aps5_f04 (ans)

# aps5_f04 (ans) - Econ 313.1 Wissink Fall 2004 PS#5 XtraQ...

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1) Suppose a representative firm in a perfectly competitive, constant cost industry has the following long run total cost function: lrtc(x) = 4x 2 + 100x + 100. a) What is the long run equilibrium price for this industry? b) If market demand is given by the function X D = 1000 – P, where P denotes price, how many firms will operate in this long run equilibrium? ANS: lratc = tc/x = 4x + 100 + 100/x. The minimum point on the lratc occurs where the first derivative equals zero: 4 – 100/x 2 = 0 so that x* = 5. OR you could recognize that lratc is at its minimum when lratc = mc. Note, mc=8x+100. Now set mc=lratc and solve for x* = 5. So the long run equilibrium price is the value of lratc or lrmc at x* = 5. It is 4(5) + 100 + 100/5 = 8(5)+100 = \$140 = P*. If the market demand is X D = 1000 – P, then at P* = \$140, X* = 860. Since each firm produces x* = 5 units in the long run, there will be N* = 172(= 860/5) firms operating in the long run. 2) Illustrate how all four points on the isoquant/isocost diagram at the right would be represented on
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Unformatted text preview: a diagram representing total cost functions - both long run and short-run. Assume L and K are the only two inputs in the production function. 3) Backpacks are produced with a technology described by the following production function: b = min{L, 4K}. Assume that the price of labor is \$2/unit and the price of kapital is \$50/unit. Determine the long run total cost function for producing widgets efficiently. Note: K=kapital, L=labor and these are the only two inputs used in the production of backpacks. ANS: So we want L = 4K to cost minimize. So we know that: x = L and that x = 4K. Now we can get the input requirement functions: L*(x) = x and K*(x) = x/4. Now just use this information and the input prices in the definition of the cost function, lrtc=wL*(x) + rK*(x), to get: \$lrtc = \$2x+\$50(x/ 4) = \$14.50x. Econ 313.1 - Wissink - Fall 2004 PS#5 – XtraQ - ANSWERS L K x x 1 2 A B C D A C B D l r t c s r t c _ 1 s r t c _ 2 x _ 1 x _ 2 \$...
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