ECO380H1 Lecture Notes - Lecture 3: Subgame, Root Mean Square, Production Function

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5 Aug 2018

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ECO380: Tutorial Problem Set - For Wednesday, March 21, 2018

1. [Dixit] So far, we have worked through the idea of NE in the second part of the Dixit game - T=2,

once ¯

K1has already been chosen and rms are making their nal output decisions (and F2 is deciding

whether or not to enter, or set q2= 0 and avoid the xed costs F2). The production function is

qi= min{Li, Ki}- producing one unit of output requires one unit of labor and one unit of capital.

The demand for the rms' output is given by P= 50 −0.01Q. The wage rate is w= 10, and the rental

rate of capital is r= 10.

(a) Solve for F2's BR function.

(b) We know that if F2 decides to produce output, then the equilibrium of this subgame will be at

some point on F2's BR function. (It will also be on F1's BR function, but we don't know what

that is until we know ¯

K1!) Given that knowledge, solve for Π2(q1)- Firm 2's prots, as a function

of Firm 1's quantity (assuming F2 produces the prot-maximizing quantity of output, larger than

0).

MC=w, and their BR function when their MC=w+r.

(d) What is the range of potential equilibrium quantities of q1that we could possibly have? (If you

get stuck, look at the graphs from the slides - there's a limit to how low or how high q1could be

in equilibrium, regardless of where ¯

K1is.)

(e) Given that range, what is the corresponding range of (before xed costs) prots to Firm 2?

2. [Dixit] In the Dixit model we saw in class, we discussed two benchmark values of ¯

K1,which we referred

to as M1and V1. Assume the same production function for output (one unit of output requires one

unit of capital and one unit of labor), for generic demand P=A−BQ , wage rate w, and rental rate

of capital r.

For some choices of these parameters A, B, w, r there is no possibility for predatory behavior: de-

pending on the value of F2Firm 2 either does not enter following monopoly behavior, or cannot be

prevented from entering. Only some combination of parameters A, B, w, r will make it possible that

there are some values of F2for which it is possible to prevent entry in a predatory manner (not by

acting like a monopolist).

What condition on A, B,w, and rmust be true for it to be possible for Firm 1 to prevent Firm 2 from

enterering for some values of F2? Here you are looking for an equation that these parameters must

satisfy if there are going to be any values of F2for which predatory behavior is possible. How does

this condition change if we instead needed two units of capital and one unit of labor to produce one

unit of output?

(To get you started: remember that the only time that behavior is predatory is when Firm 1 chooses

some ¯

K1∈(M1, V1]. So, they can only do something predatory if that interval is nonempty!)

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Document Summary

R t r s t t s p = 50 0. 01q r t s w = 10 t r t r t t s r = 10 . T st t t r s r t s s t r s t t r q1 . T t r t s t rr s r r sts r ts t r . T t t r r r p = a bq r t w r t r t . R s s t s r t rs a, b, w, r t r s ss t r r t r r . R t r t r s t r t rs a, b, w, r t ss t t t r r s s f2 r t s possible t r t tr r t r r t .

Related Questions

Suppose the process of producing lightweight parkas by Polly's Parkas is described by the function:

q = 40K^0.6 (L-40)^0.4

where q is the number of parkas produced, K the number of computerized stitching-machine hours, and L the number of person-hours of labor. In addition to capital and labor, $10 worth of raw materials are used in the production of each parka.

Note that Eq/EK = 40(0.6) K ^-0.40 (L-40)^0.4 and Eq/EL = 40K ^0.6 (0.4) (L-40) ^-0.60

By minimizing cost subject to the production function, derive the cost-minimizing demands for K and L as a function of output (q), wage rates (w), and rental rates of machines (r).

The cost-minimizing demands for K and L are:

Use these results to derive the total cost function: that is, costs as a function of q, r, w, and the constant $10 per unit materials cost. The total cost function (TC) is:

2. This process requires skilled workers, who earn $64 per hour. The rental rate on the machines used in the process is $16 per hour. At these factor prices, what are total costs as a function of q.

Total costs (TC) are:

Does the technology exhibit decreasing, constant, or increasing returns to scale?

3. Polly's Parkas plans to produce 2000 parkas per week. At the factor prices given above, how many workers should the firm hire (at 40 hours per week) and how many machines should it rent?

The firm should use ___ worker hours.

