Question

1.Suppose a firm is producing concrete with the following Cobb-Douglas production function:c = s0.5g 0.5where c is pounds of concrete, s is pounds of cement, and g is pounds of gravel. It wants to produce 100 pounds of concrete.

a.What is the marginal product of cement (i.e., by how much does output change if you add more cement, holding gravel constant)? What is the marginal product of gravel (i.e., by how much does output change if you add more gravel, holding cement constant)? How does the marginal product of gravel change if you add more cement? If you add more gravel? Show this mathematically (Hint: think second derivative)

b. Calculate the marginal rate of technical substitution (MRTS).Plot five different combinations of cement and gravel that would allow you to produce 100 pounds of concrete.Draw a series of isoquantson the same axes.

c.Suppose the firm is using 100 pounds of cement and 100pounds of gravel to produce the 100 pounds of concrete. Suppose further than the price of cement is $10 per pound and the price of gravel is $5 per pound. How much does it cost to produce 100 pounds of concrete? Find a combination of gravel and cement that would allow you to produce 100 pounds of concrete at a lower cost.

d.In this part of the question, we will look at how output changes when inputs are doubled.

i.What happens to output if the firm doubles both inputs, from 100 to 200?

ii.Suppose that the production function is instead c = s0.6g 0.6. How much output will 100 pounds of cement and 100 pounds of gravel produce now? What happens to output if the firm doubles both inputs?

iii.Suppose that the production function is instead c = s0.4g 0.4. How much output will 100 pounds of cement and 100 pounds of gravel produce now? What happens to output if the firm doubles both inputs?

iv.What does this tell you about the coefficients on a Cobb-Douglas production function?

Answer 1

c = s^0.5g^0.5

The firm wants to produce 100 pounds of concrete.

(a) Marginal product of cement( Differentiate the production function with respect to s keeping g constant) = MP_c = 0.5 g^0.5/s^0.5

Marginal product of gravel( Differentiate the production function with respect to g keeping s constant) = MP_g = 0.5 s^0.5/g^0.5

d(MP_g)/ds = 0.25/(s^0.5g^0.5)

d(MP_g)/dg = (-0.25 s^0.5)/g^1.5

(b) MRTS = (dc/ds)/(dc/dg)

MRTS = MP_c/MP_g = g/s

(c) If the price of cement is $10 per pound and the price of gravel is $5 per pound then the the cost of 100 pounds of cement and 100 pounds of gravel to produce 100 pounds of concrete is ($10 × 100) + ($5 × 100) = $1500

Now, We have to minimise the cost of producing concrete such that s^0.5g^0.5 = 100

To do this, The optimal condition is to equate the slopes of the isoquant and the price line. Therefore,

MRTS = P_s/P_g

g/s = $10/$5

g/s = $2

g = 2s

we know that, s^0.5g^0.5 = 100

Hence, 2^0.5s = 100

s = 70.711 = 71(approx.)

Therefore, g = 142(approx.)

The combination of cement and gravel that would allow us to produce 100 pounds of concrete at lower cost is (71,142).

(d) (i) c = (100^0.5)(100^0.5) = 100

When inputs get doubled from 100 to 200 then

c = (200^0.5)(200^0.5) = 200

Hence, Doubling the inputs doubles the output.

(ii) Now, The production function is c = s^0.6g^0.6

When s = 100 and g = 100 then

c = (100^0.6)(100^0.6) = 251.188 = 251(estimate)

Now, Doubling the inputs,

c = (200^0.6)(200^0.6) = 577.079

Here, Doubling the inputs more than doubles the output.

(iii) c = s^0.4g^0.4

c = (100^0.4)(100^0.4) = 39.81 = 40(approx.)

Now, Doubling the inputs

c = (200^0.4)(200^0.4) = 69.314

In case of this production function, Doubling the inputs less than doubles the output.

(iv) Doubling the inputs cause output to rise by 2^{(a + b)} where a and b are the coefficients. So, If

(a + b) = 1 then production function will exhibit constant returns to scale

(a + b)>1 then production function will exhibit increasing returns to scale.

(a + b)<1 then production function will exhibit decreasing returns to scale.