Numerical Calculation of an Asymmetric Supply Function Equilibrium with Capacity Constraints

Transcription

1 Workng Paer 5: Deartment of Economcs umercal Calculaton of an Asymmetrc uly Functon Equlbrum wth Caacty Constrants Pär Holmberg

2 Deartment of Economcs Workng aer 5: Usala Unversty March 5 P.O. Box 5 I 8-9 E-5 Usala weden Fax: UMERICAL CALCULATIO OF A AYMMETRIC UPPLY FUCTIO EQUILIBRIUM WITH CAPACITY COTRAIT PÄR HOLMBERG Paers n the Workng Paer eres are ublshed on nternet n PDF formats. Download from htt:// or from -WoPEC htt://swoec.hhs.se/uunew/

3 umercal Calculaton of an Asymmetrc uly Functon Equlbrum wth Caacty Constrants Pär Holmberg March 9, 5 Abstract Producers submt commtted suly functons to a rocurement aucton, e.g. an electrcty aucton, before the uncertan demand has been realzed. In the uly Functon Equlbrum (FE), every frm chooses the bd maxmzng hs exected roft gven the bds of the comettors. In case of asymmetrc roducers wth general cost functons, revous work has shown that t s very dffcult to fnd vald FE. Ths aer resents a new numercal rocedure that can solve the roblem. It comrses numercal ntegraton and an otmzaton algorthm that searches an end-condton. The rocedure s llustrated by an examle wth three asymmetrc frms. Keywords: suly functon equlbrum, unform-rce aucton, numercal ntegraton, olgooly, asymmetry, caacty constrant, wholesale electrcty market JEL codes: C6, D, D, L, L, L9 I want to thank my suervsor ls Gottfres and co-advsor Chuan-Zhong L for very valuable comments, dscussons and gudance. uggestons at my semnar at Usala Unversty n February 5 are also very much arecated. The work has been fnancally suorted by the wedsh Energy Agency and Mnstry of Industry, Emloyment and Communcaton. Deartment of Economcs, Usala Unversty, P.O. Box 5, E-5 Usala, weden, Phone , fax: E-mal: ar.holmberg@nek.uu.se.

4 . ITRODUCTIO The uly Functon Equlbrum (FE) was ntroduced by Klemerer & Meyer n 989 [8]. The equlbrum concet assumes that roducers submt bds smultaneously n a one-shot game. In the non-cooeratve ash Equlbrum, each roducer commts to the suly functon that maxmzes hs exected roft gven the bds of the comettors and the roertes of the uncertan demand. In 99, Bolle [] and Green & ewbery [5] observed that the set-u has many smlartes wth the organzaton of most electrcty markets. nce then, the equlbrum s often used when modelng bddng behavor n electrc ower auctons. There are a few FE aers wth other alcatons. The model can be aled to any unform rce aucton where valuatons/costs are certan and common knowledge, quantty dscreteness s neglgble and demand s uncertan. Klemerer & Meyer show that all smooth FE are characterzed by a dfferental equaton, whch n ths aer s called the KM frst-order condton. In the general case, there s a contnuum of ossble FE that fulfll ths frst-order condton [8]. However, wth caacty constrants one can often drastcally reduce the set of FE canddates []; at least for nelastc demand. The set of FE can be further reduced by allowng extreme demand outcomes,.e. that there s a ostve robablty t can be arbtrarly small that the caacty constrants of all frms but ossbly the largest bnd. Then one can show analytcally that there s a unque equlbrum n some secfc cases. One case s symmetrc roducers wth strctly convex cost functons [6]. The other s for roducers wth dentcal constant margnal costs and asymmetrc caactes []. A reservaton rce,.e. a rce ca, s needed to lmt the equlbrum rce. Inelastc demand, a reservaton rce and the ossblty of extreme demand outcomes, are all realstc assumtons for electrc ower markets, and esecally so for balancng markets [6]. But n realty, frms tycally have both non-constant margnal costs and asymmetrc roducton caactes. In ths general case, the KM frst-order condtons one for each frm consttute a system of non-autonomous ordnary dfferental equatons. To analytcally solve ths system s very dffcult and robably mossble. Baldck & Hogan [] calculate aroxmate asymmetrc FE by numercally ntegratng the system of ordnary dfferental equatons. They note that t s generally very dffcult to fnd solutons that do not volate the requrement that suly functons must be non-decreasng. The three excetons are: on-decreasng suly functons are requred by most electrcty auctons. Further, roft maxmzng frms would, under normal condtons, not submt decreasng suly functons [8].