The firm should use ___ machine hours.

What are the marginal and average costs at this level of production?

Marginal cost is $___ per unit of output.

Average cost is $___ .

bronzehornet861

The law of eventually diminishing marginal returns: (Points : 1)

a. states that each and every increase in the amount of the variable factor employed in the production process will yield diminishing marginal returns.
b. is a mathematical theorem that can be logically proved or disproved
c. is the rate at which one input may be substituted for another input in the production process
d. None of the above

The marginal product is defined as: (Points : 1)

a. the ratio of total output to the amount of the variable input used in producing the output
b. the incremental change in total output that can be produced by the use of one more unit of the variable input in the production process
c. the percentage change in output resulting from a given percentage change in the amount of the variable input X employed in the production process with Y
d. None of the above

The rate at which one input X may be substituted for another input Y in a production process, while total output remains constant, is: (Points : 1)

a. the slope of the isoquant curve
b. the marginal rate of technical substitution

c. equal to MPx/MPy

d. all of the above
e. none of the above

Marginal revenue product is: (Points : 1)

a. defined as the amount that an additional unit of the variable input adds to the total revenue
b. equal to the marginal factor cost of the variable factor times the marginal revenue resulting from the increase in output obtained
c. equal to the marginal product of the variable factor times the marginal product resulting from the increase in output obtained
d. a and b
e. a and c

Marginal factor cost is defined as the amount that an additional unit of the variable unit of the variable input adds to ____________. (Points : 1)

a. marginal cost
b. variable cost
c. marginal rate of technical substitution
d. total cost
e. none of the above

In the Cobb-Douglas production function Q=aL^?1.K^?2 ______________. (Points : 1)

a. the marginal product of labor (L) is equal to ?1
b. the average product of labor (L) is equal to ?2
c. if the amount of labor input (L) is increased by 1 percent, then output will increase by ?1 percent
d. a and b
e. a and c

The short-run cost function is: (Points : 1)

a. where all inputs to the production process are available
b. relevant to decisions in which one or more inputs to the production process are fixed
c. not relevant to optimal pricing and production output decision facilities
d. crucial in making optimal investment decisions in new production facilities
e. none of the above

Theoretically, in a long-run cost function: (Points : 1)

a. all inputs are fixed
b. all inputs are considered variable
c. some inputs are always fixed
d. capital and labor are always combined in fixed proportions

A linear total cost function implies that: (Points : 1)

a. marginal costs are constant as output increases
b. average total costs are continually decreasing as output increases
c. a and b
d. none of the above

A cost-minimizing firm has the following short-run production function:
Q=f (L,K)=72L+5L²-0.2L³

a) Briefly explain why this is a short-run production function.

b) At what level of employment would be diminishing returns set in for the variable input?

c) Fin the levels of employment that define stage two in the production process.

d)If the market-determined real wage rate is $20 determine the amount of labor a cost-minimizing firm would hire in order to minimize the total cost of production.

2) a) Briefly explain the concept of "Return to scale". What form of "Return toScale" does the following production function exhibit? Explain

b) Q=f(L,K)=L^0.3 K^0.7

Suppose a cost-minimizing firm wishes to change its scale of production and needs to know the combination of labor and capital to employ. If the first's total cost outlay is $14. Fin this combination using the information below and given that the prices of labor and capita are $1 and $3 per unit respectively.

Q= 0 1 2 3 4 5 6 7 8

TPI=0 11 20 28 35 41 45 46 46.5

TPK=0 24 45 63 78 87 93 96 97

where TPI AND TPK represent the total product of labor and total product of capital respectively.

3) Given the demand and supply schedules for semiconductors in a perfectly competitive market:

Q_d=60-10P+0.5I, Q_s=-10+4P-2C

Where I=average income of the demanders of this production

C=unit cost of the inputs used in the production of this commodity.

a) Find the expression for equilibrium price and equilibrium quantity transcended in this market.

b)Find the initial equilibrium and equilibrium quantity ifI=1000 and C=20

c) Determine the impact of an increase in income from 1000 to1200 on the initial equilibrium values.

d) Assume that an improvement in the technology of producing this commodity results in an increase in supply by a factor of 5, how will this development affect the initial equilibrium values?0

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ECO380H1 Lecture Notes - Lecture 3: Subgame, Root Mean Square, Production Function

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