Answer to Question #232702 in Microeconomics for Anna Paula Quimbra

Question

1.     The following production function has been estimated for wheat:

where, W = Wheat yield (kg/ha), F = Fertiliser (kg/ha) and L = Labour (hours/ha)

Derive an expression for the least-cost expansion path given that fertiliser costs R12 per kg and labour costs R6 per hour.                                                                                       

Answer 1

The complete question is:

 The following production function has been estimated for wheat:

$w=4F^{0.5}.L^{0.25}$

where, W = Wheat yield (kg/ha), F = Fertiliser (kg/ha) and L = Labour (hours/ha)

Derive an expression for the least-cost expansion path given that fertiliser costs R12 per kg and labour costs R6 per hour.       

Solution

At equilibrium, MRTS $= \frac{Price \space of\space fertilizer}{Price \space of\space Labor}$

$MRTS = \frac{MPF }{ MPL}$

$MPF = \frac{dW}{dF} = 4\times0.5(\frac{L^{0.25}}{F^{0.5}}) = 2\times\frac{L^{0.25}}{F^{0.5}}$

$MPL = \frac{dW}{dL} = 4\times0.25\frac{F^{0.5}}{L^{0.75}} = \frac{F^{0.5}}{L^{0.75}}$

$MRTS = 2\times\frac{L^{0.25}}{F^{0.5}} \times\frac{F^{0.5}}{L^{0.75}}$

$MRTS = 2(\frac{L}{F})$

MRTS $= \frac{Price \space of\space fertilizer}{Price \space of\space Labor}$

$2(\frac{L}{F}) = \frac{12 }{ 6} = 2$

L = F : equation of expansion path