changes as it expands it's production in relation to capital and labor or it can be said to be the tangency between Iso-cost line and the Isoquant curve.

An input is a normal input if the firm increases its proportion in its production mix as it increases production. An inferior input, on the other hand, is an input whose proportion decreases as the firm switches to other inputs at higher production level. The expansion path slopes away from an inferior input i.e. it has negative slope. But in case of a normal input, the expansion path has a positive slope.

If given a production function,

$Q\ (\ L , \ K \ )\ =\ 50L^{ 0.5}K^{0.5}\\ Tangency\ condition\\ MRTS_{L,K}=\frac{MP_{L}}{MP_K}=\frac{K}{L}=\frac{w}{r}\\ \implies\ \frac{w}{r} \cdot L\\ \implies\ This\ is\ the\ equation\ for\ expansion\ path\\ Where\ MRTS\ is \implies Marginal\ Rate\ of\ Tecnical\ substitution\\ Isoquant\ constraint\\ 50L^{ 0.5}K^{0.5}=Q_{0}\\ \implies 50L^{ 0.5}(\frac{wL}{r})^{0.5}=Q_{0}\\ \implies L \cdot (Q, w,r)=\frac{Q}{50} \cdot (\frac{r}{w})^{0.5}=K \cdot (Q, w,r)=\frac{Q}{50} \cdot (\frac{w}{r})^{0.5}\\ K \cdot (Q, w,r)=\frac{Q}{50} \cdot (\frac{w}{r})^{0.5}------>Answer$