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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 10 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 (a) Consider a company with a single division. Derive an expression for the breakeven output, conducting sensitivity analysis and analyzing when the breakeven output will be positive. Answer: If a company has a single division then profits are: = R C = PQ TVC TFC Since AVC = TVC/Q, this implies that TVC = (AVC)Q and thus: = PQ (AVC)Q TFC = (P AVC)Q TFC We get breakeven output by setting = 0, which implies that: = (P AVC)Q TFC = 0 Breakeven Q = TFC/(P AVC) 1 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain The breakeven quantity is a ratio and will be positive only if both the numerator and denominator are positive, or, negative. Because a negative breakeven output makes no sense, the strict definition is: Breakeven Q = max {TFC/(P AVC), 0} Because TFC > 0, note how the breakeven output is > 0 (i.e. produce some output) only if the contribution margin (i.e. P AVC) is positive. If instead P AVC < 0, then the breakeven output will be negative which, being a nonsensical answer, means the company should cease production, which makes perfect sense from ECO 100. Recall the shut down rule for the short run: if P < AVC the company should cease production. Of course, saying P < AVC is the same as (P AVC) < 0, so that if the contribution margin is negative, the firm should shut down and continue to incur fixed costs this strategy minimizes losses by avoiding all avoidable (i.e. variable) costs. Observe too that the expression for breakeven output is a constant elasticity function in TFC and (P AVC): Breakeven Q = TFC/(P AVC) Breakeven Q = TFC (P AVC)1 Now for sensitivity analysis: ceteris paribus, as: TFC then breakeven Q . This is intuitive: with higher fixed costs, the company needs to sell more to breakeven. In fact, a 1% increase in TFC will lead to a 1% increase in breakeven quantity. (P AVC) then breakeven Q . This is intuitive: with higher contribution margin (i.e. gross profits per unit), the company needs to sell less to breakeven. In fact, a 1% increase in (P AVC) will lead to a 1% decrease in breakeven quantity. P then breakeven Q . This is intuitive: with higher prices, the company needs to sell less to breakeven. AVC then breakeven Q . This is intuitive: with higher variable cost per unit (i.e. AVC), the company needs to sell more to breakeven. (b) Consider a company with two divisions. Derive an expression for the breakeven output, conducting sensitivity analysis and analyzing when the breakeven output will be positive. 2 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Let's label the two divisions "1" and "2" and examine the breakeven output of division 1. Now profits are: = 1 + 2 = R1 + R2 TVC1 TVC2 TFC = P1 Q1 + P2 Q2 AVC1 Q1 AVC2 Q2 TFC = (P1 AVC1 )Q1 + (P2 AVC2 )Q2 TFC Break even profits occur when = 0. Thus: = (P1 AVC1 )Q1 + (P2 AVC2 )Q2 TFC = 0 Breakeven Q1 = {TFC (P2 AVC2 )Q2 }/(P1 AVC1) The breakeven quantity is a ratio and will be positive only if both the numerator and denominator are positive, or, negative. Because a negative breakeven output makes no sense, the strict definition is: Breakeven Q1 = max {{TFC (P2 AVC2 )Q2 }/(P1 AVC1), 0} Note how the breakeven output is > 0 (i.e. produce some output) only if: Either: {TFC (P2 AVC2 )Q2 } > 0 (i.e., division 2's gross profits do not cover total fixed costs) and the contribution margin of division 1 (i.e. P1 AVC1) is positive. The intuition is that if gross profits from division 2 are insufficient to cover fixed costs and division 1 is earning a profit per unit, then division 1 should produce output. Or: {TFC (P2 AVC2 )Q2 } < 0 (i.e., division 2's gross profits cover total fixed costs) and the contribution margin of division 1 (i.e. P1 AVC1) is negative. At first glance this seems puzzling. ECO 100 tells us that a company should shut down in the short run if P < AVC. Yet here, even if P < AVC, we're saying that division 1 should continue producing! What's happening here is that there is another division "helping" division 1 since {TFC (P2 AVC2 )Q2 } < 0 means TFC < (P2 AVC2 )Q2 so that division 2 is generating enough income to cover TFC. More rigorously, rewrite the expression for division 1's breakeven output: (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 Now: {TFC (P2 AVC2 )Q2 } < 0 means that the bold terms below are positive: 3 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 At the same time, with P1 AVC1 negative, we see that the expression above can only sum to zero if (Break even Q1) > 0. This explains why even if P1 < AVC1, division 1 should produce some output, provided division 2 is generating enough income to cover TFC. Note: To really get a deep understanding, consider