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Unformatted text preview: University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain ECO 204 Summer 2009 S. Ajaz Hussain Practice Problems 16 Solutions Please help improve the course by sending me an email about typos or suggestions for improvements Note: Please don't memorize these solutions in the expectation that similar questions will appear on tests and exams. Instead, try to understand how to derive the answer as you'll be tested on techniques and applications, not on memorization. Moreover, tests and exams will cover topics and techniques that may not be in these practice problems. You are urged to go over all lectures, class notes and HWs thoroughly. Question 1 Ajax Microchip Corporation (AMC) manufactures and sells microchips. In July 2009, it produced and sold 1,000 microchips at a price of $150/microchip. AMC's cost function is estimated to be: C = 100 + 38Q Where Q is in `000s of units (i.e. 1,000 units would be Q = 1). Assume AMC has the capacity to produce 10,000 microchips a month. (a) For July 2009, management estimates that if price per microchip was dropped from $150 to $140, sales would increase by 50% while if price per microchip was raised from $150 to $160, sales would decrease by 50%. Based on this information, can you "estimate" AMC's demand curve? Express the demand curve with Q in `000s of units. Answer: Currently, at P = $150, Q = 1. Assume the demand curve is linear. This is a plausible assumption because when price increases by $10 demand falls by 50% or 500 units while when price decreases by $10 demand rises by 50% or 500 units. Thus, either way, the rate of change is the same implying the demand curve is linear: 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain A linear demand curve had the form: y = intercept + slope x. Here, y is price, x is quantity (this time expressed in `000s of units) so that we have to solve for a and b in: P = a b Q When P = $150, Q = 1 and when P = $140, Q = 1.5. This gives us two equations in two unknowns: (Equation 1) 150 = a b (1) (Equation 2) 140 = a b (1.5) Subtract the 2nd equation from the 1st equation to get: 10 = 0.5 b b = 20 From any equation you can check that a = 170. Thus, the demand equation is: 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain P = 170 20Q (b) If AMC wanted to maximize revenues, calculate the optimal price and quantity. Answer: To maximize revenues, set MR = 0. Using the short cut for linear demand this is: MR = 170 40Q Setting MR = 0: 170 40Q = 0 40 Q = 170 Q = 170/40 Q = 4.25 or 4,250 units. For price, use the demand curve: P = 170 20Q = 170 20(4.25) = $85 (c) Calculate the price elasticity for your answer in part (b). Obviously you should get E = 1. Answer: We can use two formulas for computing E: arc and point. Given a price and a quantity, we should use the point E formula: E = (dQ/dP)(P/Q) Now, the demand equation is: P = 170 20Q. To get dQ/dP we have to rearrange it: P = 170 20Q Q = 8.5 0.05 P This implies dQ/dP = 0.05. 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Thus E = (dQ/dP)(P/Q): E = 0.05(85/4.25) = 1 This makes sense because we have a linear demand curve, where revenues are maximized at the midpoint where E = 1. (d) Calculate the average variable cost and interpret it. Answer: The AVC is: AVC = TVC/Q The cost function was C = 100 + 38Q from which the TVC = 38Q. Now: AVC = TVC/Q AVC = 38Q/Q AVC = 38 Thus, the variable cost per unit is always $38. If this were an accounting course, our contribution analysis calculations (as in the Prestige Telephone Company) would be spot on. (d) Calculate the marginal cost and interpret it. Answer: The MC is: MC = dC/dQ Starting at any output, MC is the cost comprising of variable costs of producing the next unit. The cost function was C = 100 + 38Q from which: MC =dC/dQ MC = 38 Note this is the same as MC = dTVC/dQ = 38 which shows that MC is the variable cost of 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain producing another unit. Observe that in this question MC = AVC which is always true of a linear cost function. So for example, had the cost function been: C = 100 + 38Q2 Then: AVC = 38Q while MC = 76Q. (e) If AMC wanted to maximize profits, calculate the optimal price and quantity. Answer: To maximize profits, set MR = MC. From above: MR = 170 40Q and MC = 38 Setting MR = MC: 170 40Q = 38 40 Q = 132 Q = 132/40 Q = 3.3 or 3,300 units For price, use the demand curve: P = 170 20Q = 170 20(3.3) = $104 Observe how the profit maximizing price > revenue maximizing price. This is to be expected when a company has variable costs: when maximizing profits