Question 7025X MATHEMATICAL METHODS IN ECONOMICS

04 Jun Question 7025X MATHEMATICAL METHODS IN ECONOMICS

Question
7025X MATHEMATICAL METHODS IN ECONOMICS

Brooklyn College

Spring 2015

Assignment 3

1. Consider the firm with a single factor of production defined implicitly by the relation

???? = ????

3 + 4????

where z is the variable input and q is the output. The firm faces the following average revenue

function:

???? = 10 ? 2????

Calculate the elasticity of the firm’s total sales revenue with respect to the amount of labor used

when q=2.

2. The equation ????

2 + ????

2 + ????

2 + ???????? + ???????? + ???????? + ???? + ???? + ???? ? 1 = 0 has one solution (x,y,z)=(1,-1,-

1). Check that the equation does indeed define z as a function of x and y at this point. Calculate the

partial derivatives of z with respect to x and y at this point.

Hint: Define the equation as F(y,x,z). Take the partial derivatives of z with respect to x and y, implicitly, that is,

solve ????????

????????=?

????????

????????

and ????????

????????=?

????????

????????

. Check if ????????

????????

is non zero in order to see that the implicit function exists. Do all the

necessary substitutions.

3. Suppose you are examining a monopolist firm, with the following average revenue and average cost

functions:

????????(????) = ?????

3 + 12????

2 ? 30???? + 1000 and

????????(????) = 2???? + 1000 ?

100

????

where s are the average sales per day. If the monopolist chooses to produce nothing (s=0), the total

costs become negative.

i. Write an expression for the Profits as a function of s.

ii. Determine the most profitable level of output. Show that this output level does indeed lead

to a maximum rather than a minimum by checking the second order conditions.

4. This question is not mandatory, but is extra credited: The following three equations define x, y

and w as functions of z.

???????? ? ???? = 0

???? = ????

3 + 3????

????

3 + ????

3 = 2????????

Find and expression for

????????

????????, and evaluate it at the point where w=z=1. Hint: Make use of Jacobian

matrix to solve this

Tutorials for this Question