the circumstance when division 1's breakeven quantity is 0. This will happen anytime when the expression: {TFC (P2 AVC2 )Q2 }/(P1 AVC1) < 0 Which happens either when the ratio is (+/) or (/+). Take the case of a (+/) ratio: {TFC (P2 AVC2 )Q2 } > 0 (i.e., division 2's gross profits do not cover total fixed costs) and the contribution margin of division 1 (i.e. P1 AVC1) is negative. Intuitively, given that division is incurring losses on a per unit basis combined with the fact that division 2 is not covering fixed costs, means that the company can breakeven by shutting down division 1. More rigorously, rewrite the expression for division 1's breakeven output: (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 Now: {TFC (P2 AVC2 )Q2 } > 0 means that the bold terms below are negative: (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 At the same time, with P1 AVC1 negative, we see that the expression above can only sum to zero if (Break even Q1) < 0 which, from: Breakeven Q = max {{TFC (P2 AVC2 )Q2 }/(P1 AVC1), 0} implies that division 1 should cease production. Next, take the case of a (/+) ratio. This implies {TFC (P2 AVC2 )Q2 } < 0 (i.e., division 2's gross profits cover total fixed costs) and the contribution margin of division 1 (i.e. P1 AVC1) is positive. More rigorously, rewrite the expression for division 1's breakeven output: (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 4 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Now: {TFC (P2 AVC2 )Q2 } < 0 means that the bold terms below are positive: (P1 AVC1 )(Break even Q1 ) + (P2 AVC2 )Q2 TFC = 0 At the same time, with P1 AVC1 positive, we see that the expression above can only sum to zero if (Break even Q1) < 0 which, from: Breakeven Q = max {{TFC (P2 AVC2 )Q2 }/(P1 AVC1), 0} implies that division 1 should cease production. Observe too that the expression for breakeven output is a constant elasticity function in: {TFC (P2 AVC2 )Q2 } and (P1 AVC1): Breakeven Q1 = {TFC (P2 AVC2 )Q2 }/(P1 AVC1) Breakeven Q1 = {TFC (P2 AVC2 )Q2 } (P1 AVC1)1 Now for sensitivity analysis: ceteris paribus, as: TFC then breakeven Q1 . This is intuitive: with higher fixed costs, the company needs to sell more in division 1 to breakeven. P1 AVC1 then breakeven Q1 . This is intuitive: with higher division 1 contribution margin (i.e. gross profits per unit), the company needs to sell less in division 1 to breakeven. In fact, a 1% increase in P1 AVC1 will lead to a 1% decrease in division 1's breakeven quantity. P1 then breakeven Q1. This is intuitive: with higher division 1 price, the company needs to sell less in division 1 to breakeven. AVC1 then breakeven Q1 . This is intuitive: with higher variable cost per unit (i.e. AVC), the company needs to sell more in division 1 to breakeven. P2 AVC2 then breakeven Q1 . This is intuitive: with higher division 2 contribution margin (i.e. gross profits per unit), the company needs to sell less in division 1 to breakeven. P2 then breakeven Q1. This is intuitive: with higher division 2 price, the company needs to sell less in division 1 to breakeven. 5 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain AVC2 then breakeven Q1 . This is intuitive: with higher variable cost per unit in division 2 (i.e. AVC), the company needs to sell more in division 1 to breakeven. (c) Compare the breakeven quantity when a company has a single division versus when the company has multiple divisions. When is it easier to breakeven: when a company has a single or multiple divisions? Answer: We discussed how companies often have multiple divisions to diversify risk. For example, a beer company might (and do) purchase a wine company because the two products move differently through the business cycle, which cancels out or diversifies the risk. Now, we'll see another reason to have multiple divisions: that it is easier to break even when there are multiple divisions than a single division provided the combined profits of the other divisions is positive. Here is why. The breakeven quantity for division 1 when it is the only division is: Single division breakeven Q1 = {TFC}/(P1 AVC1) whereas the expression for multiple divisions is: Multiple division breakeven Q1 = {TFC (P2 AVC2 )Q2 }/(P1 AVC1) Notice that if division 2 is profitable, i.e. (P2 AVC2 )Q2 > 0 (which is to say P2 > AVC2) then: {TFC (P2 AVC2 )Q2 }/(P1 AVC1) < {TFC}/(P1 AVC1) If division 2 profitable Multiple divisions breakeven Q1 < Single division breakeven Q1 If division 2 is profitable, it helps division 1 through a lower division 1 breakeven output. Of course, if division 2 is losing money, we have the opposite result that: If division 2 not profitable Multiple divisions breakeven Q1 > Single division breakeven Q1 because now division 1 has to "help" division 2. (d) Consider a company with ten divisions. Derive an expression for the breakeven output. 