Midterm1ans

# A prove that the given production function exhibits

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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

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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.
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