Assignment questions The questions in this section contribute to your assignment grade. Starred questions are at the level of exam questions. Notes: . There are notes on this application on the canvas course page. Cost, Demand, Revenue, and Profit functions are defined there. . In the questions below, we will measure time t during the year in months with t = 0 the beginning of the year (Western calendar, January 1 just after midnight) and t = 12 the end of the year (December 31 just before midnight).. (*'kf Doctor Roberts sets up a factory producing Cherry Blossom Widgets (CBW). This factory is the only producer (monopoly). The xed costs of the factory are $7,000 per month and the price to produce a CBW is $4. A market study done at a particular time of year predicts that the demand (1(1)) for CBW is 9(2)) = 50-17 where q is the demand in thousands of CBW per month when p is the sale price in dollars. Let as be the factory production in thousands of CBW per month that meets demand at price p. (a) (1 mark) Write the cost function C (:3) in thousands of dollars per month. (b) (1 mark) Write the revenue function R(a:) in thousands of dollars per month. (c) (1 mark) Write the prot function P(as) in thousands of dollars per month. (d) (1 mark) Determine the optimal prot per month and the production level at which it occurs. Give units for your answers. . (***jf) Doctor Roberts notices that the demand for CBW is not constant during the year. It is highest in the Spring (23 = 3) and lowest in the Fall (t = 9). A consulting rm, Math Knights Inc., is hired and they predict a timedependent demand curve of the form q(p) = (1 + Asin(7rt/6)) (50 - p). with A a constant in the interval (0,1) that they estimate to be A = 1/4 using market research. (a) (2 marks) Describe briey why the form of the demand model is reasonable in its behaviour in t and p. (b) (2 marks) At what times of the year could the market survey from question #2 have been done? (c) (4 marks) If the CBW production is xed at as = 20 (20,000 widgets per month) and the price adjusted over time to the demand curve, what is the rate of change of revenue with respect to time on February 1 (t = 1)? Make sure to give units to your answer. . (3 marks) (*vr) Using the time dependent demand model from #3 above, determine the pro duction that optimizes prot as a function of time t. . (2 marks) (*rzr) If there is a one-month delay between production and sale, how should the optimal production as a function of time be adjusted? Math 1000 Applications 1: Costs, Revenue, Demand, Prot Brian Wetton June 2023 1 Prot Note 1 A widget is a. term used in economics for a generic produced good. Consider a small factory run by Doctor Bob producing widgets and selling them to wholesalers. Let a: be the number produced every month and Pm) be the monthly prot. To make the numbers more manageable, we will consider a: to be in thousands of widgets per month, and P in thousands of dollars per month. Suppose that an internal company research report predicts 13(3) = 45.? +169: 6. (1) Some simple modelling that could arrive at such a prediction is shown in the next section. For now, let us take this as given. A graph of P is shown in Figure 1. Note 2 The gure was generated with a mathematical tool called MATLAB that UBC students can use for free. You could also use a freeware gmphlng tool collect DESMOS. We can remark on a few features of interest on this graph. Note that P (0) = -6. That is, with no widgets produced, Dr. Bob will lose $6,000 per month. These are called xed costs and could include for example the mortgage, taxes, and upkeep on the factory building. However, there is a range of production that gives a positive prot, so the business is viable. The production :51 is when the factory can shift to being protable. This production value is called the break-even point. To nd the break-even point and the maximum prot and where it occurs, we complete the square for P: Pas) = 10 4(a: 2? to see that the maximum prot of 10 (remember that is $10,000 per month) occurs with production of :12 = 2, $2,000 widgets per month. We can compute the break-even points 331,2 = 2 i \\(10/4 as 0.4189,3.5811 Proflt as a functlon of production f Maximum Profit 10 Breakeven X1 /. Fixed costs ..