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A Cobb-Douglas production function for tables is Q=2K^0.5 L^0.5

a) Can any tables be built without tools?

b) Can the production of tables be mechanized?

c) There are currently 100 units of capital/tools, and in the short run no more tools can be either acquired or sold. What is the short-run production function?

d) If capital/tools rents for $100 per unit per day, and labor can be hired for $200 per unit per day, what are the fixed costs and variable costs? What is the total cost function?

e) What is the average cost function? To minimize average costs, how many tables should be produced?

f) What is the long term (meaning: capital/tool level is flexible) cost-minimizing amount of labor and capital to produce 50 tables?

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28 Sep 2019

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Related questions

Use the following information on a hypothetical short-run production function to answer questions A-C.

The price of labor is $20 per day.

Ten units of capital are used each day, regardless of output level.

The price of capital is $50 per unit.

(A). Calculate the marginal product (MP) and average variable product (AP) of each unit of labor input.
(B). Calculate total (TC), average total (ATC), average variable (AVC), and marginal costs (MC).
(C). Can you tell where diminishing marginal returns sets in?

Labor (L) Output (Q) MP AP TC ATC AVC MC
5 120 --         --
6 140            
7 155            
8 165            
9 168            

The daily demand for travel at a small regional airport is initially QD = 1000 - p; where quantity is in passengers and price is in dollars per passenger. There are initially 10 price-taking firms serving this market and each of them has the production function: q = sqrt (lk) where q is in passengers, and labor and capital are in some appropriately specified units. The price of a unit of labor is w = 100 per day; the price of a unit of capital is v = 100 per day, and this industry has constant costs. Each firm initially has a capital stock of k0 = 80. day; and this industry has constant costs. Each firm initially has a capital stock of k0 = 80.

(a) Derive the short-run cost function, SC(q), for each firm in the market.

(b) Derive the short-run market supply function, QS.

(c) Find the short-run competitive equilibrium price, p*, and quantity, Q*. How much labor, L*, does the industry employ at the competitive equilibrium?

(d) Derive the long-run cost function, C(q), of each firm in the industry.

(e) Describe the long-run market supply curve. Does the short-run competitive equilibrium at (Q*; p*) also correspond to a long-run competitive equilibrium? Explain why or why not.

Now, imagine that a catastrophic event causes the daily demand for air travel in this market to fall unexpectedly to ~QD = 750 - p Continue to assume that each of the 10 firms has a capital of k0 = 80 in the short run.

(f) Find the new short-run competitive equilibrium price, p**, and quantity, Q**, in this market. How much labor, L**, does the industry employ in the new short-run competitive equilibrium?

(g) What is the new long-run competitive equilibrium price, p***, and quantity, Q***? Describe what must happen as the industry moves from the short-run competitive equilibrium at (Q**; p**) to the long-run competitive equilibrium at (Q***; p***). How much labor L*** does the industry employ at the new long-run equilibrium?

A film studio in Hollywood produces movies according to the function (yes, they can also produce fractions of movies... Think of half a movie as a B-movie or so.) q = F(K, L) = K0.5L 0.5 /100.(reads as K to the power of 0.5 times L to the power of 0.5 divided by 100) In the short run, capital (studios, gear) is fixed at a level of 100. It costs $4,000 to rent a unit of capital and $1,000 to hire a unit of labor (actors, stuntmen, camera crew, etc.).

(a) What is the fixed cost? What is the variable cost as a function of output q?

(b) What is the marginal cost (MC) and the average cost (AC) of a movie? What is the average variable cost and average fixed cost?

(c) Where do the average and marginal cost curves intersect? What is the derivative of the AC curve and what value does it take at the intersection? What does it tell you about the minimum average cost?

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