it has to "cover" the (variable) cost of production and must therefore charge a higher price than the revenue maximizing case. (f) Calculate the price elasticity for your answer in part (e). Obviously you should get E > 1. Answer: We can use two formulas for computing E: arc and point. Given a price and a quantity, we should use the point E formula: 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain E = (dQ/dP)(P/Q) Now, the demand equation is: P = 170 20Q. To get dQ/dP we have to rearrange it: P = 170 20Q Q = 8.5 0.05 P This implies dQ/dP = 0.05. Thus E = (dQ/dP)(P/Q): E = 0.05(104/3.3) 1.6 This makes sense because we have a linear demand curve, where profits are maximized on the elastic (top half) of the demand curve. Put another way, the company cannot maximize profits when demand is inelastic. Here is why: suppose demand is inelastic. If P then R . But P leads to Q which in turn leads to TVC. Thus by P we have R and TVC which implies that . Thus, to maximize profits, having ruled out the inelastic portion of the demand curve we must be either on the mid point or the top half. But the mid point is where revenues are maximized. Thus, profits must be maximized on the top half. (g) Suppose AMC's CEO Ajax uses company funds to buy himself a nice hair piece. This hair piece, while very nice, raises AMC's TFC to $1,000,000,000. Should AMC raise prices to "cover" the higher fixed cost? Answer: No. No. No. This is one argument that confuses lots of managers and accountants. It is true that higher MC which consists of variable but not fixed costs will lead to higher prices. But this is because production and therefore price is related to variable costs. On the other hand, no matter what the output and price is the TFC is fixed (or in the accountants' nomenclature "unavoidable"). Thus, when maximizing profits: = R TVC TFC Companies should price to maximize the gap between R and TVC. This will occur happen when MR = MC. 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain The "math" way to see this is to note that TFC never entered the calculations for optimal prices. It is a constant and thus drops out of the derivatives. Hence, even if TFC = $1,000,000,000 AMC should continue charging P = $104 and will expect to sell 3,300 units. Of course, AMC profits will be lower but the price and output will remain unaltered. This gives you a clue about when there will be a change in the price: either from a change in demand and/or a change in variable costs. (h) Now suppose AMC's capacity is 3,000 units (i.e. Q = 3). Solve for the profit maximizing price and output. Answer: From above, we had to maximize profits, set MR = MC: MR = 170 40Q and MC = 38 Setting MR = MC: 170 40Q = 38 40 Q = 132 Q = 132/40 Q = 3.3 or 3,300 units However, capacity is only 3,000 units. Thus, AMC will produce Q = 3 and the price will be: P = 170 20Q = 170 20(3) = $110 This is a higher price than before since AMC is producing less. (i) Given your answer in part (h) what is the value (before cost of capacity) of adding another unit (microchips, not Q = 1 which is 1,000 units) of additional capacity? Answer: To find value of adding capacity solve the problem again as a constrained optimization "Lagrange" problem. Setup Lagrangian: L = R TFC TVC [Q 3] 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain The first order conditions (FOCs) are: L/Q = 0 MR MC = 0 Using expressions for MR and MC from above: 170 40Q 38 = 0 132 40Q = 0 The second FOC is: L/ = 0 Q = 3 Substitute in first FOC to get: 132 40Q = 0 132 40(3) = 0 132 120 = 0 132 120 = 0 = 12 What does this mean? Look at the Lagrangian again: L = R TFC TVC [Q 3] L = R TFC TVC [Q Capacity] By the envelope theorem what is the impact on profits from adding another unit of capacity? It is simply = L/Capacity = . Thus, at Q = 3, = 12. Thus, if we raise capacity by a 1,000 units (Q = 1), profits (before cost of capacity) will increase by 12. As long as the cost of adding this capacity is < 12 this is a profitable proposition. Thus, if you add a unit (Q = 1/1000) of capacity, profits will increase by 0.012. 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (j) What is the optimal capacity of AMC? Answer: The optimal capacity is reached at the point where there is no additional value to adding capacity. That is, when = 0. From the 1st FOC above: 132 40Q = 0 Let = 0. Thus: 132 40Q = 0 Q = 132/40 = 3.3 This makes complete sense: when there was no capacity constraint, the optimal output was 3.3. When asked for the optimal capacity, we get the logical answer that it should not be more than 3.3. Nice. 9 ...
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