GSB420 Class 2 Optimization Homework NAME _____________________ I. Optimization A leasing office has 200 apartments to rent. If they rent X apartments, their monthly profit (P), in dollars, is given by P 1.6 X 2 480 X 2500 1. Find the number of apartments to be rented to maximize the profit. 2. Calculate the maximum profit. 3. Prove the maximum profit using the 2nd derivative. II. Advertising. A company estimates that it will sell N units of product after spending $X thousand on advertising (minimum of 5 thousands to maximum of 25 thousand dollars) as given by N = -2X3 + 90X2 - 750X + 2000 1. What is the amount of advertisement to give the maximum number of units N? 2. What is the maximum sales (N)? III. Constrained Optimization A homeowner wants to enclose an 800-square-foot rectangular in her yard in which her dog can run. Three sides are to be of wire mesh, the other of stone. The wire mesh costs $8 a running foot; the stone, $24 a running foot. What dimensions will minimize the cost? IV. Budgeting for least cost A manufacturing company produces two models of television set, X units of model A and Y units of model B per week, at a cost (in dollars) of C(X,Y) = 6X2 + 12 Y2 If it is necessary (because of shipping considerations) that X + Y = 90 How many of each type of set should be manufactured per week to minimize cost? What is the minimum cost? V. Multivariate Optimization Your company produces two types of wheat, type X and type Z. You are under contract to deliver 12 tons of wheat. Here is the cost function for the types of wheat: C 3 X 2 4 XZ 9 Z 2 8Z 36. Find the combination of wheat that minimizes the cost of fulfilling this contract. GSB420 CLASS 2 Math (CH3) Page 1 of 12 Math Chapter 3. Optimization Linear Equation Nonlinear Equation GSB420 CLASS 2 Math (CH3) Y = X2 Graph from desmos.com Page 2 of 12 GSB420 CLASS 2 Math (CH3) Y = -X2 Page 3 of 12 GSB420 CLASS 2 Math (CH3) Page 4 of 12 Example: Sales and Advertisement Given the following relationship between sales (S) in millions of dollars and advertisement expenditures (A) in millions of dollars: GSB420 CLASS 2 Math (CH3) Page 5 of 12 Example: Profit Maximization Given the following profit function () in millions of dollars and output (Q) in units of thousands: Q3 = 4 5Q + 3Q 3 2 GSB420 CLASS 2 Math (CH3) Page 6 of 12 GSB420 CLASS 2 Math (CH3) Page 7 of 12 Optimization The rate of change of per capita consumption of a certain type of cheese in the United States from 1982 through 2002 can be modeled as Q(X) = -0.0011 X2 + 0.02 X +0.2399 pounds per person per year where x is the number of years since 1970. 1) According to the model, when was the per capita consumption of the cheese growing and when was declining? 2) Find the point that corresponds to the time when the per capita consumption of this cheese was greatest. GSB420 CLASS 2 Math (CH3) Page 8 of 12 Multivariate Optimization Assume that a firm sells the same product in two different markets. Q1 is the quantity sold in Market 1 and Q2 is the quantity sold in Market 2. The firm's profit () function is given below: = 5 8Q1 + 10Q2 Q12 3Q22 + 4Q1Q2 Identify the levels of output, Q1 and Q2, that will MAXIMIZE the profit. GSB420 CLASS 2 Math (CH3) Page 9 of 12 An Example of a Lagrangean (Multiplier) Method: Your company sells DVDs in two regions: 1 and 2. Your sales staff came up with the following relationships between sales and advertising expenses in each of these regions: S1 = 50 + 10 A1 A12 S 2 = 20 + 5 A2 0.5 A22 where: S i = sales revenue in Region \"i\" (millions of $) A i = advertising expenditures in Region \"i\" (millions of $) Your boss (or you) provides a total advertising budget of $7 million. Identify the advertising expenses for each region that will maximize your company's total sales (from these two regions). GSB420 CLASS 2 Math (CH3) Page 10 of 12 Everything is the same as above. However, the advertising budget has a new constraint of $7.1 million. That is, solve the following equation for A1, A2, , S1, S2, and S. Maximize S = S1 + S 2 = (50 + 10 A1 A12 ) + (20 + 5 A2 0.5 A22 ) Subject to: A1 + A2 = 7.1 GSB420 CLASS 2 Math (CH3) Page 11 of 12 Constrained Optimization The following is the Sales (S) equation based on two types of advertising expenditures - newspaper advertising expenditure (X) and magazine advertising expenditure (Y) S = 200 X + 100Y 10 X 2 20Y 2 + 20 XY Assuming the total advertising budget is restricted to 20, find X and Y that maximize the sales (S), and identify the value of maximum sales (S) and the corresponding value. GSB420 CLASS 2 Math (CH3) Page 