Posted: February 8th, 2019
Brooklyn College 7025X Assignment 3 MATHEMATICAL METHODS IN ECONOMICS Spring 2015
7025X MATHEMATICAL METHODS IN ECONOMICSBrooklyn CollegeSpring 2015Assignment 31. 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 revenuefunction:???? = 10 ? 2????Calculate the elasticity of the firmâs total sales revenue with respect to the amount of labor usedwhen 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 thepartial 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 thenecessary substitutions.3. Suppose you are examining a monopolist firm, with the following average revenue and average costfunctions:????????(????) = ?????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 totalcosts 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 leadto 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, yand 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 Jacobianmatrix to solve this