These curves do not cross c does this production

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These curves do not cross. c) Does this production function exhibit increasing, decreasing, or constant returns to scale? (3 pts) (As with all questions, you must show your working, which in this case should be an algebraic proof.) L AP L MP L
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10 Suppose both L and K are increased by a factor of t. The new production function can be written as 0.4 10( )( ) New Q tK tL = . This equation can be rewritten as 1.4 0.4 1.4 10 New Old Q t KL t Q = = . Notice that the production increases by a factor of 1.4 t when inputs are increased by a factor of t. Therefore, the production function shows increasing returns to scale. Many students will use a number (like 2) instead of t. That is fine. d) Calculate the marginal rate of technical substitution (MRTS) for this firm when it is employing 40 units of K and 32 units of L. Show this MRTS on a graph that contains a single isoquant containing the point K = 40 and L = 32. Please place L on the horizontal axis and K on the vertical axis. The isoquant does not need to be to scale. (4 pts) The MRTS, which is the slope of an isoquant, is equal to the negative of the ratio of the marginal products. With L on the horizontal axis 0.6 0.4 4 10 L K MP KL MRTS MP L = − = − (using the marginal product expressions from part (a)) With K = 40 and L =32 it follows that 0.6 0.4 4 0.4 0.4*40 / 32 0.5 10 KL K MRTS L L = − = − = − = − and 0.4 0.4 10 10(40)(32) 1600 Q KL = = = . The specific task is to graph a 1600 q = isoquant and show that the slope of the isoquant is equal to -0.5 when K = 40 and L =32. The diagram is just a standard isoquant diagram. e) Suppose there is technological progress that allows the firm to produce 10% more output with the same inputs. Show in a diagram what happens to the isoquants. (5 pts) K 0 Q = Q 0 (before change) Q = Q 0 (after change) K
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11 Before the change the firm produces Q 0 units of output (upper isoquant in diagram) with input combination L 0 and K 0 . After the change that same isoquant is relabeled 1.1Q 0 . After the change the lower (dashed) isoquant now represents output Q 0 (i.e., 10 percent less output than 1.1Q 0 ). Thus, the technological change has shifted the isoquants inward. Question 5: Cost A local entrepreneur from B.C. recently purchased a hand-bag producing factory for the price of $75,000. The owner also spent $25,000 in renovating the factory. If the entrepreneur decides not to produce output, the best alternative use of the factory is to lease it or rent it to another potential entrepreneur for $15,000 per year. The cost of production is constant at $10/bag and the prevailing market price is $15. The annual fixed cost of operating the factory is $5,000. a) Derive the formula for average total cost (AC) of producing the bags, Q, per year. Explain why you omitted, if you did, some of the costs described above from your AC calculation. (4 pts) The non-sunk annual fixed cost, FC = Opportunity cost of not renting the factory + Fixed cost of operation = 15,000 + 5,000 = 20,000 Therefore, total annual cost of production, C = 20,000 + 10Q and AC = 20,000/Q + 10.
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