6 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: By now a pattern should've emerged and you can guess the answer must be: Breakeven Q1 = {TFC (Pi AVCi )Qi }/(P1 AVC1) Where the sum is for i = 2, .., 10 (i.e. the other 9 divisions). If you don't see this, here is the more rigorous way. Total profits are: = 1 + i Where the sum is from i = 2, .., 10, or: i = 2 + 3 + .. + 10 Back to total profits: = R1 + Ri TVC1 TVCi TFC = P1 Q1 + Pi Qi AVC1 Q1 AVCi Qi TFC = P1 Q1 + (Pi AVCi)Qi AVC1 Q1 TFC Break even profits occur when = 0. Thus: = P1 Q1 + (Pi AVCi)Qi AVC1 Q1 TFC = 0 Breakeven Q1 = {TFC (Pi AVCi)Qi }/(P1 AVC1) Will division 1's breakeven output be lower with 10 divisions than 1? Yes, so long as the combined gross profits of the other 9 divisions is positive. Notice this allows for some of the other 9 divisions to have losses. For the division 1's breakeven output to be lower for 10 divisions versus 1 division, all that is required is that the combined profits of the other 9 divisions is positive. Question 2 Is the Prestige Telephone Company's commercial customers in a perfectly competitive market? 7 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: No. In a perfectly competitive market the price is given by the "market", i.e. firms take the price from someplace else, typically an "exchange". The case does not mention any such market. In fact, there is another clue that the market is not competitive. In a competitive market, if you were to raise your price above market price, no other firm would follow you and your sales would be zero. On the other hand, if you were to lower your price below the market price, every firm would follow you and your sales would probably increase (we'll examine this issue in detail in perfect competition). Questions 2 (a) and (b) in the case tell you that given the March 2003 commercial price of $800/hr, if the company were to raise prices to $1,000/hr, commercial sales would drop by 30% (not 100%) and if the company were to lower prices to $600/hr, commercial sales would increase by 30%. Either way, changing prices has an impact on sales, so that PTC has a downward sloping demand curve. In fact, as a study question, you can easily derive the equation of PTC's demand curve. This will be very useful later when we do revenue and profit maximizing prices. Question 3 One of the questions in the PTC case was whether a commercial price of $1,000, with an expected 30% decrease in commercial hours would boost profits. We saw that this did not result in greater profits. (a) Suppose commercial hours are raised from $800/hr to $1,000/hr. What would the expected decrease in commercial hours have to be in order for the price increase to be profitable? Answer: With rising prices, which results in lower sales, variable costs will be lower. PTC hopes that revenues will change (increase or decrease) in a way that profits increase. To calculate the decrease in commercial sales needed for profitability, recall that AVC = $28/hr and that in March 2003, there were 138 commercial hours. Denote the new number of hours by "X". We want: (New P New AVC)(New Q) (Old P Old AVC)(Old Q) > 0 What made the analysis of PTC tractable is that New AVC = Old AVC because the TVC function is linear, so that AVC is a constant. Thus, we want: (1,000 28)(X) (800 28)(138) > 0 8 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain (1,000 28)(X) > (800 28)(138) X > {(800 28)(138)}/{1,000 28) X > 106,536/972 X > 109.6 Given the current number of commercial hours is 138, this represents almost a 20% decrease in commercial sales. Hence, if commercial prices increase from $800/hr to $1,000/hr, then as long as the decrease in commercial sales is at most (approx) 20%, this will generate profits. The intuition is that as P , then Q , so that TVC , i.e. "cost savings". Now, revenues can increase or decrease. If revenues increase, then for sure, profits . If you derive the demand curve see question 1 you'll note that revenues cannot increase going from a price of $800/hr to a price of $1,000/hr because these prices are on the elastic portion of the demand curve. In fact, revenues will go down, i.e. "revenue losses". To ensure that PTC profits increase, we want to make sure that the cost savings > revenue losses, or to put it differently, output can't fall too much. (b) Suppose commercial hours are raised from $800/hr to $X/hr. Given an expected decrease in commercial hours of 30%, what is X in order for the price increase to be profitable? Answer: By raising prices, which results in lower sales, variable costs will be lower. PTC hopes that revenues will change (increase or decrease) in a way that profits increase. To calculate the increase in commercial prices needed for profitability, recall