# Isoquants and MRTS: Solved Numerical Questions

1- The production function of a firm is given by Q=4K^(0.75)L^(0.25). Assume that the wage rate is equal to $60 and the price of capital per unit is equal to$20. Now, find out the least cost combination of labor and capital and total cost of production for producing 200 units of output.

Solution:
Given Q=4L^(0.75)K^(0.25)
w=$60, r=$20 and Q=1000

Required equilibrium conditions:

MRTS_(LK)=\frac{MP_L}{MP_K} and

MRTS_(LK)=\frac{w}{r}

Or, \frac{MP_L}{MP_K}=\frac{w}{r}.....(i)

Equation (i) requires MP_L and MP_k. So let us first calculate  MP_L and MP_K from the given production function.

MP_L=\frac{dQ}{dL}

=\frac{d(4L^(0.75)K^(0.25))}{dL}

=3L^(-0.25)K^(0.25)......(ii)

MP_K=\frac{dQ}{dK}

=\frac{d(4L^(0.75)K^(0.25))}{dK}

=L^(0.75)K^(-0.75)......(iii)

Plotting equation (ii) and (iii) in equation (i) we, get the value as;

\frac{3L^(-0.25)K^(0.25)}{L^(0.75)K^(-0.75)}=\frac{w}{r}

Or,\frac{3K^(0.75)K^(0.25)}{L^(0.75)L^(0.25)}=\frac{w}{r}

Or, \frac{3}{L}=\frac{w}{r}

Given w=$60 and r=$20

Or, \frac{3K}{L}=\frac{60}{20}

Or, \frac{K}{L}=1

Or, K=L

Given the production function Q=4L^(0.75)K^(0.25), now let us put the value of  K  and Q in production function, we get.

Or, 200=4L^(0.75)L^(0.25)
Or, 200=4L
Or, L=50
\because\L=K
\therefore\ K=50

Hence, least cost combination of input for producing 200 units of output is, L=50 and K=50

Total cost for the production of 200 units is as;

C=wL +rK

Putting the required values in the above cost function, we get the total cost of producing 200 units of output.
=60(50)+20(50)
=3000+1000
=$4000 2- Given the production function Q=10K^(0.4) L^(0.6), wage rate w=$15,  and price per unit of capital r=$10, Find out: a) Optimal combination of inputs for producing 400 units of output, b) Minimum cost of production, Solution: #### a) Calculation of optimal combination of inputs Given the production function, Q=10K^(0.4)L^(0.6).....(i) Let us calculate marginal productivity of labor by taking derivative of production function with respect to labor. MP_L=\frac{dQ}{dL} =\frac{d(10K^(0.4)L^(0.6))}{dL} =6K^(0.4)L^(-0.4)......(ii) Let us calculate marginal productivity of capital by taking derivative of production function with respect to capital, MP_K=\frac{dQ}{dK} =\frac{(10K^(0.4)L^(0.6))}{dK} =4K^(-0.6)L^(0.6)......(iii) Required equilibrium conditions: MRTS_(LK)=\frac{MP_L}{MP_K} and MRTS_(LK)=\frac{w}{r} Or, MRTS_(LK)=\frac{MP_L}{MP_K}=\frac{w}{r} Putting the values in the above equation we get the optimal input combination as, \frac{6K^(0.4)L^(-0.4)}{4K^(-0.6)L^(0.6)}=\frac{w}{r} Or, \frac{6K^(0.4)K^(0.6)}{4L^(0.4)L^(0.6)}=\frac{w}{r} Or, \frac{6K}{4L}=\frac{w}{r} Or, \frac{6K}{4L}=\frac{15}{10} Or, 60K=60L Or, K=L......(iv) Putting the value of L in production function, we get. Q=10K^(0.4)K^(0.6) Q=10K......(v) Putting the units of given quantity in equation (v) we get the units of capital hired to produce. Or,400=10K Or, K=40 \therefore\K=40 Putting the value of K in equation (iv) we get the number of labor employed to produce. 40=L \therefore\L=40 Hence the optimal input combination for the production of 400 units of output is L=40 and K=40. #### b) Calculation of minimum cost of production Minimum cost of production is calculated from the cost function as given below. C=wL+rK Let us substitute wage rate 15 for w, labor hired 40 for L, capital price per unit 10 for r and units of capital hired 40 for K and we get the minimum cost of production. C=15×40+10×40 C=600+400 C=$1000

3- Given production function Q=K^(0.4) L^(0.6), capital per hour unit price  r=$8 and wage rate per hour w=$12,
find out:
a) Find the optimum combination of inputs at minimum cost,
b) Which budget $900,$1200 or $1500 is the cost minimizing budget for the production of 60 units of output. Solution: #### a) Calculation of optimum combination of inputs Given the production function, Q=K^(0.4)L^(0.6).....(i) Let us calculate marginal productivity of labor by taking derivative of production function with respect to labor. MP_L=\frac{dQ}{dL} =\frac{d(K^(0.4)L^(0.6))}{dL} =0.6K^(0.4)L^(-0.4)......(ii) Let us calculate marginal productivity of capital by taking derivative of production function with respect to capital, MP_K=\frac{dQ}{dK} =\frac{(K^(0.4)L^(0.6))}{dK} =0.4K^(-0.6)×L^(0.6)......(iii) Required equilibrium conditions: MRTS_(LK)=\frac{MP_L}{MP_K} and MRTS_(LK)=\frac{w}{r} Or, MRTS_(LK)=\frac{MP_L}{MP_K}=\frac{w}{r} Putting the values in the above equation we get the optimal input combination as, \frac{0.6K^(0.4)L^(-0.4)}{0.4K^(-0.6)L^(0.6)}=\frac{w}{r} Or, \frac{0.6K^(0.4)K^(0.6)}{0.4L^(0.6)×L^(0.4)}=\frac{w}{r} Or, \frac{0.6K}{0.4L}=\frac{w}{r} Or, \frac{0.6K}{0.4L}=\frac{12}{8} Or, 48K=48L Or, L=K......(iv) Given the production function, Q=K^(0.4)L^(0.6) Putting the value of L, and Q to be produced in production function, we get the hours of capital hired. 60=K^(0.4)K^(0.6) 60=K \therefore\K=60 Putting the value of K in equation (iv) we get the hours of labor hired for production. L=K L=60 \therefore\L=60 Hence, optimum combination of inputs, L=60 and K=60 #### b) Calculation of cost minimizing budget, Cost function for the equilibrium is symbolized as, C=wL+rK Putting the required values in the above cost function, we get the minimum cost for the production of 60 units of output. C=12(60)+8(60) C=720+480 C=$1200

Hence cost minimizing budget is $1200. Budget$900 is less than $1200, and not sufficient to produce 60 units. Similarly budget$1500 is more than \$1200 and not desirable..