Questions -
Q1. Short-run Production and Costs:
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Labor
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Output
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AP
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MP
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FC
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VC
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TC
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AFC
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AVC
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ATC
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MC
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1
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4
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4
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4
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120
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60
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180
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30
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15
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45
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15
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2
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10
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5
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6
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120
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3
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18
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4
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28
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5
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35
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6
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40
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7
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45
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8
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48
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9
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50
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10
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50
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a) Fill-in the missing amounts in the above table.
b) Using graph paper or a computer, draw a total product graph with Output on the y-axis and Labor on the Y-axis. Describe and indicate the various stages of production.
c) Using the total product curve, show graphically where the AP is equal to the MP.
d) Using graph paper, draw a total cost curve with Total Cost on the y-axis and output on the x-axis on the same graph draw a variable cost curve. Explain the shape of the Total Cost curve and the Variable Cost curve. What separates the two curves?
e) Using the total cost and variable cost curves show graphically where the ATC is equal to the MC and where the AVC is equal to the MC.
Q2. Isoquants: Draw a set of isoquants for...
a) a software firm with technology such that each software program produced is created by exactly two employees and one computer.
b) a home security firm with technology such that either a night watchman or a security camera can prevent thefts equally well.
c) a gardening service with technology such that either a bigger lawnmower or more gardeners could allow for a greater amount of property to be cared for--at a given level of quality-but with diminishing returns to both gardeners and power equipment.
d) Which one of the above examples will have a corner solution? Briefly explain why.
3. Cobb-Douglas: You are the production manager for A-1 Baseball Hats, a top-end producer of the finest major league hats from all the favorite big-league teams. Suppose the production function for baseball hats is.
a) What type of production function is this? What are the returns to scale?
b) Find the expressions for average product of capital and marginal product of capital (using calculus) and average product of labor and marginal product of labor.
c) Using some graph paper or a computer, carefully graph a series of isoquants for the production function for q = 10, 15, and 20 baseball hats. Plot at least 5 points for each isoquant. (Hint: Rearrange the production function so that K is on the left and everything else on the right. Then use the given value for q and create a value for L. Solve for K.)
d) Suppose the firm has available only $1000 to produce hats and that the price of labor is $5 per unit and the price of capital is $10 per unit. Using the Lagrangian Multiplier method, find the optimal combination of labor and capital to maximize q. Approximately how many hats will be produced?
Q4. Perfect Competition: Domenic grows almonds. The market supply for almonds is Qs = -4 +3P and demand is Qd = 60 - 5P. Domenic's cost function is 40 - 2q + 0.1q2, where q is his output of almonds in pounds.
a) Derive the average cost function, the marginal cost function, and the average fixed cost function, and graph them on graph paper.
b) What is the market price of a pound of almonds?
c) What is Domenic's profit-maximizing output of almonds? (Use calculus.) What are profits? Use Π = (P - AC)q and show in a diagram.
d) The market for oats is perfectly competitive, and thus the market price must fall to the minimum of average cost for each firm in the long run. Since it is assumed that all firms have the same cost structure, what will be the long-run price of oats? What will be Domenic's profits in the long run?
Q5. Perfect Competition: You are an olive oil maker. Your production function is q = 120K - ¼K2 + 100L - ¼L2 and the price of your olive oil is $10 per bottle. Capital equipment costs $20 per unit and labor costs $50 per unit.
a) Create a profit function for your olive oil business. If you are unconstrained in your production budget, what is your profit-maximizing use of capital, K* and labor, L* in the long run? (When all inputs are variable.) What are your profits?
b) Now assume that your production budget is constrained to $10,000, which you must allocate between capital and labor. Set up a constrained Lagrangian profit maximization problem. What are your new optimal levels of K and L? (K* and L*) What are your profits?