Question

1.) A firm manufactures bottles with the following technology: q =f (L) = 100√L, where q denotes bottles and L, the number of workers.

1. (a) How many workers are needed to produce q = 500 bottles? If the firm doubles that amount of workers, will they be able to double the amount of output?

2. (b) Compute the marginal productivity of labor, MPL(L) = f0(L).

3. (c) Show that MPL(L) is decreasing.

2. In problem 1, imagine that wage is equal to 10. The firm’s cost function, c(q), is obtained as follow: If L(q) is the amount of labor required to produce q bottles, then c(q) = 10L(q) — the total amount they spend on wages for the required amount of labor to produce q bottles.

(a) Compute the formula for c(q).

(b) Compute the marginal cost, MC(q) = c0(q). utility parameter.

(c) Show that MC(q) is increasing.

Answer 1

Ans (1)

Ans (a) The Production function given in the question is as follows:-

q=f(L) =100$\small \sqrt{L}$

Here, in this question, it is asked to calculate the number of workers that can produce q=500 bottles which can be calculated in the following manner:-

Putting the value of q in the production function we get:-

500 = 100$\small \sqrt{L}$

Squaring both the sides we get:-

(500)² =(100)²*L

So, L=500*500/100*100

L = 25 workers

Thus, 25 workers are required to produce 500 bottles.

However, if the firm doubles the number of workers from 25 to 50, the number of Bottles that can be produced can be calculated as follows:-

q= f(L)

= 100*$\small \sqrt{25}$

= 100*5=500 units

Thus, if the firm doubles the number of workers, the level of output will remain the same .i.e 500 units.

Ans(c) The Marginal Productivity of Labour is defined as the change in the output associated with a change in that factor holding other factors into production as constant. Thus, the Marginal Productivity of Labour be calculated in the following manner:-

Change in Output (Y)/Change in Labour (L)

=$\small \Delta Y/\Delta L$

=500 units -500 units /50-25

=0/25=0

Thus MPL(L)= f0(L)

The Marginal Productivity of Labour is directly related to the costs of production. However, the costs can be divided between fixed and variable costs. Fixed Costs are the costs which remain fixed irrespective of the level of output whereas variable costs are the ones which vary with the level of output. The concept of Diminishing Marginal Productivity of Labour lies in the fact that by increasing one input and by keeping other inputs at the same level will initially increase the output, but any further increase in the input will have a limited effect and will eventually have no effect or negative effect on the output.

When the level of output q was 500 units, the number of workers required to produce that output was 25. However, if the firm doubled the number of workers to 50, the output level remained the same.i .e. 500 units. Thus, with a change in the number of workers, there was no change in the level of output. , thereby explaining the reason behind a decreasing MPL(L).

ANS (2)

Ans (a) A cost function expresses production costs in terms of the amount produced. It specifies the cost to produce q units.

The cost function that is given in the question is as follows:-

c(q)=10 L(q)

Where the wage is given as 10 and L(q) is the amount of Labour required to produce q bottles.

Thus, the formula for c(q) can be written as follows:-

c(q) =w^*L^*(q)

= 10*L(q)

In order to produce 500 units, we need 25 workers who are paid a wage at a rate of 10.

Putting these figures in the above cost function we get:-

c(q) = 10*25 workers *500 units =125000

However, if the number of workers is double the Total Cost will come out as:-

c(q) =10*50 workers *500 units =250000

Ans(b) Marginal Cost refers to the increase or decrease in the Total Production Cost if the output is increased by one more unit. It is calculated as:-

Marginal Cost = Change in costs /Change in Quantity

= 250000-125000/500-500= An infinite value.

Ans (c) When the number of workers was 25 the Total Cost was 125000, however, when the number of workers doubled up from 25 to 50, the Total cost increased to 250000.

The marginal Cost curve is U shaped initially when a firm increases its output, The Total Cost, as well as Variable Cost, starts to increase at a diminishing rate. As the Marginal Product of the variable input decreases due to the Law of Diminishing Marginal Returns, a firm must hire more of the variable input to get the same increase in output. Thus, in order to produce the same level as the output of 500 units, the firm has to employ more labor at an increased cost and thereby MC(q) also increases.