A prove that the given production function exhibits

Info icon This preview shows pages 3–4. Sign up to view the full content.

View Full Document Right Arrow Icon
a) Prove that the given production function exhibits constant return to scale (6 marks). 1 2 1 2 2 0 L K MP L and 1 2 3 2 0 L MP K L L b) Show that, when w is the labour wage and r is the capital rent, the optimal ratio between capital and labour is 1 4 K L . (6 marks) In equilibrium, the optimal bundle requires L K MP w MRTS MP r With 1 1 2 2 1 1 2 2 2 1 2 ; 2 8 4 L K K L K w MP MP MRTS L r L K
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
The optimal bundle requires 1 4 K L c) Compute the optimal number of labour and capital used to produce 16 units of output. (6 marks) Optimal bundle is one equation. The second equation is 1/2 1/2 ( , ) 4 Q K L K L =32. Substitute (1) into (2) we get 2L = 16 or L = 16 and K = 4. d) Suppose capital is fixed at 4 K in the short run. Derive (if exists) the short run total cost, short run average total cost, short run marginal cost, short run variable cost and short run fixed cost as a function of Q . (7 marks) When K is fixed at 4 in the short run, the short run production function becomes: Q = 8L 1/2 or L = Q 2 /64 SRTC(Q) = rK + wL = 4*8 + 2*Q 2 /64 = 32 + Q 2 /32 SRATC(Q)=32/Q + Q/32 SRMC(Q)=Q/16 SRVC(Q) = Q/32 SRFC = 32 e) Use a diagram to prove that the long run total cost is never greater than the short run total cost. (5marks) Another way to show this is as per figure 10-22 on page 299 where long run average cost curve is an envelope of all the short run average cost curves.
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern