Homework 6TaxesEcon 101Prof. Guse, W & L UniversityDecember 5, 20081. Unit Tax. Suppose that the market for cigarettes is represented by the following demandand supply functionsQd(p) = a − bpQs(p) = dp − cwhere a, b, c and d are all non-negative numbers withab>cd. Suppose the governmentimposed a tax τ per unit where 0 < τ <ab−cd.(a) What is market clearing price for cigarettes, P0, and quantity Q0, before the tax? AN-SWER setting demand equal to supply and solving for price we have ...Qd(p) = Qs(p)⇒a − bp = dp − c⇒a + c = p(d + b)⇒p =a + cd + bPlugging that price back into either the demand or supply curve will yield the equilibriumquantity traded ...1Qd(p =a + cd + b) = a −b(a + c)d + b=ad − bcd + b(b) What is the elasticity of demand, ǫ, and the elasticity of supply, σ at the pre-tax equi-librium point? ANSWER. B y definition ǫ =∆QdP∆P Qd. Since the demand curve is linear,−b always represents∆Q∆P. Therefore ǫ =−bPQ. Plugging in the equilibrium price andquantity derived in the previous question we haveǫ =−bPQ=−ba+cd+bad−bcd+b=−b(a + c)ad − bcSimilarlyσ =dPQ=da+cd+bad−bcd+b=d(a + c)ad − bc(c) What price (Pd) do consumers pay after the tax is imposed? ANSWER. In the newtax equilibrium, we must haveQd(PD) = Qs(PD− τ)⇒a − bPD= d(PD− τ) − c⇒a + c + dτ = PD(d + b)⇒ PD=a + c + dτd + b2(d) What price (Ps) do producers get? ANSWER Since, in the tax equlibrium, PS= PD− τwe must havepS=a + c + dτd + b− τ=a + c − bτd + b(e) What is the excess burden of this tax? ANSWER The new quantity traded in themarket in the tax equilibrium is Qtax= Qd(p = fraca + c + dτd + b) = a −b(a+c+dτ )d+borQtax=ad−bc−bdτd+b. The difference between this quantity and the old (pre-tax) equilibriumquantity isbdτd+b. The DWL or excess burden of the tax is equal to one half of the taxrate time this difference in quantity or DW L =bdτ22(d+b).aPS=a+c−bτd+bQuantitySupplyDemand(packs of cigarettes)Price($ / pack)ad−bcd+bad−bc−bdτd+b(Qtax) Q0PD=a+c+dτd+bP0=a+cb+dabcdFigure ??. An excise tax. The height of the purple line represent the tax rate,τ. The area of the cyan-shaded area is the DWL or excess burden of the tax.Here we see an example where the slope of the S curve is significantly shallowerthan the demand curve (d > |b|). Hence the burden on consumers is higherthan the burden on producers.(f) What share of the tax burden do consumers bear? (i.e. what isPd−P0τequal to?)ANSWER. Given the calculations made above, the consumers’ price went up bya+c+dτd+b−a+cd+bordτd+b. HencePd−P0τ=dd+b.(g) What would have predicted based on the formula for consumer tax burden share givenin class. (i.e.Pd−P0τ=σσ+|ǫ|.) We would have predicted...3Pd− P0τ=σσ + |ǫ|=d(a+c)ad−bcd(a+c)ad−bc+ |−b(a+c)ad−bc|=d(a + c)d(a + c) + | − b(a + c)|=dd + bThe same as calculated in the previous part!(h) How much revenue does the government earn? ANSWER τQtax=τ (ad−bc−bdτ )d+b.HINT for all parts:If you are having a difficult time answering these questions in termsof the parameters, a, b, c, and d, try setting them equal to sp eci fic values. For exampleset a = 20, b = 1, c = 0 and d = 3. In either case, discuss how changing b and d changethe answers to the questions about excess burden and consumers’ share of the tax burden.For example what happens to your answers if b goes from 1 to 2, or if d goes from 3 to4.ANSWER. From above the excess burden is equal tobdτ22(d+b). Notice that if we hold thevalue of d constant and lower b toward 0, the excess burden goes to 0. Intuitively, this isbecause lowering the value of b toward zero is akin to making the demand curve go towardperfectly inelastic (vertical). When the demand is perfectly vertical, the quantity traded afterthe tax does not change and so there can be no DWL. Similarly for supply w.r.t. changes inthe value of d.2. Monopoly Pricing. Air Helios has a monopoly on flights from St. Petersburg to Madrid.They face the following marginal willingness to pay curve.MW T P (Q) = 2000 − 10Qwhere Q is the number of round-trips per week. The firm’s cost isC(Q) = 6000 + 200Q + 5Q24This means that the firm marginal cost is MC(Q) = 200 + 10Q.(a) If Air Helios is a profit maximizer and must set one price to charge for every ticket it sells,what will that price be. ANSWER We need to calculate the marginal revenue functionand set it equal to MWTP in order to find the profit maximi zing quantity. Then plugthat quantity into the MWTP curve to find the profit-max price. For a linear demandcruve, marginal revenue has the same vertical intercept and twice the slope. Thereforewe have MR(Q) = 2000−20Q. Setting this equal to MC we get Qπ−max= 60. Pluggingthat into MWTP we get Pπ−max= $1400.(b) How many round-trip tickets will it sell? ANSWER 60. See above.(c) What is the firms average cost per ticket at the profit-maximizing ticket amount? AN-SWER. Average Cost is cost divided by quantity. Therefore AC(Q) =6000Q+ 200 + 5Q.Evaulating this at the profit maximizing quantity, we get AC(Q = 60) =600060+200+5∗ 60or $600 per ticket.(d) How much profit will the firm earn? ANSWER Π = (P − AC) ∗ Q = (1400 − 600) ∗ 60or $48K.(e) What is the dead-weight loss for this monopoly? ANSWER The efficient quantity iswhere MC = MWTP, so Qeff= 90. The DWL is the difference between the MWTPand MC over the efficient unit not provided (i.e. from Q=60 to 90). The value of thisloss is therefore1260 ∗ (1400 − 800).(f) From the cost function we see that this firm’s fixed cost amount $6000 per week. If adifferent airline had higher fixed costs but faced the same demand curve for its flights,how would that affect its behavior (in terms of price and quantity) and its profit com-pared with Air Helios. ANSWER The profit would of course be lower by the differencein fixed cost, but the behavior - quantity and price - would be exactly the same, at leastthe short run.3. Market Externality The demand for electricity is given byQd(p) = 1000 − 2000Pwhere Q is energy demand in kilowatt-hours per day and prices are quoted in dollars perkilowatt-hour. Supp ose that all electricity is produced using coal and that the private marginal5cost of electricity for all coal-fired power plants is $.05/ kw-h. Meanwhile the marginal costof environmental damages from coal-fired electricity production is M EC(Q) = $.00025Q.(a) What is the Marginal Social Cost of electricity? ANSWER M SC(Q) = .05 + .00025Q.(b)