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(solution) Working on number #1. I graphed it and I'm unsure as to how the

Working on number #1. I graphed it and I'm unsure as to how the graph should be structured

Intermediate Microeconomics Fall 2016 Homework 2 - Part A

Exercise 1:

The following production plans are feasible:

(10, ?10, ?70) ; (10, ?20, ?40) ; (10, ?35, ?35) ; (10, ?40, ?20) ;

(10, ?80, ?10) ; (20, ?40, ?80) ; (20, ?50, ?60) ; (20, ?80, ?40) .

The negative numbers stand for inputs and the positive one for output. Also,

technology is convex and scaling up is possible (for scaling parameter t > 1).

a. In a graph with the two inputs in each axis, plot all 8 production plans.

b. Create 4 new feasible production plans according to the following specifications:

i. A combination of (10, ?20, ?40) and (10, ?40, ?20), where the weight

on the 1st and 2nd plans is (A + 6)/20 and (14 ? A)/20 respectively.

ii. A combination of (20, ?40, ?80) and (20, ?80, ?40), where the weight

on the 1st and 2nd plans is (B + 6)/20 and (14 ? B)/20 respectively.

iii. A scaling of (10, ?10, ?70) , with scaling parameter t = (11 + B)/10.

iv. A scaling of (10, ?80, ?10) , with scaling parameter t = 2,

where you should replace A and B with the last and second-to-last digits

of your ASU ID# respectively.

c. Which production plans can you eliminate for being non-efficient? Graph the

isoquants for output equal to 20 and 101 .

d. New technology is discovered. Now the production plan (20, ?80, ?30) is

feasible. Eliminate non-efficient production plans and graph the new isoquant

for y = 20.

e. Now assume that the technology exhibits constant returns to scale (for all

scaling t > 0). Eliminate more inefficient plans and graph the new isoquants

for output levels 10 and 20.

1 Keep in mind that convexity means that plans can be combined, but the resulting combination of any two plans that produce the same does not necessarily produce more than the

original plans. So you can?t just assume that it does (if you can prove it, that?s a different

story). 1 Exercise 2:

For the following production function:

y = K ? L?

a. Show whether the function exhibits constant, increasing or decreasing returns

to scale for different parameter values (of ?).

b. Consider the following three parameter values: ? = 1, ? = 1/2 and ? = 1/4.

For each of these parameter values, draw two isoquants in a graph, one for y = 1

and one for y = 2. Use the graph to show the returns to scale of the production

function.

c. Define the long-run cost minimization problem for the firm.

d. Obtain the long-run input demands, total cost and average cost all as a

function of output y, input prices w1 , w2 and technology parameter ?. Also,

obtain the short-run average costs for K = 1.

e. Graph the (long-run) average cost functions for all three cases in part (b).

They should have a very essential difference. Why? (your answer should point

towards returns to scale).

f. Consider the case of ? = 1/(3 + A), where A is the last digit of your ASU

ID#. Also, assume that the market operates under perfect competition. Obtain

long-run quantity produced and profits for the firm (as a function of parameters w1 , w2 , p). Also obtain the (long-run) unconditional input demands (as a

function of parameters w1 , w2 , p).

g. For the case of constant returns to scale in part (b) obtain the long-run

output price p (as a function of input prices w1 , w2 ) that implies zero profits for

the firm. Argue that a higher price is not consistent with perfect competition

and a lower one would induce the firm to shut down.

h*. Consider the case of ? = (2 + B)/(3 + B), where B is the second-to-last

digit of your ASU ID#. Also assume that the firm has a monopoly. The inverse

demand function is:

3+2B

p = y ?( 6+3B )

Obtain the long-run profit-maximizing quantity and price (as a function of parameters w1 , w2 ). 2 Exercise 3

For this exercise replace ?A? with the last digit of your ASU ID# and ?B? with

the second-to-last digit of your ASU ID#.

Consider the following production function:

?/? y = [(A + 1)x?1 + (B + 1)x?2 ] , where ? > 0, 1 > ? > 0 a. Determine the relationship between ? and returns to scale.

b. Define the cost-minimization problem, obtain the long-run input demand

functions and the total cost function when w1 = (A + 1) and w2 = (B + 1).

c. Determine the relationship between ? and both the average and marginal

cost function. Explain briefly why this is consistent with your answer to (a). If you are unsatisfied with the level of difficulty so far, consider solving

the following exercise (otherwise just ignore it):.

Exercise 4**:

a**. Prove that if a production function exhibits constant returns to scale then

it can be re-expressed as:

y = f (x1 , x2 ) = f1 (x1 , x2 ) x1 + f2 (x1 , x2 ) x2 .

Now consider the following production function:

y = f (x1 , x2 ) = x1? x2? ? F

0 if x1? x2? > F

,

Otherwise where y is output produced and F represents a quasi-fixed output cost. In other

words, (y + F ) can be reinterpreted as total production, but where the first F

units are worthless and the rest y is useful.

b*. Prove that if ? + ? ? 1 then the function exhibits increasing returns to

scale.

c***. Prove that if ? + ? < 1 then the function will exhibit decreasing returns

to scale (regardless of the scaling rate) only if the ratio of useful production to

total production is greater than ? + ?. 3 Intermediate Microeconomic Theory Fall 2016 Student Name: Student ID Homework 2 - Part A

Front Page

Selected Answers:

Exercise 1.b.i.

The new production plan is: Exercise 1.b.iii.

The new production plan is: Exercise 2.f.

The supply function (quantity produced as a function of w1 , w2 and p) is:

y? = Exercise 2.h.

The supply function (quantity produced as a function of w1 and w2 ) is:

y? = Exercise 3.b.

The (conditional) input demand function for input 1 (as a function of y, ?

and ?) is:

x?1 = 4

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