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# 667 Class Note for AGEC 65200 with Professor Preckel at Purdue

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This 20 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 90 views.

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Date Created: 02/06/15

Games and Market Imperfections Q The mixed complementarity MCP framework is effective for modeling perfect markets but can it handle imperfect markets I A At least part ofthe time A particular type of gamemarket structure that can be handled as an MCP is the Nash equilibrium Purdue University Ag Econ 652 Lecture 22 Nash Equilibria The description of agent behavior in a Nash setting sounds a lot like our view ofan oligopolistic market for a homogeneous good Agents make their choices knowing what impact their choices will have on the system exceptthat they assume their choices will have no effect on the choices of others In the example on endogenous price models the oligopolistic producer chose its output level based on knowledge of consumers response but assuming that other producers will not change Purdue University Ag Econ 652 Lecture 22 Nash Equilibrium cont d The key in the example we saw earlier was the symmetry of the players This symmetry allowed us to express total market production as a function of individual production QI7Y If symmetry is eliminated the determination ofthe optimal production pattern is a difficult analytical problem and in general more easily attacked numerically Purdue University Ag Econ 652 Lecture 22 3 Example Electricity Markets Electricity is an instance of a market in the process of deregulation that has substantial opportunities for exercise of market power on the part of generators Demand is inelastic Due to costs of importexport local generators have a competitive advantage I There are generally very few local generators serving a regional market Purdue University Ag Econ 652 Lecture 22 4 Example Electricity Markets Perfectly competitive models ofelectricity markets are not consistent with observed firm behavior prices charged are far above marginal costs Essential model features Spatial disaggregation demand generation interregional flows of electricity Market power on the part of generators Purdue University Ag Econ 652 Lecture 22 Example Electricity Markets Simplest model Agents Consumers described by a linear demand function Producers described by cost structure and capacity for generation costs for interregional transmission location of generation facilities and Nash behavior Purdue University Ag Econ 652 Lecture 22 Example Electricity Markets Consumers one for each region maximizing consumer s surplus minimize aiql blin 2 1161 q Where aand bare the intercept and slope ofthe inverse demand function q is market demand in market i and lambda in market i isthe price of electricity FOC s a biqi l 2 0 qa biqi if 0 Purdue University Ag Econ 652 Lecture 22 Example Electricity Markets Producers pro t maximizers Nash mig izegikw b q g k EUquot 1 d kg k V Wny Where gm is electricity generation in region iby rm k fUk is transmission of electricity from region ito region jby firm k dk is the constant marginal cost of generation and W0 is the transmission cost from itoj Purdue University Ag Econ 652 Lecture 22 Example Electricity Markets First constraint total flows from itoj cannot exceed residual capacity S 14 S Fmaxij LLy Second constraint generation cannot exceed generation capacity gik S Gmaxik ik l and nonnegativity gik 2 0 f 2 0 Purdue University Ag Econ 652 Lecture 22 9 Example Electricity Markets The firstorder conditions for gm are N gik ai biqi bigik fjik k dik Gmaxik gik ai bv39qi hrgik f k fquotfquot 55quot 0 Notice that the third term in the inequality accounts for the fact that generators are Nash players and know that their choices affect demand Purdue University Ag Econ 652 Lecture 22 10 Example Electricity Markets The firstorder conditions for fUk are N ai biqi bigik 2fmik fmk m1 N aj quj bjgjk 2fmjk fjmk m1 wj Jj20 Again note that the third term on the first and second lines reflect the Nash assumption Purdue University Ag Econ 652 Lecture 22 1 1 Example Electricity Markets And N kai biqi bigik 392fmik aj quj bjgjk fWIjk fjmk t E J Also note that the sum of outshipments minus in shipments generates lines one and two Purdue University Ag Econ 652 Lecture 22 12 Example Electricity Markets The GAMS formulation of this problem I set k Firm index 13 i Location index 14 alias ksijmn parameter ai Inverse demand intercept 1 520 2 500 3 500 4 450 parameter bi Inverse demand slope 11213141 Purdue University Ag Econ 652 Lecture 22 13 Example Electricity Markets Data for firms table dis Cost function linearterm 1 2 3 1 776 I2 837 827 3 8134 4 794 Purdue University Ag Econ 652 Lecture 22 14 Example Electricity Markets table gmaxis Generation maximum 1 2 3 1 530 2 500 500 3 212 4 300 Purdue University Ag Econ 652 Lecture 22 15 Example Electricity Markets Costs and limits relating to the transmission system table wij Transmission charges 1 2 3 4 I I1 1 12 I2 1 14 105 3 14 4 12 105 Purdue University Ag Econ 652 Lecture 22 16 Example Electricity Markets table fmaxij Maximum ow on line 1 2 3 4 1 O 80 120 120 2 80 O 140 150 3 120 140 O O 4 120 150 O O Purdue University Ag Econ 652 Lecture 22 17 Example Electricity Markets Next declare the variables I positive variables I gis Power generation by firm 5 at location i fijs Power ow from i to j generated by rm 5 qi Power demand at i lambdai Market price of power in i Purdue University Ag Econ 652 Lecture 22 18 Example Electricity Markets Notice that the maximum flow constraints from itoj are less than or equal to constraints hence the associated Lagrange multipliers should be non positive negative variables muij Shadow price on maximum ow from i to j Don t forget the limits on generation capacity by region and firm I gupis gmaxis Purdue University Ag Econ 652 Lecture 22 19 Example Electricity Markets Now declare equations equations dqi dgis dfijs sdi maxfij Firstorder conditions for q at location i Firstorder conditions for g Firstorder conditions for f Supplydemand by firm and location Maximum ow from itoj Purdue University Ag Econ 652 Lecture 22 20 10 Example Electricity Markets Now define the structure ofthe equations in GAMS notation Firstorder conditions for the consumer s problem I dqi I aibiqilambdai g o Purdue University Ag Econ 652 Lecture 22 21 Example Electricity Markets Firstorder conditions for producer s s choice of generation level in region i dgisdis aibiCtibigiis sumjfjisdj sfijsdi s diS g 0 Purdue University Ag Econ 652 Lecture 22 22 11 Example Electricity Markets Firstorder conditions for producer s s choice of transmission level from region ito regionj dfijsdis aibi qibigiys l summfmisdmsfimsdis aJbJ39QJ39bJ39gJlS l summfjmsdjsfmjsdms Wilmuil g 0 Purdue University Ag Econ 652 Lecture 22 23 Example Electricity Markets Supply demand balance for region i sdi l sumsgisdis sumjfijsdisfjisdjs g C10 Flow limits from itoj maxfij l sumsdisfijs l fmaxij Purdue University Ag Econ 652 Lecture 22 24 12 Example Electricity Markets Model and solve statements model nash dqqdggdffsdlambdamaxfmu I solve nash using mcp Purdue University Ag Econ 652 Lecture 22 25 Example Electricity Markets Solution output EQU dq Firstorder conditions for q at location i LOWER LEVEL UPPER MARGINAL 1 520000 520000 lNF 356120 2 500000 500000 lNF 401120 3 500000 500000 lNF 212000 4 450000 450000 lNF 356030 Purdue University Ag Econ 652 Lecture 22 26 13 Example Electricity Markets EQU dg Firstorder conditions for g 1 LOWER 1 512240 22 23 31 43 491 630 491 730 491 866 442060 LEVEL 512240 491 630 491 730 284000 442060 NF NF NF NF NF UPPER MARGINAL 356120 380510 170610 212000 REDEF 206030 Notice the REDEF indication for g31 this relates to the upper bounds on g with dg as a 5 Purdue University Ag Econ 652 Lecture 22 27 Example Electricity Markets EQU df Firstorder conditions forf 111 121 131 141 212 213 222 223 232 233 LOWER LEVEL 400000 NF 21000 331120 NF 20000 51760 NF 71200 276210 NF 19000 19015 NF 19000 181085 NF 580000 NF 160000 NF 1400 141295 NF 1400 1395 NF Purdue University Ag Econ 652 Lecture 22 UPPER MARGINAL 80000 REDEF 120000 REDEF 80000 REDEF 140000 REDEF 28 14 Example Electricity Markets 242 243 311 321 331 341 413 423 433 443 LOWER 51050 51050 20000 1400 50000 68800 48950 50000 LEVEL 285600 190330 628240 102880 280000 47970 285940 209670 230060 240000 UPPER MARGINAL NF 150000 REDEF NF NF NF 140000 REDEF NF NF NF 120000 REDEF NF NF NF Purdue University Ag Econ 652 Lecture 22 29 Example Electricity Markets AQNA LOWER LEVEL EQU sd Supplydemand balance location UPPER MARGINAL NF 163880 NF 98880 NF 288000 NF 93970 Purdue University Ag Econ 652 Lecture 22 30 15 Example Electricity Markets EQU maxf Maximum ow from itoj LOWER LEVEL NF NF 80000 NF NF 120000 NF 80000 NF NF 140000 NF 150000 UPPER MARGINAL 80000 120000 120000 80000 154525 140000 278335 150000 Purdue University Ag Econ 652 Lecture 22 31 Example Electricity Markets 31 32 33 34 41 42 43 44 LOWER NF NF NF NF NF NF NF NF LEVEL 140000 120000 UPPER MARGINAL 120000 140000 120000 150000 Purdue University Ag Econ 652 Lecture 22 32 16 Example Electricity Markets VAR 9 Power generation by firm 5 at location i LOWER LEVEL UPPER MARGINAL 1 1 1 11 356120 530000 1 12 EPS 1 13 EPS 1 21 EPS 1 22 380510 500000 1 23 170610 500000 1 31 212000 212000 207866 1 32 EPS 1 33 EPS 1 41 EPS 1 42 EPS 1 43 206030 300000 Purdue University Ag Econ 652 Lecture 22 33 Example Electricity Markets VARf Power flow from location itoj generated by firm 5 LOWER LEVEL UPPER MARGINAL 1 1 1 111 400000 1 121 80000 80000 310120 1 131 120000 71760 1 141 120000 120000 205010 1 212 80000 0015 1 213 80000 80000 200085 1 222 580000 1 223 160000 1 232 140000 140000 139895 1 233 140000 0005 1 242 150000 150000 234550 Purdue University Ag Econ 652 Lecture 22 34 17 Example Electricity Markets 1 243 1 311 1 321 1 331 1 341 1 413 1 423 1 433 1 443 LOWER LEVEL UPPER 150000 120000 140000 120000 140000 120000 150000 MARGINAL 241380 608240 101480 280000 2030 354 740 160720 280060 Purdue University Ag Econ 652 Lecture 22 240000 35 Example Electricity Markets ACHN LOWER LEVEL 356120 401120 212000 356030 VAR q Power demand at location i UPPER MARGINAL NF NF NF NF Purdue University Ag Econ 652 Lecture 22 36 18 Example Electricity Markets LOWER LEVEL 1 163880 NF 2 98880 NF 3 288000 NF 4 93970 NF Purdue University Ag Econ 652 Lecture 22 VAR lambda Market price in region i UPPER MARGINAL 37 Example Electricity Markets LOWER LEVEL 11 NF 12 NF 13 NF 14 NF 21 NF 154525 22 NF 23 NF 278335 24 NF Purdue University Ag Econ 652 Lecture 22 VAR mu Shadow price on max flow i to UPPER MARGINAL EPS EPS 120000 EPS EPS EPS 38 19 Example Electricity Markets LOWER LEVEL UPPER MARGINAL 31 NF 120000 32 NF EPS 33 NF EPS 34 NF EPS 41 NF EPS 42 NF 150000 43 NF EPS 44 NF EPS Purdue University Ag Econ 652 Lecture 22 39 Example Electricity Markets The MCP framework can also be used to formulate certain types of Games Imperfect markets I It cannot be used for problems with lndivisibilities and Uniqueness of solutions can be difficult to establish Purdue University Ag Econ 652 Lecture 22 40 20

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