Question

6.7 The production function

Q=KaLb where 0≤ a, b≤1 is called a Cobb-Douglas production function. This function is widely used in economic research. Using the function, show the following:

a. The production function in Equation 6.7 is a special case of the Cobb-Douglas.

b. If a+b=1, a doubling of K and L will double q.

c. If a +b < 1, a doubling of K and L will less than double q.

d. If a +b > 1, a doubling of K and L will more than double q.

e. Using the results from part b through part d, what can you say about the returns to scale exhibited by the Cobb-Douglas function?

6.8 For the Cobb-Douglas production function in Problem 6.7, it can be shown (using calculus) that MPK =aKa-1Lb MPL =bKaLb-1 If the Cobb-Douglas exhibits constant returns to scale (a +b=1), show that

a. Both marginal productivities are diminishing.

b. The RTS for this function is given by RTS=bK/aL

c. The function exhibits a diminishing RTS.

6.9 The production function for puffed rice is given by q=100√KL where q is the number of boxes produced per hour, K is the number of puffing guns used each hour, and L is the number of workers hired each hour.

a. Calculate the q=1,000 isoquant for this production function and show it on a graph.

b. If K=10, how many workers are required to produce q=1,000? What is the average productivity of puffed-rice workers?

c. Suppose technical progress shifts the production function to q=200√KL. Answer parts a and b for this new situation.

d. Suppose technical progress proceeds continuously at a rate of 5 percent per year. Now the production function is given by q=(1,05)t q=100√KL, where t is the number of years that have elapsed into the future. Now answer parts a and b for this production function. (Note: Your answers should include terms in (1,05)t. Explain the meaning of these terms.)

7.5 A firm producing hockey sticks has a production function given by q=2√KL. In the short run, the firm’s amount of capital equipment is fixed at K=100. The rental rate for K is v=$1, and the wage rate for L is w=$4.

a. Calculate the firm’s short-run total cost function. Calculate the short-run average cost function.

b. The firm’s short-run marginal cost function is given by SMC=q/50. What are the STC, SAC, and SMC for the firm if it produces 25 hockey sticks? Fifty hockey sticks? One hundred hockey sticks? Two hundred hockey sticks?

c. Graph the SAC and the SMC curves for the firm. Indicate the points found in part b.

d. Where does the SMC curve intersect the SAC curve? Explain why the SMC curve will always intersect the SAC at its lowest point.

7.3 The long-run total cost function for a firm producing skateboards is TC =q3-40q2+430q where q is the number of skateboards per week.

a. What is the general shape of this total cost function?

b. Calculate the average cost function for skateboards. What shape does the graph of this function have? At what level of skateboard output does average cost reach a minimum? What is the average cost at this level of output?

c. The marginal cost function for skateboards is given by MC= 3q2 -80q +430 Show that this marginal cost curve intersects average cost at its minimum value.

d. Graph the average and marginal cost curves for skateboard production.

Answer 1

Q 6.7.

a)

Production function is given by:

where

This is an example of Cobb Douglas production function due to the following properties:

And also for cross product

Let us double the inputs K and L and see how that affects the production function Q.

b) If a+b = 1,

i.e. doubling L and K will double Q

c) If a+b < 1

i.e. doubling L and K will less than double Q

d) If a+b > 1

i.e. doubling L and K will more than double Q

e)

Part b - Constant returns to scale

Part c - Decreasing returns to scale

Part d - Increasing returns to scale