Games and Market Imperfections

Transcription

1 Games and Market Imperfectons Q: The mxed complementarty (MCP) framework s effectve for modelng perfect markets, but can t handle mperfect markets? A: At least part of the tme A partcular type of game/market structure that can be handled as an MCP s the Nash equlbrum Lecture 22 1 Nash Equlbra The descrpton of agent behavor n a Nash settng sounds a lot lke our vew of an olgopolstc market for a homogeneous good Agents make ther choces knowng what mpact ther choces wll have on the system except that they assume ther choces wll have no effect on the choces of others In the example on endogenous prce models, the olgopolstc producer chose ts output level based on knowledge of consumers response, but assumng that other producers wll not change Lecture

2 Nash Equlbrum (cont d.) The key n the example we saw earler was the symmetry of the players Ths symmetry allowed us to express total market producton as a functon of ndvdual producton (Q=ny) If symmetry s elmnated, the determnaton of the optmal producton pattern s a dffcult analytcal problem and, n general, more easly attacked numercally Lecture 22 3 Electrcty s an nstance of a market n the process of deregulaton that has substantal opportuntes for exercse of market power on the part of generators Demand s nelastc Due to costs of mport/export, local generators have a compettve advantage There are generally very few local generators servng a regonal market Lecture

3 Perfectly compettve models of electrcty markets are not consstent wth observed frm behavor prces charged are far above margnal costs Essental model features: Spatal dsaggregaton demand, generaton, nter-regonal flows of electrcty Market power on the part of generators Lecture 22 5 Smplest model Agents Consumers descrbed by a lnear demand functon Producers descrbed by cost structure and capacty for generaton, costs for nter-regonal transmsson, locaton of generaton facltes, and Nash behavor Lecture

4 Consumers one for each regon maxmzng consumer s surplus mnmze a q b q / 2 + λ q q where a and b are the ntercept and slope of the nverse demand functon, q s market demand n market, and lambda n market s the prce of electrcty 2 FOC s: a b q + λ 0, q ( a b q + λ ) = 0 Lecture 22 7 Producers proft maxmzers (Nash) mnmze g k f jk n [ a bq ] g + d + where g k s electrcty generaton n regon by frm k, f jk s transmsson of electrcty from regon to regon j by frm k, d k s the (constant) margnal cost of generaton, and w j s the transmsson cost from to j + j k = 1 = 1 k g k n w j f n jk ( f jk f jk ) Lecture

5 Frst constrant, total flows from to j cannot exceed resdual capacty S s= 1 f Fmax : µ js j Second constrant, generaton cannot exceed generaton capacty g k Gmax k : β j k and, non-negatvty g k 0, f 0 jk Lecture 22 9 The frst-order condtons for g k are g k a bq b g ( Gmax k g k k + N j = 1 ( f jk ) a b q b g ) + d ( f ) + d (Notce that the thrd term n the nequalty accounts for the fact that generators are Nash players and know that ther choces affect demand) k f + jk N jk k + f + jk k = 0 Lecture

6 The frst-order condtons for f jk are a + bq + b gk + a + j bjq j bj g + w µ 0 j j m= 1 jk ( f m= 1 ( f ) ) (Agan note that the thrd term on the frst and second lnes reflect the Nash assumpton) N + mk N f mjk mk f jmk Lecture And, f jk a a + b q j bjq j + b g b j g k jk + + N m= 1 N m= 1 ( f ( f mk mjk f f mk jmk ) ) (Also note that the sum of out-shpments mnus nshpments generates lnes one and two) Lecture

7 The GAMS formulaton of ths problem: set k Frm ndex / 1*3 / Locaton ndex / 1*4 / ; alas (k,s),(,j,m,n) ; parameter a() Inverse demand ntercept /1 520, 2 500, 3 500, /; parameter b() Inverse demand slope /1-1, 2-1, 3-1, 4-1 /; Lecture Data for frms table d(,s) Cost functon lnear term ; Lecture

8 table gmax(,s) Generaton maxmum ; Lecture Costs and lmts relatng to the transmsson system table w(,j) Transmsson charges ; Lecture

