eco204_HW_12_solution

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Unformatted text preview: University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain ECO 204 2008‐2009 Ajaz Hussain HW 12 Solutions Question 1 Suppose the Ontario government levies a fine on tobacco companies for misleading smokers about the dangers of smoking (a similar measure was passed in the US a few years ago). If the fine is collected as a lump sum tax, should tobacco companies raise the price of cigarettes to compensate for the fine? Answer: The fine is collected as a lump sum tax which will show up on the books as an increase in TFC. From Lecture 14 ‐‐ where we used mathematical, graphical and intuitive arguments ‐‐ an increase in TFC does not, and should not, impact optimal output and therefore price. Question 2 In Lecture 14 we discussed the envelope theorem which gives us a simple technique for evaluating the change in the optimized objective when a parameter changes. In this question, you will reproduce some of the results in Lecture 14. Suppose a company ‐‐ using ECO 204 and ECO 220 tools ‐‐ estimates the demand and cost functions to be linear: P(Q) = a ‐ bQ C(Q) = TFC + c Q Where P is in $, Q is in ‘000s and a, b, TFC, c are positive parameters 1 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain (a) Interpret the parameters a, b, TFC, c. Answer: The parameters a and b appear in the equation: P(Q) = a ‐ bQ. Observe that the parameter a is the intercept of the demand curve and therefore the market’s maximum willingness‐to‐pay (max WTP) for the good; b is the slope of the demand curve. The parameters TFC and c appear in the equation: C(Q) = TFC + cQ. Here TFC is intercept of the cost curve and therefore the firm’s total fixed cost (TFC); c is the slope of the cost curve. Since the cost curve is linear, c is dC(Q)/dQ or simply the MC. (b) Assuming there is no opportunity cost and unlimited capacity, derive an expression for the firm’s profits with Q as the decision variable. Answer: With Q as the decision variable and profits as the objective, we express profits in terms of Q: ∏(Q) = R(Q) ‐ C(Q) → ∏(Q) = P(Q)Q ‐ C(Q) Note we have expressed price as P(Q) to denote the fact that price is a function of Q. → ∏(Q) = (a ‐ bQ)Q ‐ {TFC + cQ} → ∏(Q) = aQ ‐ bQ2 ‐ TFC ‐ cQ This is the objective in terms of Q. Note that we haven’t performed the optimization yet‐‐ hence this is not the optimized objective. (c) What is the optimal Q expressed as a function of the parameters a, b, TFC and c? Answer: To maximize profits, set: d∏(Q)/dQ = 0. This yields: d∏(Q)/dQ = a ‐ 2bQ ‐ 0 ‐ c = 0 → 2bQ = a ‐ c → Q = (a ‐ c )/2b This is the optimal decision variable. Observe how it is a function of a, b and c. That is: Q(a, b, c) = (a ‐ c )/2b 2 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain As an exercise, verify this formula is right by using the parameters a, b, c and TFC from HW 11. (d) Use the envelope theorem to gauge the impact on the optimized profits from an increase of 1 unit from each the parameters a, b, TFC and c, holding all other parameters constant. (That is, what is the impact of raising the parameter a by 1 unit holding all other parameters constant). Answer: Without the envelope theorem, we’d substitute Q = (a ‐ c )/2b into the objective ∏(Q) = aQ ‐ bQ2 ‐ TFC ‐ cQ to express profits only in terms of parameters: ∏(Q(a, b, c) | a, b, c) = aQ(a, b, c) ‐ b Q2(a, b, c) ‐ TFC ‐ cQ(a, b, c) → ∏(Q(a, b, c) | a, b, c) = a{(a ‐ c )/2b} ‐ b {(a ‐ c )/2b}2 ‐ TFC ‐ c{(a ‐ c )/2b} This is the optimized objective. Then we’d differentiate the optimized objective with respect to a, b and c respectively. That sounds awful ‐‐ as you’ll see in the nasty algebra in part (g). But we have the envelope theorem. It tells us we simply should differentiate the objective with respect to a, b, c and TFC. The objective is: ∏(Q) = aQ ‐ bQ2 ‐ TFC ‐ cQ From which: d∏(Q)/da = Q d∏(Q)/db = ‐Q2 d∏(Q)/dc = ‐Q d∏(Q)/dTFC = ‐1 In these expressions which “Q” do we use? We use the optimal Q = (a ‐ c )/2b. The expressions above can also be re‐expressed as: d∏(Q)/da = (a ‐ c )/2b d∏(Q)/db = ‐{(a ‐ c )/2b}2 d∏(Q)/dc = ‐ {(a ‐ c )/2b} d∏(Q)/dTFC = ‐1 3 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain (e) Interpret the results in part (d). Answer: We had: d∏(Q)/da = Q d∏(Q)/db = ‐Q2 d∏(Q)/dc = ‐Q d∏(Q)/dTFC = ‐1 where Q is the “optimal Q”. To understand these, recall what you’ve seen in MATH 133. Say, for example, y = 3x. Then: dy/dx = 3 This says “if you increase x by 1 unit, y will increase by 3 units”. You can see this by re‐arranging dy/dx = 3 as: dy = 3 dx ↔ The change in y = 3 The change in x If the change in x is +1 unit, it implies that the change in y is: The change in y = 3 * (1) = +3 The argument works in reverse as well: suppose x decreases by 1 unit‐‐ then y will decrease by 3 units. Using this reasoning: d∏(Q)/da = Q means: • • If the parameter a (i.e. the max WTP) increases by 1 unit, profits will increase by Q, the optimal output (= (a ‐ c )/2b). If the parameter a (i.e. the max WTP) decreases by 1 unit, profits will decrease by Q, the optimal output (= (a ‐ c )/2b). d∏(Q)/db = ‐Q2 means: 4 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain • • If the parameter b (i.e. the slope of the demand curve) increases by 1 unit, profits will decrease by Q2, the optimal output squared (=(a ‐ c )/2b)2. If the parameter b (i.e. the slope of the demand curve) decreases by 1 unit, profits will increase by Q2, the optimal output squared (=(a ‐ c )/2b)2. d∏(Q)/dc = ‐Q means: • • If the parameter c (i.e. MC) increases by 1 unit, profits will decrease by Q, the optimal output (=(a ‐ c )/2b). If the parameter c (i.e. MC) decreases by 1 unit, profits will increase by Q, the optimal output (=(a ‐ c )/2b). d∏(Q)/dTFC = ‐1 means: • • If the parameter TFC (i.e. total fixed cost) increases by 1 unit, profits will decrease by 1 unit. If the parameter TFC (i.e. total fixed cost) decreases by 1 unit, profits will increase by 1 unit (f) If you, the manager had a budget to change one of the parameters (for the same cost) which one parameter would you pick and what would you do? Answer: If the manager wants to maximize profits, she can do the following with the parameters a, b, c and TFC: • • a: She can spend money to raise the parameter a (i.e. the max WTP) by 1 unit, in which case profits will increase by Q, the optimal output (= (a ‐ c )/2b). b: She can spend money to decrease the parameter b (i.e. the slope of the demand curve) by 1 unit, in which case profits will increase by Q2, the optimal output squared (=(a ‐ c )/2b)2. c: She can spend money to decrease the parameter c (i.e. MC) by 1 unit, in which case profits will increase by Q, the optimal output (=(a ‐ c )/2b). TFC: She can spend money to decrease the parameter TFC (i.e. total fixed cost) by 1 unit, in which case profits will increase by 1 unit. Parameter Change Effect on Profits • • 5 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain ↑ a by 1 unit ↓ b by 1 unit ↓ c by 1 unit ↓ TFC by 1 unit ↑ ∏ by Q units ↑ ∏ by Q2 units ↑ ∏ by Q units ↑ ∏ by 1 unit If Q > 1, the reducing TFC is the least attractive way to spend the budget (the increase in profits will only be 1 unit). Next, note that raising max WTP (parameter a) will have the same effect on profits as reducing MC by 1 unit. Of course, in real life, it’s easier and more predictable to reduce MC than to raise max WTP. Finally, reducing b (the absolute slope of the demand curve, which makes the demand curve flatter) has the greatest impact on profits. Dollar for dollar, if Q > 1, the manager should aim to reduce b, followed by either a or MC, and lastly by reducing TFC. (g) Suppose you didn’t know the envelope theorem. What is the impact on the optimized