| .J I. l. _. .L _ 0 0.5 1 1.5 2 2 5 3 3.5 4 Figure 1: Prot as a. function of production. using the square completed form of P or using the quadratic formula. on the original form. Note 3 Later in the term, we will use Calculus to nd the maximum prot. 2 Costs, Revenue, and Demand Let us look at the kind of modelling that could lead to the prot function in Equation (1). We introduce the demand function p(a:) that relates the price in dollars per widget p to the number of widgets produced. Of course, Doctor Bob could produce as many widgets as they want and try to sell them at any price 39. However, they would be foolish to sell widgets at a lower price than they could or to price their widgets so high that they do not all sell. We could think of the inverse function (the) which is the maximum number of widgets that could be sold each month at a price p. When written 33(p), a: is usually called the demand, which production should be matched to. If you nd the concept of demand function confusing, there is an accompanying video with further explanation. Market research predicts a demand function 2(1)) = 5 - 10/4- Note that 37(1)) is decreasing: we will always expect that demand decreases as the price increases. We will see below that we will have a choice of using 55' or p as the independent variable for the prot function. Let us choose as here to match the form in the previous section and convert m(p) above to p(a:): p(a:) = 20 4:13. (2) Note 4 In this example, the algebra is equally easy whether we use a; or p as the independent uariable. In other cases, one might be much easier to use. We can now compute the revenue function T'(:L'), the amount of money com- ing in every month from selling the production of m widgets: r(1)= 22pm) = n20 43.") = 20m 41-2. Producing widgets has costs C(33), and it is estimated that C(33) = 6 + 49:. The \"6\" is the xed costs discussed in the last section, and 456 represents variable costs that are a constant $4 per widget in this case. We can now put together this information and derive the Prot function in (1). Prot is simply revenue minus costs PU") = NSC) C(56) = 9310(56) C(93) = 4:c2 + 1631: 6. Note 5 Word problems will give you enough information to determine the de- mand a'(p) or p(a:) as well as the costs C(50). Then the prot combines these PM = new} - C(93) PUD) = 33(1));0 - C(m(p))- Break-euen points can then be computed with root nding and maaimum prot points can be computed using calculus techniques we will learn later in the course. 3 Problems Note 6 There will be additional problems of this type in the Applications IV notes on optimization, where we will use calculus to nd the optimal prot. Problem 1 (*Vk'z) Try nding P(p) and check that the maximum value of this function is also 10 and corresponds to p = 10(2) = 12. Problem 2 (***) Doctor Bob leaves the operation of their factory to their oldest child {a graduate of the combined Economics and Mathematics BA program at UB C j and becomes the manager of a hotel. The hotel has 100 rooms which have upkeep of $20 per day if they are occupied and $10 if not. They experiment with changing the cost of the hotel rooms. When they charge $200 per night, they have an average occupancy of 60. When they charge $250 per night, occupancy goes down to 50. Assuming a linear demand function, determine the room charge that will maximize prot. 4 Solutions Solution 1 We were given 27(1)) = 5 p/4 and C(33) = 6 + 4.1:. Now PUD) = MUD) - 00105)) = P(5 -iJ/4) - (6 + 4(5 - 39/41)) = '392/4 + 6P - 26- We complete the square Pia) = (p2 24:2 +104)/4 = c(p 12)\" 4W4 =10 (p -12)2l4 from which we can see that the optimal price is $12 per widget leading to an optimal prot P(12) = 10 (that is, $10,000 per month). Solution 2 The costs are do) = 1000 + 103: where a: is the number of occupied rooms ($10 per room plus another SM if they are occupied). We are given two points on the demand curve.- 33(200) = 60 and 3(250) = 90. With the assumption that who) is linear, we nd it has slope _ 5060 _ m' 250200 ' l 5 (negative as expected). Using the point slope formula, we have 3:03) = 60 - (p 200)/5 = 100 - p/S. We construct the prot function Hp) = pw(p)-c{a:(p)) = p(100-p/5}-(1000+10(100-p/5)) = -p2/5+102p-2000. Completing the square we have P(p) = -(p2+510p+10000)/5 = -((p-255)2-55025)/5 = -(p-255)2/5+11005. From this last expression, we can see that the optimal room charge is p = 255 (that is, $255 per night). Although not requested in the wording of this question, we can add that this corresponds to an occupancy of $0.55) = 49 and a daily profit of $11,005