Question

3. (5 points) Consider a manufacturing company whose daily production output (number of daily produced goods) is described by the Cobb-Douglas production function 20x1/2,1/2, where x is the number of units of labour (number of working hours) and y is the number of units expended on capital (number of machine hours). A unit of labour costs $10 (hourly wage is $10) and a unit of capital costs $40 (cost per machine hour is $40). (a) (2 points Assume that the company can spend daily $1,600 in total costs for labour and capital. What are the optimal numbers of units of labour and capital? (b) (2 points) Assume that the company needs to have a daily production output of 1,000. What are the optimal numbers of units of labour and capital? (c) (1 point) In 3(a), we fixed the input (costs for labour and capital) and optimized the output (produced goods). In 3(b), we fixed the output and optimized the input. Would it make sense to optimize output and input independently in two separate optimization problems without constraints? Explain why or why not.

Answer 1

ANSWERS Let 'X' be the number of unit of labour and Let y be the number of unit of experded, or copital, la Given production function is f(3,4) = 20x%. y'a 919,y) - 108+ 40y = 1600 By using the Lagrange multiplier method of = xog fx = xgx 10th. y2 = 10x = 22. y' ax fy = xgy → boxyzyl2 = 40% 272.442242 0 By dividing 0 and then we get, al2.cyle - 4 212.yl2 2.5° = 4 x=44 put the bc value in 968,y) equation then we get 10 (44) + 40y = 1600 4 = 20 23 420) -80 :. 80 units of labour and 20 units of optimum output capital gives

= (b). Here f(,y) = 10x+ 404 g(x,y) = 2032.942 =1000 fx = 19x + 10 = (10312 y'3) 4 1=7x7 y2 0 fy= ngy → 40 - (102'2 yl2) ► dx?? y V2 → By dividing equations © and then we get se"?jk all2. y 12 2.5' =4 x=44 By putting values in 96,4) then 20(44)"? yY2- 1000 404 = 1000 y= 25, 2= 100 :. 1000 units of labour and 25 units of labou capital are grequired for optimum. input. les AS per my opinion we cannot seperate input and output for a process AS holistic approach we should not treat them independeren Thanking you a - Hook