that AVC = $28/hr and that in March 2003, there were 138 commercial hours. Denote the new price by "X". We want: (New P New AVC)(New Q) (Old P Old AVC)(Old Q) > 0 What made the analysis of PTC tractable is that New AVC = Old AVC because the TVC function is linear, so that AVC is a constant. Thus, we want: (X 28)(0.7)(138) (800 28)(138) > 0 (X 28)(0.7)(138) > (800 28)(138) (X 28)(0.7) > (800 28) 9 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain X 28 > (800 28)/0.7 X > 28 + (800 28)/0.7 X > $1,131 (approximately) If a 30% decrease in sales was associated with a price increase to at least $1,131 would a strategy of raising commercial prices result in greater profits. (c) Suppose commercial hours are raised from $800/hr to $1,000/hr. What would the expected decrease in commercial hours have to be in order for the price increase to eliminate losses? Answer: The proposals analyzed in the PTC case only asked if each would result in greater profits, not whether these would eliminate losses. In March 2003, PTC had losses of 21,438. To calculate the decrease in commercial sales needed to eliminate losses, recall that AVC = $28/hr and that in March 2003, there were 138 commercial hours. Denote the new number of hours by "X". We want: (New P New AVC)(New Q) (Old P Old AVC)(Old Q) > 21,438 What made the analysis of PTC tractable is that New AVC = Old AVC because the TVC function is linear, so that AVC is a constant. Thus, we want: (1,000 28)(X) (800 28)(138) > 21,438 (1,000 28)(X) > 21,438 + (800 28)(138) X > {21,438 + (800 28)(138)}/{1,000 28) X > 127,974/972 X > 131.67 Given the current number of commercial hours is 138 this represents almost a 4% decrease in commercial sales. Hence, if commercial prices increase from $800/hr to $1,000/hr, then as long as the decrease in commercial sales is at most (approx) 4%, PTC will eliminate losses or be "in the black". Question 4 From the analysis of the PTC case, can you say anything about PTC's returns to scale? Returns? 10 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain Answer: Recall that returns to scale is a long run concept. It measures the impact on output of increasing all inputs (i.e. scaling inputs up). We know that if: RTS > 1, increasing returns to scale, AC decreases with output RTS = 1, constant returns to scale, AC is constant with output RTS < 1, decreasing returns to scale, AC increases with output You might be tempted to take PTC's cost function: C = 197,820 + 28Q and look at: AC = C/Q AC = 197,820/Q + 28 And conclude that because AC is falling with Q, that PTC has increasing returns to scale. However, that is incorrect. This is because, in order to reach to such conclusion, you need to examine the behavior of long run AC. Since the PTC's cost function has a fixed cost, it is a short run cost function. Even if you had the long run AC, you cannot deduce returns to scale. As we discussed in class, long run AC's behavior depends on returns to scale and/or other factors such as input prices. Be careful about what we just said. For example, increasing RTS implies AC declines with output but not the other way around. Turning to returns: this is a short run concept. It measures the impact on output of increasing all variable inputs, holding fixed inputs constant. We know that if: Returns > 1, increasing returns, AVC decreases with output Returns = 1, constant returns, AVC is constant with output Returns < 1, decreasing returns, AVC increases with output Assuming input prices are constant, from: Cpower = 179 + 4Q 11 University of Toronto, Department of Economics, ECO 204. Summer 2009. S. Ajaz Hussain AVCpower = 4 so that there are constant returns in power Coperations = 21,600 + 24Q AVCoperations = 24 so that there are constant returns in operations In fact the overall cost function exhibits constant returns as well: C = 197,820 + 28Q and look at: AVC = TVC/Q AVC = 28 If the underlying production technology was CobbDouglas, you could label all fixed inputs by "k" and write the production function as: Q = Power Operations k where Q is hours of data service and = = 1 (i.e. constant returns in power and operations) and where nothing can be said about returns to scale (i.e. the value of + + ). Notice the constant returns result could also stem from a complements technology: Q = min(Power, Operations, k) Put simply, knowing the production technology will tell you the cost function and/or its properties but not the other way around. 12 ...
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This note was uploaded on 05/02/2011 for the course ECO 204 taught by Professor Hussein during the Fall '08 term at University of Toronto Toronto.
 Fall '08
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 Economics, Microeconomics

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