Question

QUESTION 7

1. Determine which of the following statement below are correct. Multiple statements may be correct.

   A firm's production function is equal to Q = K^(1/2) L^(1/2) and the Marginal Product of Labor is equal to: MP(L) = 1/2 * K^(1/2) * L^(-1/2) Capital is currently fixed at 81 units. Labor increases from 100 units to 256 units. As a result, the Marginal Product of Labor falls from 0.45 to 0.28. This result means that the productivity of the last unit of labor falls from 0.45 units to 0.28 units. As the MP(L) falls, the Average Productivity of Labor will also fall.

   A firm's production function is equal to Q = K^(1/2) L^(1/2) and the Marginal Product of Labor is equal to: MP(L) = 1/2 * K^(1/2) * L^(-1/2) Capital is currently fixed at 256 units. Labor increases from 100 units to 256 units. As a result of labor increasing from 100 units to 256 units: (1) Output Increases from 160 units to 256 units (2) AP(L) decreases from 1.6 units per worker to 1 unit per worker (3) MP(L) decreases from 0.8 units to 0.5 units for the last worker employed

   A firm's production function is equal to Q = K^(1/2) L^(1/2) and the Marginal Product of Labor is equal to: MP(L) = 1/2 * K^(1/2) * L^(-1/2) You know that capital (K) is currently fixed at 100 units. When labor increases from 49 to 81 units, the Average Product of Labor declines from 1.43 to 1.11.

   A firm's production function is equal to Q = K^(1/2) L^(1/2) and the Marginal Product of Labor is equal to: MP(L) = 1/2 * K^(1/2) * L^(-1/2) You know that capital (K) is currently fixed at 100 units. When labor (L) increases from 36 to 49 units, the MP(L) of labor increases from 0.43 to 0.5.

Answer 1

1).

So, here the production function is “Q=K^0.5*L^0.5”, => MPL = dQ/dL = (1/2)*K^0.5*L^(-0.5). Now, “K=81” and “L” increases from “100” to “256”.

So, MPL= (1/2)*K^0.5*L^(-0.5), MPL= (1/2)*81^0.5*100^(-0.5) = (1/2)*(9/10) = 0.45.

MPL= (1/2)*K^0.5*L^(-0.5), MPL= (1/2)*81^0.5*256^(-0.5) = (1/2)*(9/16) = 0.28. So, as “L” increase from “100” to “256”, => MPL decreases from “0.45” to “0.28”. Now, given the production function the “MPL” as well as the “APL” are downward sloping, => as “L” increases both “MPL” and “APL” decreases. So, the 1^st statement is “TRUE”.

2).

So, given the production function the “MPL” is given by, “MPL = 0.5*K^0.5*L^(-0.5)”. Now, “K=256” and “L” increases from “100” to “256”.

=> Q = K^0.5*L^0.5, => Q = 256^0.5*100^0.5 = 160 and for “L=256” the corresponding “Q” is given by.

=> Q = 256^0.5*256^0.5 = 256, => “Q” increases from “160” to “256”.

Now, APL is given by, “APL = Q/L”. So, for “L=100” the corresponding APL is given by.

=> APL = 160/100 = 1.6 and for “L=256” the APL is given by, “APL = 256/256 = 1”, => APL is falling from “1.6” to “1”.

Now, the MPL is given by, “MPL = 0.5*K^0.5*L^(-0.5)”. So, for “L=100” the MPL is given by.

=> MPL = 0.5*256^0.5*100^(-0.5) = 0.5*(16/10) = 0.8. Now, for “L=256” the MPL is given by.

=> MPL = 0.5*256^0.5*256^(-0.5) = 0.5*(16/16) = 0.5, => “MPL” is falling from “0.8” to “0.5”. So, the 2^nd statement is also “TRUE”.

3).

Now, given the production function the output is given by for “L=49”, => Q=K^0.5*L^0.5=100^0.5*49^0.5=70. Now, the level of output is given by for “L=81”, => Q=K^0.5*L^0.5=100^0.5*81^0.5=90. So, here the “APL” is given by, => APL = 70/49 = 1.43 and for “L=81” the APL = 90/81=1.11, => the 3^rd statement is also “TRUE”.

4).

Now, given the production function the MPL is given by.

=> MPL = 0.5*K^0.5*L^(-0.5), which is decreasing in “L”, => as “L” increases “MPL” decreases. So, here as “L” increases from “36” to “49” the “MPL” must decrease, => the 4^th statement is “FALSE”.