AECO 300 Assignment 2

AECO 300 Assignment 2

Instructor: Mingda Zhang

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1.Draw the isoquant map for the technolog 𝑞=√𝐿𝐾. Be precise.

2.Consider the production function f(L,K)=L+K. Suppose K is fixed at 2. For this function, the marginal product of labor MPLis 1. (a) Find the algebraic expression for the average product of labor APL. (b) Graph the total product of labor function TPL, the average product of labor APL,and themarginal product of labor MPL. Does your answer to part b) violate the rule describing the relationship between average and marginal values? Explain

3.For each of the following production functions, determine if the technology exhibits increasing, decreasing, or constant returns to scale.

(a)𝑓(𝐿,𝐾)=2𝐿+𝐾

(b)𝑓(𝐿,𝐾)=√𝐿+√𝐾

(c)𝑓(𝐿,𝐾)=𝐿𝐾

(d)𝑓(𝐿,𝐾)=√𝐿𝐾+𝐿+𝐾

4.A firm uses capital and labor to produce output according to the production 𝑞=4√𝐿𝐾

(a)Find the marginal product of labor (MPL) andmarginal product of capital (MPK).

(b)If the wage w=$1/labor-hr. and the rental rate of capital r=$4/machine-hr., what is the least expensive way to produce 16 units of output?

(c)What is the minimum cost of producing 16 units?

(d)Show that for any level of output, q, the minimum cost of producing q is $q.

(e)Find the Total cost function.

5.A competitive firm has a short-run total cost curve STC(q) = 0.1q2+10q + 40.

(a) Identify SVC and SFC.

(b)Find and plot the SAC and SAVC curves.

(c)For this function, the SMC curve is given by SMC(q) = 0.2q +10. Include this curve in your diagram for part b).

(d)Write the equation for the firm’s short-run supply curve and indicate it in the above graph.

(e)Find the firm’s break-even price pe(at which profits are zero) and its shut-down price ps.

6.N.S. textbook problem 8.6(Picture is attached at the end)

7.The following figure depicts the short-run per unit cost curves for a competitive firm. Ifthe market price is p0, indicate the firm’s revenue, STC, SVC and profits.

8.For the firm in problem 7, indicate the firm’s short-run supply curve. What will be the effect of an increase in fixed costs on the firm’s supply curve and on its profits?

9.Suppose there are two technologies for producing steel. Under technology A, a firm’s short-run total cost curve is 𝑆𝑇𝐶𝐴(𝑞)=12𝑞2+100𝑞+10, and using technology Bit is 𝑆𝑇𝐶𝐵(𝑞)=2𝑞2+6. Assuming there are 100 firms using technology A and 400 using B, determine the short-run market supply curve.

10.Suppose there are only two firms in the marker, firm A and firm B. They produce identical products. Firm A and firm B have the same constant marginal cost, 𝑀𝐶𝐴=𝑀𝐶𝐵=𝐴𝐶𝐴=𝐴𝐶𝐵=25.The market demand function is given by 𝑄=400−4𝑃.

a.If the firms practice under the Bertrandmodel, what will be the Nash equilibrium market priceand output level?

b.If these twofirms practice under the Cournotmodel. Write the best response functions for firm A and firm B.

c.What are the Cournotoutput level for each firm?What is the market output level? What is the market price?

d.Graph the best response functions of firm A and B, and label the CournotNash equilibrium in the graph.

e.Calculate the profits for each firm in the Cournotmodel.

f.If these twofirms cooperate and act as a monopolist. What is the monopoly price and monopoly output level.

g.Is the monopoly outcome stable? If firm A operates under the monopoly outcome, what is firm B’s choice of output level?

h.Graph the monopoly outcome, cournot outcome and perfect competition outcome in one graph. (Hint, the y-axis should be price, x-axis should be quantities)

i.If the firms practice under the Stackelbergmodeland firm A moves first, what is the output levels for firm A and firm B?

j.Calculate the profits for each firm in the Stackelbergmodel and find that first mover has an advantage by earning more p

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