State whether the following claim is true or false, and briefly explain your answer. The following production function exhibits increasing returns to scale:

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PART I: PRODUCTION FUNCTIONS

  1. State whether the following claim is true or false, and briefly explain your answer. The following production function exhibits increasing returns to scale:
  • Q= I, +1(1
  1. Erin’s admission to a prestigious music school depends on the total score (Q) the admissions office decides to assign her. The school evaluates both her academic performance in high school and her skills as a musician. Hence, Q will be determined by the hours she spends on school work (S) and the hours she spends playing piano (P), her favorite instrument, every day. We can represent the total score using the following production function:
  • Q=SP 1 .

Answer the following:

  • (a) To be admitted Erin needs a score of at least 2. Draw an isoquant curve for a total score of Q = 2 by placing school work, S, on the y-axis and piano practice, P, on the x-axis. To simplify, compute P for the points P = [1, 2, 4] and connect the dots.
  • (b) In a day there are 24 hours and Erin knows that she needs to sleep at least 8 hours. Hence, assuming that she can allocate maximum 16 hours to work school and piano practice, write this constraint and provide a representation of this constraint on your graph.
  • (c) Erin really wants to be admitted to her dream school, she prefers to focus two times as long on school work than on piano practice (S=2P), and she wants time to socialize outside of her duties. Is this possible? (Remember she has only 16 hours available; she needs to sleep too)
  • (d) Find Erin’s marginal productivity of school work, piano practice, and her MRTS of piano practice for school work (MPp/MPs).
  1. For each production function listed below, determine MPL, MPK, and MRTSL,K (i.e., MPL /MPK)
  • (a) Q = -,1.4 L K2
  • (b) Q = L1/2+K1/3
  • (c) Q = 3 L3+K2

 

PART II: OPTIMAL INPUTS

  1. We will study the case of a cost-minimizing soy farmer. The fanner uses two inputs – fertilizer (f) and irrigation water (w) – to produce soy (S). Under normal circumstances, her production function is given by S = min{2f, – that is, she needs to use f and w in the same proportion, needing twice as much fertilizer as she needs water The price of one unit of fertilizer is rf and one unit water is r,,.. Suppose rf = $3 and rp, = $6.
  • (a) Under normal circumstances, what is the cost minimizing bundle of inputs (fi;’, sirp needed to produce 4 units of S? How much does this bundle cost?
  • (b) The current year has been drier than usual and more irrigation is required to produce the same amount of crop. Her production function in this case is S = min{ f, w}. What is her cost minimizing bundle of inputs in the case of drought (f:, led needed to produce 4 units of S? How much does this bundle cost?
  • (c) To counter the effect of drought, an agricultural R&D company has developed a new type of seed which requires less fertilizer f to yield the same amount of soy. The production function under this new technology is given by R = min{2f, 2w}. What is the highest price our farmer would be willing to pay for this technology if she still wants to produce 4 units of S?
  1. Assume that a restaurant can sell their specialty dish with either purple rice or brown rice: these two inputs are perfect substitutes. The restaurant wants to minimize costs and so the manager is required to determine the optimal combination of white and brown rice given the input costs and the production function of their specialty dish

Q = 2P +B (where Q represents the number of dishes they can serve).

  • (a) What are the restaurant’s MPp and MPB? What about its MRTSps?
  • (b) Assume that the price of purple rice is pp = $6, while the price of brown rice pB = $2. What combination of inputs would you suggest to the manager? Please, explain your reasoning by using the marginal product per dollar of expenditure of the two inputs.
  • (c) Assume now that there is a shock in the brown rice market that increases prices up to pa = $6 (while the cost of purple rice is still also pp = $6). Would you suggest the manager to change the input allocation from part (b)? Again, please provide an explanation involving the relation between the marginal product per dollar of expenditure of the two inputs.
  1. A textile firm produces t-shirts while minimizing costs. It’s production of t-shirts is generated by the production function is Q = 4LK. The price of labor services is w, and the price of capital services is r. Suppose that w = $2, r = $8, and the firm’s total cost is $64.
  • (a) What are the firm’s MPL, MPO
  • (b) What is the firm’s optimal input combination of L and K given a cost of 64? How many t-shirts are they producing?
  • (c) Suppose new management takes over the company, and they intend to significantly increase production, poten-tially increasing total costs as well. The CEO announces that they will now produce Q = 1024 t-shirts using the same technology. Find (a) the new optimal choices of K and L given what the company intends to produce, and (b) compute the new level of their production costs.
  • (d) What can you conclude about the returns to scale of the firm’s production function after the analysis in point (c)? Assume that each good can be sold at a price of $10. What is the effect of the CEO’s decision on profits? Would you agree with his choice?