OPMT 5701-Online Final Assignment Winter 2014 1. You are an assembler of specialty computer terminals with a modest amount of monopoly power. Suppose that your average revenue per unit depends on how many terminals per day you wish to sell, and is given by AR(q) = q 3 + 12q 2 30q + 1000 where q is sales per day. Suppose further that your average cost of production is given by 100 q AC(q) = 2q + 1000 Notice that your total costs are negative if you choose to produce nothing. This is because you receive a grant from the government for setting up in Surrey, B. C. (a) Write out an expression for your pro…ts as a function of output, q. (Hint AR(q) is also the price) (b) Determine the most pro…table level of output. Show that this output level does indeed lead to a maximum rather than a minimum by checking the second order conditions. (Hint, you should …nd three critical points, two should be relative maximums. All q vaules should be between 0 and 10) 2. A …rm has two markets, with the demands given by P1 = 240 2Q1 P2 = 180 Q2 The …rm has an option of (a) setting up two factories and producing goods 1 and 2 separately, or (b) having one factory and producing the goods jointly. For option (a) the cost functions are 0:5Q21 + 5000 C(Q1 ) = C(Q2 ) = Q22 + 5000 For option (b) the joint cost function is C(Q1 ; Q2 ) = 0:5Q21 + Q22 Q1 Q2 + 12000 (a) Determine the prices, quantities and pro…t when the …rm uses to separate factories (b) Determine the prices, quantities and pro…t when the …rm uses joint production. (c) What is the best option? (Hint: show that joint production is better than separate production) 3. A monopolist o¤ers two di¤erent products, each having the following market demand functions q1 = 14 q2 = 24 1 4 p1 1 2 p2 (try (try p1 = 56 p2 = 48 4q1 ) 2q2 ) The monopolist’s joint cost function is C(q1 ; q2 ) = q12 + 5q1 q2 + q22 The monopolist’s pro…t function can be written as = p1 q1 + p2 q2 C(q1 ; q2 ) which is the function of four variables: p1 ; p2 ; q1 ;and q2 . Using the market demand functions, we can eliminate p1 and p2 leaving us with a two variable maximization problem. (a) Find the optimal quantities, prices and total pro…t (b) check second order conditions 1 4. Analyzing di¤erent types of utility functions Consider the case of a two-good world where both goods, x and y are consumed. Let the consumer, Beaufort, have the utility function U = U (x; y). Beaufort has a …xed money budget of B and faces the money prices Px and Py . Therefore Myrtle’s maximization problem is Maximize U = U (x; y) Subject to B = Px x + Py y For EACH utility function listed below, i ii iii iv 2 U = (xy) 1=2 U = (xy) U = 2 ln x + ln y U = x2=3 y 1=3 do the following: (a) Find the MRCS (or MRS) of each utility function. What do you notice when you compare the MRCS of (i) and (ii)? What about the MRCS of (iii) and (iv)? (b) Suppose the budget is initially B = 24, and Px = 6; Py = 6: Find use the Lagrange method to …nd the optimal x; y , value for U (c) Let B increase from 24 to 36. Find the NEW optimal x and y , value for U B B b a and y = : Hint: for U = xa y b the answers for x and y should be x = a + b Px a + b Py (you should be able to verify your answers using this formula, but you need to derive the answers from the Lagrange) 5. A person faces two constraints: a budget constraint (B Px x + Py y) and a coupon constraint (C cx x + cy y). The person wants to maximize utility but cannot violate either constraint. Let’s suppose the utility function is of the form U = x y 2 . Further, let B = 100; Px = Py = 1 while C = 120 and cx = 2; cy = 1:The (a) On the same graph, carefully draw the two constraints: 100 = x + y and 120 = 2x + y. Shade in the region that satis…es BOTH constraints (hint it will be "kinked") solve for where the two constraints cross. (b) Determine the x and y that maximizes utility Do this by solving two lagrange equations, one for each constraint Z = xy 2 + Z = xy 2 + 1 (100 x 2 (120 2x y) and y) . Which constraint(s) are binding? Draw a graph of your answer (c) suppose the person’s utility function is now U = x2 y: Redo part (a) Hint, watch the Lecture Video on Inequality Constraints (Kuhn-Tucker) 6. There are two …rms, each with the same total savings functions: T S1 = 200e1 0:5(e1 )2 and T S2 = 250e2 2(e2 )2 ; where e1 and e2 are the emissions from …rm 1 and …rm 2 respectively. The two …rms want to maximize their savings from emissions, subject to whatever regulatory constraints they face. Currently, both …rms are constrained by a pollution standard limiting their emissions to e1 = e2 = 48. 2 The government decides to implement a new "Cap and Trade" program that will allow the …rms to trade emission credits from their initial allocation of 48 credits each. However, any trade of emissions must maintain the air quality constraint at the two receptors that are set up in the region the …rms operate. The national air quality constraint limits particles in the air to P = 120. If particles rise (P > 120) means air quality falls. Each receptor has it’s own di¤usion equation that links emissions to particles in the air. At receptor one, the formula is P1 = 1:5e1 + e2 At receptor two the di¤usion equation is P2 = e1 + 1:5e2 The economic and environmental problem is to M ax T S1 + T S2 subject to P1 = 1:5e1 + e2 120 P2 = e1 + 1:5e2 120 (a) Set up the Lagrangian for this problem and solve for the optimal e1 and e2 : (b) Determine which constraint(s) are binding. (c) Given that both …rms begin with 48 emission credits each, determine the amounts traded and the trade ratio. (d) Graph your answer carefully, labelling all the relevant points. 3