9 table fmax(,j) Maxmum flow on lne ; Lecture Next declare the varables postve varables g(,s) Power generaton by frm s at locaton f(,j,s) Power flow from to j generated by frm s q() Power demand at lambda() Market prce of power n ; Lecture

10 Notce that the maxmum flow constrants from to j are less than or equal to constrants hence, the assocated Lagrange multplers should be nonpostve negatve varables mu(,j) Shadow prce on maxmum flow from to j; Don t forget the lmts on generaton capacty by regon and frm g.up(,s) = gmax(,s) ; Lecture Now declare equatons equatons dq() Frst-order condtons for q at locaton dg(,s) Frst-order condtons for g df(,j,s) Frst-order condtons for f sd() Supply-demand by frm and locaton maxf(,j) Maxmum flow from to j ; Lecture

11 Now defne the structure of the equatons n GAMS notaton: Frst-order condtons for the consumer s problem dq().. -a()-b()*q()+lambda() =g= 0 ; Lecture Frst-order condtons for producer s s choce of generaton level n regon dg(,s)$d(,s).. -a()-b()*q()-b()*(g(,s) +sum(j,f(j,,s)$d(j,s)-f(,j,s)$d(,s))) +d(,s) =g= 0 ; Lecture

12 Frst-order condtons for producer s s choce of transmsson level from regon to regon j df(,j,s)$d(,s).. a()+b()*q()+b()*(g(,s) +sum(m,f(m,,s)$d(m,s)-f(,m,s)$d(,s))) -a(j)-b(j)*q(j)-b(j)*(g(j,s) +sum(m,f(j,m,s)$d(j,s)-f(m,j,s)$d(m,s))) +w(,j)-mu(,j) =g= 0 ; Lecture Supply demand balance for regon sd().. sum(s,g(,s)$d(,s) - sum(j,f(,j,s)$d(,s)-f(j,,s)$d(j,s))) =g= q() ; Flow lmts from to j maxf(,j).. sum(s$d(,s),f(,j,s)) =l= fmax(,j) ; Lecture

13 Model and solve statements model nash / dq.q,dg.g,df.f,sd.lambda,maxf.mu / ; solve nash usng mcp ; Lecture Soluton output: ---- EQU dq Frst-order condtons for q at locaton INF INF INF INF Lecture

14 ---- EQU dg Frst-order condtons for g INF INF INF INF REDEF INF (Notce the REDEF ndcaton for g(3,1) ths relates to the upper bounds on g wth dg as a <) Lecture EQU df Frst-order condtons for f LOWER LEVEL UPPER MARGINAL INF INF REDEF INF INF REDEF INF INF REDEF INF INF INF REDEF INF. Lecture

15 LOWER LEVEL UPPER MARGINAL INF REDEF INF INF INF REDEF INF INF INF REDEF INF INF INF. Lecture EQU sd Supply-demand balance locaton LOWER LEVEL UPPER MARGINAL 1.. +INF INF INF INF Lecture

16 ---- EQU maxf Maxmum flow from to j 1.1 -INF INF INF INF INF INF INF INF Lecture INF INF INF INF INF INF INF INF... Lecture

17 ---- VAR g Power generaton by frm s at locaton EPS EPS EPS EPS EPS EPS EPS Lecture VAR f Power flow from locaton to j generated by frm s Lecture

18 Lecture VAR q Power demand at locaton INF INF INF INF. Lecture

19 ---- VAR lambda Market prce n regon INF INF INF INF. Lecture VAR mu Shadow prce on max flow to j 1.1 -INF.. EPS 1.2 -INF.. EPS 1.3 -INF INF.. EPS 2.1 -INF INF.. EPS 2.3 -INF INF.. EPS Lecture

20 3.1 -INF INF.. EPS 3.3 -INF.. EPS 3.4 -INF.. EPS 4.1 -INF.. EPS 4.2 -INF INF.. EPS 4.4 -INF.. EPS Lecture The MCP framework can also be used to formulate certan types of Games Imperfect markets It cannot be used for problems wth Indvsbltes, and Unqueness of solutons can be dffcult to establsh Lecture