Appendix to Ch. 4: A Calculus Approach to Individual Behavior Lecture Notes
Hand in the following questions only as part of Assignment 3:
Qs 1, 2, 5, 8, 9, 10, 11.
Q1.A firm has decided through regression analysis that its sales (S) are a function of
the amount of advertising (measured in units) in two different media, television (x) and magazines (y):
S(x, y) = 100 – x2 + 30x – y2 + 40y
(a)Find the level of TV and magazine advertising units that maximizes the firm's sales.
(b)Suppose that the advertising budget is restricted to 31 units. Determine the level of advertising (in units) that maximizes sales subject to this budget constraint. (c)Give an economic interpretation for the value of the Lagrangian Multiplier obtained in part (b) above.
(d)The marketing department of the firm is lobbying to have the advertising budget increased to 40. Do you agree with the marketing department? If not, what advertising budget would you recommend and why?
Q2.The SFU Wellness Center is an organization providing help to distressed students on campus during the two weeks exam period at the end of every semester. You’ve been hired as a business consultant to help the Center develop a hiring policy so that it can provide the most meaningful student service (S) possible. You’ve determined that service can be described as a function of Medical (M) and Counseling (C) staff input as follows:
S = M + 0.5C + 0.5MC – C2
The staff budget for the SFU Wellness Center for the coming semester is $1,200.00. Each member of the medical staff costs $60 and each member of the counseling staff costs $30. (a)Determine the optimal combination of medical and counseling staff for the SFU Wellness Center. (b)Solve for and interpret the Lagrangian multiplier.
(c)If the Center were required to break even, what fee per service would you recommend that they charge?
Q3.An individual faces the following utility function:
U = 20X + 10Y- X2 – 2Y2 + 2XY
(a) Calculate her utility-maximizing choice of X and Y if both goods are free? (b) Calculate her utility-maximizing choice of X and Y if Px = Py = $1, and her income is $50? (c) Give an economic interpretation to the value of the Lagrangian multiplier.
(d) What income level maximizes her total utility from X and Y?
Q4.Tom’s income is $480and he spends it on two goods, X and Y. His utility function is U = XY. Both X and Y sells for $8 per unit. (a)Calculate Tom’s utility-maximizing purchases of X and Y. (b)How will his utility change if his income decreases by $2.00?
(c)If the price of Y increased to $11.52, with no change in the price of X, by how much would his income have to increase to enable him to maintain his initial level of utility as in part (a) above?
Q5.Moe’s income is $320 per week and he spends it on two goods, X and Y. Good X costs $8 and good Y costs $4 per unit. His utility function is U = 4.5XY. (a)Calculate Moe’s utility-maximizing purchases of X and Y.
(b)Calculate Moe’s constrained utility-maximum if his income decreases by $2.00?
(c)If the price of Y doubles, with no change in the price of X, by how much would his income have to increase to enable him to maintain his initial level of utility (as in part (a) above)?
(d) Using your answers from parts (a) and (c), calculate Moe’s arc income elasticity of demand for good X.
Q6.A firm produces two goods: widgets (X) and woozles (Y). Its profit function is given by:
= 55X – 2X2 – XY – 3Y2 + 100Y
and its maximum output capacity is X + Y = 17.
(a)Use the Lagrangian method to calculate the output mix the firm should produce. (b)Estimate the effects on profits if output capacity is expanded by 1 unit. (c) You are hired by the firm to evaluate a proposal from its Engineering Department that the firm’s output capacity be expanded to 33. Do you agree with the proposal?...