Answer to Question #219849 in Microeconomics for Sips

Question

how that the cost function for a firm with the constant returns Cobb–Douglas production function y = Az1^a z2^1-a is given C( p, y) = yp1^a p1^1-a B , where B is a function of A and a only. Sketch the cost curves. Derive the conditional input demands

Answer 1

$C( p, y) = yp1^{a} p1^{1-a} B\\ Average\space cost = \frac{C}{y} = p1^a p1^{1-a} B\\ Marginal \space Cost (MC) = \frac{dC}{dy} = p1^a p1^{1-a} B\\$

When a rise in inputs like capital and labor will cause the same proportional increase in output is constant returns to scale. Here, the costs curves ATC and AVC at CRS are at their minimum as there is no intention to move from this point. The MC curve cuts both the ATC and AVC at it minimum.

Finding profit function:

$Profit = TR - C$

$Profit = py -yp1^a p1^{1-a} B$

$Replacing y = Az1^a z2^{1-a}$

$Profit = p( Az1^a z2^{1-a} ) -( Az1^a z2^{1-a} )p1^a p1^{1-a} B$

For input z1:

$MRTS =\frac{dProfit}{dz1} =pa Az1^{a-1 }z2^{1-a} - aAz^{1a-}1 z2^{1-a} p1^a p2^{1-a} B \\ = \frac{p1}{p2} z1^{a-1} (pa A z2^{1-a} - aA z2^{1-a} p1^a p2^{1-a} B )\\ z1^{a-1 } =\frac{ p1 }{ p2}( pa A z2^{1-a} - aA z2^{1-a} p1^a p2^{1-a} B )\\ z1 = {\frac{p1 }{ p2}( pa A z2^{1-a} - aA z2^{1-a} p1^a p2^{1-a} B )} ^{\frac{1}{(a-1)}}$

Conditional input demand for $z1 = {\frac{p1 }{ p2}( pa A z2^{1-a} - aA z2^{1-a} p1^a p2^{1-a} B )} ^{\frac{1}{(a-1)}}$

For input z2:

$\frac{dProfit}{dz2} =p(1-a) Az1^{a-1} z2^{-a }- (1-a)Az1^{a-1} z2^{-a }p1^a p1^{1-a} B =\frac{ p1}{p2}\\ z2^{1-a} = \frac{p1}{p2} (p (1-a) Az1^{a-1} - (1-a) Az1^{a-1}p1^a p1^{1-a} B) \\ z2 = {\frac{p1}{p2} (p (1-a) Az1^{a-1} - (1-a) Az1^{a-1}p1^a p1^{1-a} B)}^{\frac{1}{(1-a)}}$

$z2 = {\frac{p1}{p2} (p (1-a) Az1^{a-1} - (1-a) Az1^{a-1}p1^a p1^{1-a} B)}^{\frac{1}{(1-a)}}$

Conditional input demand for z1$= {\frac{p1}{p2} (p (1-a) Az1^{a-1} - (1-a) Az1^{a-1}p1^a p1^{1-a} B)}^{\frac{1}{(1-a)}}$