Answer to Question #204454 in Financial Math for Thabiso Mohlakoana

Question

5. Parexel Co. produces and sells cell phone batteries. Its production function is 𝒒 = 𝟑𝟐𝟏𝑲𝟎.𝟑𝑳𝟎.𝟑, while its cost function is 𝒄 = 𝟐𝟏𝟎𝑲 + 𝟗𝟎𝑳. Parexel can sell its batteries at 𝒑 = 𝟔𝟑.

a. Does this production function exhibit increasing, decreasing or constant returns to scale? Explain briefly.

[2]

b. Does this production function exhibit decreasing returns to capital? Explain briefly. [2]

c. Use implicit differentiation to find the marginal rate of technical substitution, 𝑴𝑹𝑻𝑺. [2]

d. Write down an expression for the isoquant if 𝒒 = 𝟐𝟎𝟒𝟖. [2]

e. Write down an expression for the labour elasticity of production, 𝒆𝑳. [2]

f. Which levels of 𝑳 and 𝑲 satisfy the the first-order and second-order conditions for the maximisation of

Parexel’s profit? [8]

g. Write down the levels of profit, revenue, cost and output at the profit maximising levels of 𝑲 and 𝑳. [2]

Answer 1

a

$q_0=321K^{0.3}L^{0.3}$

multiply each input by m

$q=321(m)^{0.6}K^{0.3}L^{0.3}\\m^{0.6}q_0$

i.e output increases by less than m, hence decreasing return to scale

b.

$MP_k=(321)(0.3)L^{0.3}K^{-0.7}\\\frac{\delta MP_k}{\delta k}=(321)(0.3)(-0.7)L^{0.3}K^{-1.7}<0$

This implies MPk is decreasing

hence Diminishing marginal return

c.

$MRTS=\frac{\frac{\delta q}{\delta L}}{\frac{\delta q}{\delta K}}$

$=\frac{(321)(0.3)K^{0.3}L^{-0.7}}{(321)(0.3)K^{-0.7}L^{0.3}}$

$MRTS=\frac{K}{L}$

d.

$q=2048\\\implies2048=321K^{0.3}L^{0.3}\\\implies K^{0.3}L^{0.3}\\\implies K^{0.3}L^{0.3}=6.4$ Isoquant

e.

labor elasticity

$\in_L=(\frac{l}{q})(\frac{dq}{dl})$

$=\frac{l}{(321)K^{0.3}L^{0.3}}((321)(+0.3)K^{0.3}L^{-0.7})$

$\in=0.3$

f.

Profit will be mximized when

MPk=210

$(321)(0.3)K^{-0.7}L^{0.3}=210...................................(1)$

amd MPl=90

$(321)(0.3)K^{0.3}L^{-0.7}=-90..................................(2)$

by $\frac{(1)}{(2)}$

$\frac{L}{K}=\frac{21}{9}=\frac{7}{3}$

$L=\frac{7}{3}K$

from cost function

$2048=321(K^{0.3})(\frac{7}{3})^{0.3}K^{0.3}\\\implies K^{0.6}=8.2\sqrt2$

$K=33.52\\L=78.22$

g.

$cost (C)=210K+90L\\=210(33.52)+90(78.23)= 1079$

$Revenue=P\times Q= (63)(2048) =129024\\profit=TR-TC\implies129024-1079=14945$

From the question q is given as 2048 therefore,

$\\output=2048$