profits from an increase of 1 unit of the parameter a? (Of course, you should have the same answer as in part (d)). Answer: From above, the optimized objective was: ∏(Q(a, b, c) | a, b, c) = a{(a ‐ c )/2b} ‐ b {(a ‐ c )/2b}2 ‐ TFC ‐ c{(a ‐ c )/2b} To see how this reacts to an increase in the parameter a, we differentiate ∏(Q(a, b, c) | a, b, c) with respect to a. The algebra will be tractable if we expand ∏(Q(a, b, c) | a, b, c): ∏(Q(a, b, c) | a, b, c) = a{(a ‐ c )/2b} ‐ b {(a ‐ c )/2b}2 ‐ TFC ‐ c{(a ‐ c )/2b} → ∏(Q(a, b, c) | a, b, c) = a2/2b ‐ ac/2b ‐ b {(a2 ‐ 2ac + c2)/4b2} ‐ TFC ‐ ac/2b + c2/2b → ∏(Q(a, b, c) | a, b, c) = a2/2b ‐ ac/b ‐ (a2 ‐ 2ac + c2)/4b ‐ TFC + c2/2b → ∏(Q(a, b, c) | a, b, c) = a2/2b ‐ ac/b ‐ a2/4b + 2ac/4b ‐ c2/4b ‐ TFC + c2/2b → ∏(Q(a, b, c) | a, b, c) = a2/2b ‐ ac/b ‐ a2/4b + ac/2b ‐ c2/4b ‐ TFC + c2/2b → ∏(Q(a, b, c) | a, b, c) = a2/2b ‐ ac/2b ‐ a2/4b ‐ TFC + c2/4b → d∏(Q(a, b, c) | a, b, c)/da = 2a/2b ‐ c/2b ‐ 2a/4b 6 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain → d∏(Q(a, b, c) | a, b, c)/da = a/b ‐ c/2b ‐ a/2b → d∏(Q(a, b, c) | a, b, c)/da = a/2b ‐ c/2b → d∏(Q(a, b, c) | a, b, c)/da = (a ‐ c)/2b But we know that Q = (a ‐ c)/2b. So: d∏(Q(a, b, c) | a, b, c)/da = Q Which is what we had above: see how much work the envelope theorem saves us? (h) Use the envelope theorem to gauge the impact on the optimized profits with price as the decision variable. Answer: With P as the decision variable and profits as the objective, we express profits in terms of P (see HW 11 again): ∏(P) = R(P) ‐ C(P) → ∏(P) = PQ(P) ‐ C(P) → ∏(P) = PQ(P) ‐ {TFC + cQ} Now P(Q) = a ‐ bQ Thus Q(P) = (a ‐ P)/b → ∏(P)= P(a ‐ P)/b ‐ TFC ‐ c((a ‐ P)/b) → ∏(P) = aP/b ‐ P2/b ‐ TFC ‐ ac/b + cP/b This is the objective in terms of P. By the envelope theorem, the impact on the optimized objective due to an increase in the parameter a is: → d∏(P)/da = P/b ‐ c/b Or: → d∏(P)/da = (P‐ c)/b → d∏(P)/da = (a ‐ bQ ‐ c)/b = (a ‐ c)/b ‐ Q When we say “Q” here, we mean the optimal Q, which we know is Q = (a ‐ c)/2b. Hence: → d∏(P)/da = (a ‐ c)/b ‐ (a ‐ c)/2b = (a‐ c)/2b = Q 7 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain Note: whether we express profits in terms of Q or P, the impact on optimized profits to the parameter a (or any parameter for that matter) is the same. Math is groovy. Question 3 In Lecture 14 we discussed how if a monopolist could charge a lower price than competitive markets, provided the monopolist had a low enough MC (see transport example with railways vs. horses). Suppose all perfectly competitive companies have a constant MCc. Suppose also that the market has a linear market demand curve P(Q) = a ‐ bQ. Suppose the monopolist has a constant MCM. What must be true about MCM if the monopoly price (PM) is lower than the competitive price (PC). Hint: See Lecture 14. Answer: The competitive firm’s price must be equal to MC (because every firm sets MR = MC and in perfect competition MR = P, so that P = MC): PC = MCc The monopolist sets: MRM = MCM By the “same intercept and twice the intercept rule”: → a ‐ 2bQM = MCM → QM = (a ‐ MCM )/2b Thus: PM = a ‐ b QM → PM = a ‐ b (a ‐ MCM )/2b → PM = a ‐ (a ‐ MCM )/2 → PM = a ‐ a/2 + MCM /2 → PM = a/2 + MCM /2 → PM = (a + MCM )/2 If the monopolist charges a lower than under competition: PM < PC 8 University of Toronto, Department of Economics, ECO 204 2008‐2009 S. Ajaz Hussain → (a + MCM )/2 < MCc → MCM < 2 MCc ‐ a But the parameter a is the max WTP. Hence: PM < PC ↔ MCM < 2 MCc ‐ a Hence, if the monopolist and perfect competition have the same (linear) demand and constant marginal costs, a monopoly will charge a lower price if it’s MC is less than twice the MC of competition minus the maximum WTP. 9 ...
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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.

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