Question

Consider the case of a firm that produces output x (sold at price p) using a production function x = A*/*klaße, where / is labor, kis capital, and e is energy (for example, oil or electricity). a) What is the interpretation of A? b) Under what condition(s) does the production function exhibit constant returns to scale? Is it homogeneous? Are the marginal products of inputs increasing, constant, or decreasing? c) Set up the profit maximization problem for the firm. d) Find the optimality conditions for all the inputs in the production function. e) Find the input demand functions for labor, capital, and energy. f) Find the optimal supply function for x. g) Set up the cost minimization problem. Show the relevant Technical Rates of Substitution. h) Assume that the market is populated by N consumers and M firms and that the production function is the one given above. Do you expect the market to provide the most efficient allocation? Under what conditions would the allocation be inferior to the one provided by a benevolent social planner?

Answer 1

a) A* is the productivity parameter of production function . If production technology advances , it reflects in production function as an increase in A*.

b) A production function F (x,y,z) is said to show constant returns to scale if it is homogeneous of degree 1 i.e., if all factors are increased by t units , the output also increases by t units. Let F(x,y,z) = x

F(tx, ty, tz) = t F(x,y,z)= tx

In this case production function is F(l,k,e) = A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$

F(tl, tk,ek) = A^* (tl)^$\alpha$ (tk)^{1-$\alpha$-$\beta$} (te)^$\beta$ = t^{$\alpha$-(1-$\alpha$-$\beta$)-$\beta$}A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ = t F(l,k,e)

Hence the production function always show constant returns to scale ,in any conditions.

Marginal product of labour MP_{​​​​​L} = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l = $\alpha$ A^* l^$\alpha$-1 k^{1-$\alpha$-$\beta$}e$\beta$ = $\alpha$ A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /l = $\alpha$ F(l,k,e)/l

If $\alpha$ >0 the marginal product of labour decreases as more and more labour are emloyed.

If $\alpha$ = 0 Marginal product of labour is constant = 0

If $\alpha$ <0 Marginal product of labour is negative and increasing

Similarly , marginal product of capital = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k = (1- $\alpha$ -$\beta$)A^* l^$\alpha$k^{-$\alpha$-$\beta$}e$\beta$ =(1- $\alpha$ -$\beta$) A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /k = (1-$\alpha$-$\beta$) F(l,k,e)/k

If 1- $\alpha$ -$\beta$>0 the marginal product of capital decreases as more and more labour are emloyed.

If 1- $\alpha$ -$\beta$= 0 Marginal product of capital is constant = 0

If 1-$\alpha$-$\beta$<0 Marginal product of capital is negative increasing

Marginal product of energy MP_​​​e = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e= $\beta$ A^* l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$^-1 = $\beta$ A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /e= $\beta$ F(l,k,e)/e

If $\beta$ >0 the marginal product of energy decreases as more and more labour are emloyed.

If $\beta$ = 0 Marginal product of energy is constant = 0

If $\beta$ <0 Marginal product of energy is negative and increasing

c) Let us denote wages by w , cost of energy by c and cost of using capital by r

Profit = Revenue - total cost

Profit = px - ( wl - ce - rk )

Profit =  pA^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ - wl - ce - rk is the profit maximisation problem

d) Optimality condition for all inputs in the production function is derived by first order condition of profit maximisation

$\partial$(profit)/$\partial$l = 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l - w = 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l = MP_{​​​​​L}​

Therefore optimality condition for labour input is MP_​​L​ = w ..

Similarly we get

$\partial$(profit)/$\partial$k = 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k - r= 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k= MP_{​​​​​K}

Therefore optimality condition for capital. input is MP_​​K= r

$\partial$(profit)/$\partial$e= 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e - c= 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e= MP_e

Therefore optimality condition for energy input is MP_​e= c

Answer 2

a) A* is the productivity parameter of production function . If production technology advances , it reflects in production function as an increase in A*.

b) A production function F (x,y,z) is said to show constant returns to scale if it is homogeneous of degree 1 i.e., if all factors are increased by t units , the output also increases by t units. Let F(x,y,z) = x

F(tx, ty, tz) = t F(x,y,z)= tx

In this case production function is F(l,k,e) = A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$

F(tl, tk,ek) = A^* (tl)^$\alpha$ (tk)^{1-$\alpha$-$\beta$} (te)^$\beta$ = t^{$\alpha$-(1-$\alpha$-$\beta$)-$\beta$}A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ = t F(l,k,e)

Hence the production function always show constant returns to scale ,in any conditions.

Marginal product of labour MP_{​​​​​L} = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l = $\alpha$ A^* l^$\alpha$-1 k^{1-$\alpha$-$\beta$}e$\beta$ = $\alpha$ A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /l = $\alpha$ F(l,k,e)/l

If $\alpha$ >0 the marginal product of labour decreases as more and more labour are emloyed.

If $\alpha$ = 0 Marginal product of labour is constant = 0

If $\alpha$ <0 Marginal product of labour is negative and increasing

Similarly , marginal product of capital = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k = (1- $\alpha$ -$\beta$)A^* l^$\alpha$k^{-$\alpha$-$\beta$}e$\beta$ =(1- $\alpha$ -$\beta$) A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /k = (1-$\alpha$-$\beta$) F(l,k,e)/k

If 1- $\alpha$ -$\beta$>0 the marginal product of capital decreases as more and more labour are emloyed.

If 1- $\alpha$ -$\beta$= 0 Marginal product of capital is constant = 0

If 1-$\alpha$-$\beta$<0 Marginal product of capital is negative increasing

Marginal product of energy MP_​​​e = $\partial$ (A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e= $\beta$ A^* l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$^-1 = $\beta$ A*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ /e= $\beta$ F(l,k,e)/e

If $\beta$ >0 the marginal product of energy decreases as more and more labour are emloyed.

If $\beta$ = 0 Marginal product of energy is constant = 0

If $\beta$ <0 Marginal product of energy is negative and increasing

c) Let us denote wages by w , cost of energy by c and cost of using capital by r

Profit = Revenue - total cost

Profit = px - ( wl - ce - rk )

Profit =  pA^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ - wl - ce - rk is the profit maximisation problem

d) Optimality condition for all inputs in the production function is derived by first order condition of profit maximisation

$\partial$(profit)/$\partial$l = 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l - w = 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$l = MP_{​​​​​L}​

Therefore optimality condition for labour input is MP_​​L​ = w ..

Similarly we get

$\partial$(profit)/$\partial$k = 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k - r= 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$k= MP_{​​​​​K}

Therefore optimality condition for capital. input is MP_​​K= r

$\partial$(profit)/$\partial$e= 0 or $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e - c= 0

And we know from part b that $\partial$ ( A^*l^$\alpha$k^{1-$\alpha$-$\beta$}e$\beta$ )/$\partial$e= MP_e

Therefore optimality condition for energy input is MP_​e= c