DOC PREVIEW
OSU ECON 4001.03 - Sol5

This preview shows page 1-2 out of 5 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 5 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Homework 5 Solution1 Econ 4001.03, Fall 2013 Prof. Lixin Ye Homework 5 Solution 1. True and False, and explain briefly. 1) If marginal product is greater than average product, total product must be increasing. True. When marginal product is greater than average product, the average product must be increasing. Hence the total product will also be increasing. 2) You decide to purchase a new car for $12,000. Upon driving the car off of the lot, the resale value of the car falls to $9,000. The opportunity cost of using the car is $12,000. False. The opportunity cost of using the car is $9,000. 2. Identify the returns to scale (increasing, constant, or decreasing) for the following production functions. 1) 22100Q KL= ( ) ( )( )222 2224 224Given 1, then100 100 100 aaQ aK aLaKaLa KLaQ>==== Since an increase in all inputs by a factor of aproduced a more than proportional increase in output  4aa, the production function exhibits increasing returns to scale. Remark: Alternatively, you may simply check what happens when all inputs are doubled (as we did in the lecture), that is, let 2a= in your computation. 2) 113322QK L=+2 ( ) ( )( )11331 1 113 3 3311 133 313Given 1,222222aaaaaQ aK aLQ aK aLQa K LQ aQ>=+=+=+= Since an increase in all inputs by a factor of aproduced a less than proportional increase in output <aa31, the production function exhibits decreasing returns to scale. 3. Suppose a firm has the production function 50Q KL= 1) If the wage rate is $10 per unit of labor and the rental rate of capital is $5 per unit of capital, how much capital and labor should the firm employ in the long run to minimize the cost of producing 40,000 units? First, 50LMP K= and 50KMP L=. Use the tangency condition to determine the optimal capital-labor ratio. 50 5010 55 102LKMP MPwrKLKLKL==== Next, use this result with the production function to determine the optimal quantities of capital and labor. 25040,000 50(2 )40020Q KLLLLL==== Finally, since 2KL=, 20(2) 40.K == 2) Using the solution in part 1), what will the firm’s long-run total cost be?3 10(20) 5(40)400TC wL rKTCTC=+=+= 4. Consider a production process where capital and labor are perfect complements – two units of capital are required for each unit of labor to produce four units of output. 1) Derive the production function for this production process. This production process can be characterized with a fixed proportions production function. 2min( ,2 )Q KL= 2) If the wage rate is $5 per unit of labor and the rental rate of capital is $8 per unit of capital, how much capital and labor should the firm employ to minimize the cost of producing 100 units? With the fixed proportions production function there is no tangency condition. First we have the production constraint condition: 100 2min( ,2 )50 min( ,2 )KLKL== (1.1) We also have the “most efficient production” condition: 2KL= (1.2) Substituting (1.2) into (1.1), we have 50/2 25KLK=== 3) What will the total cost be to produce the 100 units using the quantities of capital and labor determined in part 2)? 5(25) 8(50)525TC wL rKTCTC=+=+= 5. Suppose a firm has the production function4 1/2 1/4 1/4Q K LM The wage rate 16w, rental rate 2r , and the price of the materials 1m . 1) Suppose in the short run, K is fixed at *K. What’s the solution to the firm’s short run cost-minimization problem? First, 1/2 3/4 1/4 1/2 1/4 3/411, 44LMMP K L M MP K L M. Tangency condition: 16 (1)16 (2)LMMP M wMP L mML  Substituting (2) into the constraint *1/2 1/4 1/4Q K LM, we have *1/2 1/4 1/4 *1/2 1/4 1/4 *1/2 1/2(16 ) 2QKLM KL L KL  We therefore have the short run (conditional) demand functions: * 2** 2*( , ) /(4 )(, ) 4 /SSL QK Q KM QK Q K 2) What is the solution to the firm’s long run cost minimization problem given that the firm wants to produce Q units of output? Note that 1/2 1/4 1/412KMP K L M. The second tangency condition is given by 18 (3)216 (4)LKMP K wMP L rKL  Substituting equations (2) and (4) into 1/2 1/4 1/4Q K LM, we have the long run (conditional) demand functions for labor: 1/2 1/4 1/4(16 ) (16 )8( ) /8Q LL LQLLQ Q5 Substituting this back to (2) and (4), we have the long run demand functions for capital and materials as follows: ( ) 16( /8) 2( ) 16( /8) 2KQ Q QMQ Q Q 3) Suppose that *10, 20QK. Compare the long run and short run demands. The long run demands for labor and materials are: (10) 10/8 1.25(10) 2(10) 20LM The short run demands for labor and materials are: (10,20) 100/(4 20) 1.25(10,20) 4(100)/20 20SSLM So the long run and short run demands coincide. The reason is that *K is fixed at the long-run optimal


View Full Document

OSU ECON 4001.03 - Sol5

Download Sol5
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Sol5 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Sol5 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?