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AEM 2500 Pigou Math Supplement 092110

AEM 2500 Pigou Math Supplement 092110 - Poe AEM 2500...

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Unformatted text preview: Poe, AEM 2500 Supplementary Quantitative Notes, 09/21/10 Demand and Supply, Marginal Benefits and Marginal Costs: From Lecture I.5 we were given to following market demand (QDM) and supply (QSM) functions in Millions of KL per quarter QDM = 112.50 – 50P QSM = 100 P In Lecture I.9, these were reexpressed as Marginal Benefit (MB) and Marginal Cost Curves, which can be calculated following the idea of “inverse demand” and “inverse supply” (or ”marginal costs” or “marginal private costs (MPC)” – there are many names used) curve presented in class corresponding to Lecture I.3 and Lecture I.4 (see also Gobbett 2): MB = 2.25‐0.02Q (derivation: QDM = 112.50 – 50 P given above (QDM/50) = (112.50/50) – P divide both sides by 50 P = 2.25 – 0.02 QDM MB = 2.25 – 0.02 Q rearranging and re‐ expressing replacing P with MB, and expressing Q generically) given above divide both sides by 100 replacing P with MPC, and expressing Q generically) MPC = 0.01 Q (derivation: QSM = 100 P (QSM/100) = P MPC = 0.01 Q Graphically these two curves are presented below, giving, with exception of the labels for each curve, the same picture as the market supply and demand. 2.50 2.25 2.00 Marginal Cost Marginal Benefit Price $AU per kL 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 0 25 50 75 100 125 Million kL per Quarter Market Equilibrium: Market equilibrium can be found by either equating Supply and Demand or Marginal Benefit and Marginal Private Costs (MPC) as presented in Gobbet #4: 112.50 – 50 P = QDM = QSM =100 P 112.50 = 150 P PM* = .75 Or: 2.25 ‐ 0.02 Q = MB = MPC = 0.01 Q 2.25 = 0.03 Q QM* = 75 (adding 0.02 Q to both sides) (from which MB = MC =.75 can be solved by plugging QM* = 75 into the respective MB and MC equations) (adding 50 P to both sides) (from which QM* =75 can be solved by plugging PM* = .75 into either supply or demand) For the remainder of this handout we will rely on the MB and MPC functions, which correspond more closely with efficiency/welfare economic concepts than do supply and demand curves. Externalities and the Social Optimum: In Lecture 1.9, we were provided with an increasing Marginal External Social Costs (MESC), or damage, function associated with the production of Q. Note that these MESC costs can take a number of forms, of which we described constant (as in Gobbet 8), increasing (as done here and in class), threshold, and maximum in the lecture notes. MESC = 0.003333Q (given) And we defined Marginal Social Costs (MSC) as the sum of Marginal Private Costs and Marginal External Social Costs. Following methods in Gobbet 8, this results in the following: MSC = MPC + MESC = 0.01 Q + 0.003333 Q = 0.013333 Q The Social Optimum (QS*) corresponding to what I have been referring to the social planner in class is found where the Marginal Benefits of Consumption equal the Marginal Social Costs of production. That is, the social planner seeks to maximize the sum of the net welfare of all the members of society; producers, consumer, and other “external” members of society not directly involved in the market transaction but who are affected negatively or positively by the production and consumption of the good. This equilibrium is derived as follows (see Gobbet 8): 2.25 ‐ 0.02 Q = MB = MSC = 0.013333 Q 2.25 = 0.033333 Q Q*S = 67.5 (Drawing from equations above) (Adding 0.02 Q to both sides) (Dividing through by 0.013333) Graphically this is indicated by the quantity corresponding to the intersection of the blue (marginal benefits) and orange (marginal social costs) lines: 67.5 Finding the Pigouvian Tax: The Pigouvian Tax is defined as a per unit tax equal to the MESC at the social optimum. To find this value, plug QS* = 67.5 into the MESC equation: MESC = 0.003333 (67.5) = 0.225 (which conveniently equals our distortionary tax used in Lecture 1.8.) Based on this result you would set a Pigouvian Tax on every unit of good Q produced. This would be depicted graphically as follows, where the height, i.e. the per unit value, of the tax is 0.225 as calculated above. 67.5 The total tax collected would be the area of the parallelogram, which equals the per unit tax multiplied by the number of units sold in the with‐tax market ($0.225 * 67.5 = $15.1875). Marginal Net Benefits: As given in the readings (Harris p. 91 ‐ 93) the Marginal Net Benefits (MNB) measures the vertical difference between the MB and MC curve, and can be calculated by subtracting the MPC (in Harris this is Ps) from the marginal benefits (in Harris this is PD). Harris describes this as a way of compressing information about both supply and demand into one curve. Calculations are further demonstrated in Gobbet #9. MNB = MB – MPC = 2.25‐0.02 Q ‐ 0.01 Q = 2.25 – 0.03 Q As we discussed in class, at Q=0, MNB = 2.25 (which is MB at zero minus the MPC at zero) and at the market equilibrium QM* = 75, MNB =0 as MB = MPC. The MNB for the producers and consumers operating in the market can be represented graphically as follows. Producers will continue to produce and sell additional units to consumers as long as the MNB are greater than zero (i.e. transacting an additional unit represents a Pareto improvement), with the maximum net benefit from private market transactions occurring where MNB = 0. This is equivalent to finding the point where MB = MPC, or MNB = 0 = 2.25 – 0.03 QM 0.03 QM = 2.25 (adding 0.03 QM to each side) (dividing through by 0.03 and solving for Q) QM* = (2.25/0.03) = 75 On this graph we can overlay the MESC, The socially optimal level of production, QS*, can be calculated by setting MNB equal to the MESC. This is interpreted as equating the MNB gained by producers and consumers in the market of each additional unit transacted to the MESC of those who are not participating in the market but are affected by the externality: simply another application of Mankiw’s principle that rational people think at the margin. 2.25 – 0.03Q = MNB = MESC = 0.003333 Q 2.25 = 0.03 Q + 0.03333 Q = 0.033333 Q (2.25/0.0333333) = 67.5 = QS* (add 0.03 Q to both sides) (divide through by 0.033333) Note that this gives the same results as derived above. Likewise, you can plug QS* = 67.5 into MESC as was done before to derive the Pigouvian tax of 0.225, defined as a per unit tax equal to marginal external social costs at the social optimum. Graphically this can be depicted as follows, with the total tax collected being the area ot the tax rectangle (0.225 * 67.5). (This is the first iteration of these supplemental notes. If you find any errors on this handout, please contact Prof. Poe at [email protected]) ...
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