12 of 12 Math Chapter 4. Application to Economics Total Revenue (TR) Total Cost (TC) Total Profit () GSB 420 CLASS 3 Homework NAME ________________________________ 1. A 300-room hotel in Las Vegas is filled to capacity every night at $80 a room. For each $1 increase in rent, 3 fewer rooms are rented. Each rented room costs $10 to service per day. That means the demand equation is Q = 540-3P (1 points each). a) Setup the equation for profit and the demand equation. b) Find the optimal level of rent that the management charge for each room to maximized profit and its profit level? c) If city raises the profit tax of 10%, find the optimal level of rent and profit. d) Instead of profit tax, city wants collect 10% of rent in each room. Find the optimal level of rent and profit. e) Compare the results between profit tax vs. tax on rent in terms of rent and output level. Who will pay the tax? f) What is the rent elasticity of the demand for room and its meaning when there is no tax? 2. The price-demand equation and the cost function for the production of the smart watch are given, respectively, by P = 200 - X/30 C(X) = 72000 + 60x where 0 <= X <= 6000 a) Find the maximum revenue b) Find the maximum profit, the production level that will realize the maximum profit, and the price the company should charge for each smart watch set. c) If government decides to tax the company $5 for each set it produces, how many set should the company manufacture to maximize its profit? What is the maximum profit? What is the price to charge to maximize profit? 3. A company produces X units of product A and Y units of product B (both hundreds per month). The monthly profit function (in thousands of dollars) is found to be P(X, Y) 4 X 2 4 XY 3Y 2 4 X 10Y 81 a) Find the profit if X = 1 and Y = 3 b) How many X and Y products should be produced each month to maximize profit? What is the maximum profit? \fGSB420 CLASS3 MATH CH4 Page 1 of 7 CLASS 3: APPLICATION TO ECONOMICS TOTAL REVENUE (TR), TOTAL COST (TC), DEMAND CURVE (D) and PROFIT () Maximization GSB420 CLASS3 MATH CH4 I. Page 2 of 7 PROFIT OPTIMIZTION 1. Defining Q to be the level of output produced and sold, let's assume that the firm's total cost (TC) function is given by the following relationship: TC = 20 + 5Q + Q2 Furthermore, assume that the demand for the output of the firm is a function of price P given by the following relationship: Q = 25 - P GSB420 CLASS3 MATH CH4 2. Page 3 of 7 Using the cost and demand functions of: TR = PQ = (25 - Q)Q = 25Q - Q2 TC = 20 + 5Q + Q2 suppose that the government imposes a 20 percent profits tax on the firm 3. Suppose that the government imposes a 20 percent sales tax (that is, a tax on revenue) on the firm. Find the profit-maximizing output, Q; the price, P; the profits level; and tax, using the original total revenue and cost functions shown below: TR = PQ = (25 - Q)Q = 25Q - Q2 TC = 20 + 5Q + Q2 GSB420 CLASS3 MATH CH4 Page 4 of 7 II Application to Labor Supply Suppose that we can divide our 24-hour day into either sleeping or leisure hours (S) or nonsleeping, thus, working hours (W). If the following equation represents our utility (=satisfaction) function, can you identify the optimal amount of sleeping hours (S) and working hours (W) that will maximize our utility (=satisfaction=U)? Max U = 2 S 2 + 4.25 SW + 2.0625W 2 GSB420 CLASS3 MATH CH4 Page 5 of 7 III Application to Demand Elasticity Calculation Demand elasticity measures how much the quantity demanded, Q, changes in percentage points, given a percentage change in a single variable, X, that is believed to affect Q. Example 1: Own Price Elasticity of Demand Suppose the follow demand relationship between the number of hamburgers (Q) and its per-unit price (P) is identified by your analyst: Q = 200 - 10P GSB420 CLASS3 MATH CH4 Example 2. Page 6 of 7 A Multivariable Demand Equation Given the following demand equation for Good Old Hotdogs (Q), Q = 100 - 3P - 4I + 5Pc where P = the price of a Hotdog; I = average disposable income of the consumers ($) for Hotdogs; and Pc = the price of a competitor's Hotdogs Given a generic point elasticity of demand as: X = Q X X Q a. Calculate the point price elasticity of demand at P=$1, given I = $10 and Pc =$2. GSB420 CLASS3 MATH CH4 Example 3. Page 7 of 7 An Exponential Demand Function Given the following demand equation for Good Old Hotdogs (Q), Q = 25P 3 I 2 A 1 Pc2 where P = the price of a Hotdogs; I = average disposable income of the consumers ($) for Hotdogs; A= advertising expense for Hotdogs; and Pc = the price of a competitor's hotdogs