Suppose the firm's cost function is given by C(Q)= 100 + Q + 2*Q^2 + 3*Q^3

Question 1:

What are the marginal and average total costs, respectively?

Given the total cost function

• C(Q)=100+Q+2Q2+3Q3C(Q)=100+Q+2Q2+3Q3

The marginal cost will be equal to:

• MC=ΔTCΔQ=1+4Q+9Q2MC=ΔTCΔQ=1+4Q+9Q2

At Q = 2, the marginal cost is equal to:

• MC=1+4(2)+9(2)2=$45MC=1+4(2)+9(2)2=$45

The average total cost will be equal to:

• ATC=C(Q)Q=100+Q+2Q2+3Q3Q=100Q+1+2Q+3Q2ATC=C(Q)Q=100+Q+2Q2+3Q3Q=100Q+1+2Q+3Q2

At Q = 2, the average total cost will be equal to:

• ATC=1002+1+2(2)+3(2)2=$67ATC=1002+1+2(2)+3(2)2=$67

Question 2:

What should a firm charge for a package of 3 t-shirts if a typical consumer's inverse demand function is P=10-Q?

This is block pricing scheme.

Block pricing yields the same level of profits as perfect price discrimination.

Equating the inverse demand curve to the marginal cost:

• 10−Q=MC10−Q=MC

Substituting for Q = 3:

• 10−Q=MC10−Q=MC

• 10−3=MC10−3=MC

Thus, the marginal cost is equal to MC = $7.

The consumer surplus between the demand curve and the marginal cost is equal to:

• CS=12(10−7)×3=$4.5CS=12(10−7)×3=$4.5

Thus, the price that the firm should charge for a package of 3 t-shirts is equal to:

• P=MC∗3+CSP=MC∗3+CS

• P∗=7∗3+4.5=$25.5P∗=7∗3+4.5=$25.5

Question 3:

If output is produced according to Q = 4l + 6k, (l is the quantity of L and k is the quantity of K) the price of K is $12, and the price of L is $6, then the cost-minimizing combination of K and L capable of producing 60 units of output is?

With a linear production function, the optimal input mix involves using some units of one input and zero units of the other input. The choice of the input to use depends on the marginal product per dollar spent on each input.

The firm will use capital only if the marginal product per dollar spent on capital is higher than the marginal product per dollar spent on labor. That is:

• MPKr>MPLwMPKr>MPLw

The firm will use labor only if the marginal product per dollar spent on labor is higher than the marginal product per dollar spent on capital. That is:

• MPLw>MPKrMPLw>MPKr

From the production function, the marginal product of labor is equal to:

• MPL=δQδL=4MPL=δQδL=4

The price of labor is w = 6. Thus, the marginal product per dollar spent on labor is equal to:

• MPLw=46=2/3MPLw=46=2/3

The marginal product of capital is equal to:

• MPK=δQδK=6MPK=δQδK=6

The price of capital is r = 12. Thus, the marginal product per dollar spent on capital is equal to:

• MPKr=612=0.5MPKr=612=0.5

Since MPLw=2/3>MPKr=0.5MPLw=2/3>MPKr=0.5, the firm will use only labor.

To produce 60 units of output, the quantity of labor that the firm will hire is equal to:

• Q=4l+6kQ=4l+6k

• 60=4l+6(0)60=4l+6(0)

• l∗=604=15l∗=604=15 units.