Finance Solutions to Problem Set #2: Optimization

Transcription

1 Finance 300 Solutions to Problem Set #: Optimization ) According to a study by Niccie McKay, PhD., the average cost per patient day for nursing homes in the US is C A. 6X.0037X We want to minimize the cost per patient by our choice of patient days (X). min A.6X.0037X X The first order necessary condition (take the derivative with respect to X and set it equal to zero remember, A is a constant) is as follows: X 0 Solving for X, we get X Further, if we take the second derivative, we get Which is always positive this guarantees a minimum! ) Suppose that you are currently selling magazine subscriptions for $36. You currently have 6,000 subscribers. You also know that for every $ you lower your price, you will gain 00 new subscribers. Fine the price to maimize your revenues from magazine subscriptions. So, we know a couple things. The slope of the demand curve that I face is equal to Further, we have one point on the demand curve. At a price of $36, I have 6,000 subscribers. Price $36 $6, Quantity

2 Plug in what we know. Solve for A Q A BP 6,000 A Q3, 00 00P Now, total revenue equals price times quantity. TR PQ Plug in the demand curve for Quantity. 3, , TR P P P P We want to maimize revenues, so take the derivative with respect to P and set it equal to zero. Now solve for P. 3, P 0 3, 00 P $ Q 3, , 600 TR $33 6, 600 $7,800 3) Suppose that you are a petroleum company. Your profits depend on your production of heating oil and gasoline. Specifically, your profits as a function of gasoline (G) and heating oil (H) is given by: Profits 60 40H 00G 0H 8G 6HG Solve for the profit maimizing choices for gasoline and heating oil. Take the derivative with respect to H and G and set them equal to zero. 40 0H 6G G 6H 0

3 So, we have two equations and two unknowns. Let s solve the first equation for G. 40 0H 6G 0 6G 40 0H 40 0 G H 6 6 G H Now, plug this into the second equation H 6H H 6H H 7.8 H.8 G H ) Consumer research for Kraft foods has discovered the following relationships between sales of TANG (Yes, the orange drink used by astronauts!) and advertising ependitures for two districts. Sales (S) and advertising (A) are in millions. S A. 0 A S 4A. A We can write Kraft s maimization problem as: Ma A A, A subject to 4A A A The lagrangian for this problem is.a $.A l A 4A. A.A A A Taking the first order conditions, we get l 3A 0 A l 4 A 0 A A A

4 Solving the above system for A and A we get A A. 3.. The value of the multiplier of. tells us that a $M increase in the advertising budget would raise sales by.m (given that we are acting optimally). ) You are designing a poster that has inch margins at the top and along each side and a 3 inch margin at the bottom. To save on costs, you want the overall poster to be as small as possible, but you need the printed area inside the margins to be 80 square inches (the shaded area below). Solve for the dimensions of the poster. y 3 Let s call the overall width and height and y. Therefore, the shaded area will have an area of 4 y We need this are to be 80 square inches. We also want to minimize the overall area which is times y. So the minimization problem is: min y y subject to 4 80 So, set up the lagrangian

5 y 4 y 80 Let s multiply the stuff in parentheses out y y 4y 0 80 Now, take the derivatives with respect to and y and set them equal to zero. y y Solve each for lambda and set equal to one another. y y 4 Now, solve for y y 4 Plug back into the constraint y 4 80 Substitute in y Now, solve for 6 y 0 4

6 6) Suppose that you are a livestock breeder. You have 00 ft. of fencing to build a rectangular area divided into two pens. (see diagram below) w Find the dimensions to maimize the enclosed area. If you were given 0% more fencing, approimately how much bigger in area could your pens be? So, we are faced with maimizing the total area A w subject to the fact that we only have 00 ft. of fencing. So, 3w 00 ma w subject to 3 w 00 Write down the lagrangian w 00 3 w Take derivatives with respect to and w 0 w 30 Solve each for lambda and set them equal w 3 w 3 Now, use the constraint

7 3w y w 60, 000 A 0% increase in fencing would be an etra 0 ft. That could increase the ,000 sq. ft. optimal area by 7) You want to build a bo with a square base and an open top that has a volume of 3 cubic inches. Find the dimensions that will minimize the surface area. h So, we know that the volume is 3 cubic inches. h 3. We want to minimize the surface area. S 4h. min 4h subject to h 3 Again, the lagrangian 4h h 3

8 Take derivatives with respect to and w 4h h Solve each for lambda and set them equal 4h 4 h h h h h Now, use the constraint h 3 4h h h 3 h 8 h h4 A h 4 3 S 4 h 4 * ) Stafford Rug Company produces wool rugs and cotton rugs. Total cost (in dollars) is given by C 7X X X 9X. We can write Stafford s minimization problem as: Min 7X X, X subject to 9X X X The lagrangian for this problem is 0.X X l 7X 9X.X X X X 0

9 Taking the first order conditions, we get 4X.X 0 8X.X 0 X X 0 Solving for X, X and we get X A A 0 percent increase in orders (i.e. an increase of rugs would raise Stafford s costs by (approimately) 7.3* = $ ) A firm s inventory of a certain commodity is depleted at a constant rate per unit time. The firm reorders amount whenever the firm s inventory level drops to zero (new orders materialize instantaneously). The annual requirement for the commodity is 00 units and the firm orders n times per year. The firm incurs two types of costs; a holding cost and an order cost. The holding cost is equal to.08 times the average amount of inventories in the warehouse (/) while ordering costs are equal to. times the number of orders made per year. Therefore, total annual costs are equal to C. 08. n Minimize inventory costs (by choice of and n) subject to the constraint that total annual demand equals 00. Min.08.n, n subject to n 00 The lagrangian for this problem is l.08.n n 00 Taking the first order conditions, we get

10 l.04 n 0 ln. 0 n A Solving for, n and we get n 4.0 0) Suppose that you have the following technology for producing output. y k l The price of labor is $ per hour and capital costs $00 per unit. You need to produce at least 00 units of output. Find k and l to minimize your total production costs. Min l 00k k, l subject to k. l. 00 The lagrangian for this problem is l l k k l Taking the first order conditions, we get.. 00.k l 0...k l 0.. k l 00 First, solve the first two epressions for lambda 00.k l.k l.... Now, set the two epressions for lambda equal to each other

11 Simplify k l.k l k l.k l k l.k l 40 l k k l Now, use the constraint. k l l l l 00 l k 40l k 79 l 3,60