Question

Consider the information in the following table for GAD's restaurant. Marginal product of capital 3,000 pizzas per oven Marginal product of labor 1,200 pizzas per worker Wage rate $300 per week Rental price of ovens $600 per week Briefly explain whether Gad is minimizing costs. If he is not minimizing costs, explain whether he should rent more ovens and hire fewer workers or rent fewer ovens, and hire more workers.

Answer 1

Let us say a production function,

Y = f(K, L)

Here, MPL = dY/dL and MPK = dY/dK

The Cost function is

C = w.L + r.K

Where, w = wage rate and r = rental price.

For a certain level of output Y°, if we minimize cost, then the Lagrange's Equation is,

J = w.L + r.K + a.[Y° - f(K, L)]

Where, a>0 is the Lagrange's Multiplier.

Now, the First Order Derivatives or FOCs are

dJ/dL = 0

or, w - a.MPL = 0

or, w = a.MPL..........(1)

dJ/dK = 0

or, r - a.MPK = 0

or, r = a.MPK..........(2)

And, dJ/da = 0

or, Y° = f(K, L).........(3)

Now, deviding (1) by (2), we get

MPL/MPK = w/r.........(4)

This condition must be satisfied for cost minimization.

Now, according to the given information in the question, we are told that,

Marginal product of capital 3,000 pizzas per oven and Marginal product of labor 1,200 pizzas per worker. Hence,

MPL = 1200 and MPK = 3000

Wage rate $300 per week and Rental price of ovens $600 per week. Hence,

w = $300 and r = $600

Hence, from the above informations, we get

MPL/MPK = 1200/3000

or, MPL/MPK = 0.4

And, w/r = 300/600

or, w/r = 0.5

Hence, we can see that,

MPL/MPK < w/r

Equation (4) is not satisfied here.

Hence, GAD is not minimizing cost.

✓ Here, we can see that, MPL/MPK is smaller than w/r. GAD need to equate these two ratios for cost minimization i.e. MPL/MPK = w/r.

Either GAD can decrease (w/r) or we can increase (MPL/MPK).

Here, w and r is fixed. Hence, GAD can not change the ratio (w/r).

But, GAD can change the (MPL/MPK) ratio. We have to increase the ratio to equate (w/r) ratio.

We know that,

• Hiring more workers (L), decreases MPL and Hiring fewer workers increases MPL.

• Hiring more ovens (K), decreases MPK and Hiring fewer ovens increases MPK.

To increase MPL/MPK, GAD have to increase MPL and decrease MPK.

Hence, GAD should hire fewer workers and more ovens.

Hope the explanation is clear to you my friend.