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..