5 symmetrc frms wth dentcal cost functons, cases wth affne solutons.e. affne margnal costs and no caacty constrants and when there are small varatons n the demand. In ths aer, I suggest, a new numercal algorthm to fnd a vald FE. It s ntended for asymmetrc frms and cost functons more general than the three secal cases mentoned by Baldck & Hogan. The equlbrum conssts of ece-wse smooth suly functons and s nsred by the unque equlbrum revously derved for asymmetrc roducers wth constant margnal costs []. ome of the analytcally derved roertes are conectured to be vald also for ncreasng margnal costs. These roertes are: Large frms have more market ower and have larger mark-us for any ercentage of the caacty. Hence, caacty constrants of smaller frms bnd earler. Let be the rce at whch the caacty constrant of frm starts to bnd. Arrange the roducers accordng to sze, startng wth the smallest frm. The caacty constrant of the second largest frm starts to bnd at the rce ca. Thus C < < K <, where C() s the aggregate cost functon. The largest roducer offers ts remanng caacty wth a erfectly elastc suly at the rce ca. All frms offer ther frst unt of ower at the lowest margnal cost, as f under Bertrand cometton, whch s n agreement wth general results for unform rce auctons [9]. To ensure an equlbrum wth the conectured roertes, the followng two assumtons are made. Frst, the larger of any two frms has weakly larger margnal cost for any ercentage of the caacty. econd, all frms has the same margnal cost at zero suly 5. The constants K are unknown, so far. Gven these constants, the termnal condtons of the,,, system of KM frst-order condtons are known and the suly functons of all frms can be solved by numercal ntegraton. The numercal ntegraton starts at the rce ca and roceeds n the drecton of decreasng rces. The ntegraton s termnated as soon as any suly functon volates the requrements, a suly functon must be non-decreasng and nonnegatve. The functon Γ(, ), K s equal to the termnated rce. In theory, all consdered FE canddates should fulfll Γ( ). K In ractce, however, one, C has to be somewhat forgvng due to numercal errors. If there s a unque equlbrum, t can be found by an otmzaton algorthm mnmzng Γ. It mght be enough to assume that the larger of any two frms has a weakly hgher margnal cost for the last unt. It should be ossble to numercally calculate asymmetrc FE for even more general cost functons. However, then adustments of the conecture mght be needed,.e. the order n whch the caacty constrants bnd. 5 It should be ossble to numercally calculate asymmetrc FE also when frms have dfferent margnal costs at zero demand, but then frms wll offer ther frst unts of ower at dfferent rces.

6 ecton ntroduces the notaton and assumtons used n the analyss of ths aer. The KM frst-order condtons of the conectured FE are resented n ecton. In ecton, the numercal algorthm s aled to an examle wth three frms. The algorthm returns one soluton that aroxmately fulflls the frst-order condton and the non-decreasng requrement. It s grahcally verfed that no frm wll fnd t roftable to devate from the equlbrum canddate. The acceted roducton of the equlbrum s neffcent, because markus are asymmetrc. The aer s concluded n ecton 5.. OTATIO AD AUMPTIO Excet for frms caactes and costs, the notaton and market assumtons are the same as n revous aers by Holmberg [6,]. There are asymmetrc roducers. The bd of each frm conssts of a ece-wse smooth.e. ece-wse twce contnuously dfferentable nondecreasng and left-hand contnuous suly functon (). In most electrcty auctons, suly functons are requred to be non-decreasng. The aggregate suly of the comettors of frm s denoted - () and the total suly s denoted (). Let be the caacty constrant of roducer. Wthout loss of generalty, we can order frms accordng to ther caacty,.e. < < K <. The total caacty s desgnated by,.e.. Let denote the rce, at whch frm chooses to offer hs last unt,.e.. Denote the nelastc demand by and ts robablty densty functon by f(). I assume that demand s always non-negatve 6. The densty functon s contnuously dfferentable and has a convex suort set that ncludes zero demand. To get a unque equlbrum, extreme demand outcomes are allowed for,.e. such that > occur wth a ostve robablty. In equlbrum ths mles that the caacty constrants of all frms, but ossbly the largest one, bnd wth a ostve robablty. The reservaton rce ensures that the demand s zero above the rce ca. Accordngly, the market rce equals the rce ca when >, f there are such demand outcomes. 6 As n [6], t s straghtforward to extend the analyss to negatve demand, whch s relevant for balancng markets. does not nclude, as suly functons are left-hand contnuous. ote that

7 All frms have ncreasng, strctly convex and twce contnuously dfferentable cost functons. Among other thngs ths mles ncreasng margnal costs. Denote the aggregated cost functon of all frms by C(). For the cost functons of the ndvdual frms, t s assumed that C ( ) C ( ), f and >. Further, ( ). C These assumtons are C made to ensure an equlbrum wth the conectured roertes. For more general cost functons, adustments of the conecture mght be necessary. The resdual demand of an arbtrary roducer s denoted by q (,). As long as the suly functons of hs comettors are not erfectly elastc at, hs resdual demand s: q,. (). THE COJECTURED FE For symmetrc roducers [6] and roducers wth asymmetrc caactes and dentcal constant margnal costs [], t has been shown that there s a unque equlbrum wth the followng roertes: All roducers offer ther frst unts of ower at the rce C. All suly functons are twce contnuously dfferentable, excet at onts where the caacty constrant of some roducer starts to bnd. There are no suly functons wth erfectly elastc segments below the rce ca and only the largest frm can have a erfectly elastc segment at the rce ca. Ths mles that all suly functons () are contnuous below the rce ca. Frms wth non-bndng caacty constrants do not have suly functons wth nelastc segments. Below the rce ca, all suly functons wth non-bndng caacty constrants fulfll the KM frst-order condton. C < < < K <. It s conectured that these roertes are true also for the asymmetrc frms studed n ths aer 8. The conecture s the bass of the numercal algorthm develoed below. 8 It s lkely that they can be roven by means of the analytcal tools used n revous work [,6,]. 5

8 .. ecessary condtons Assumng that comettors do not have erfectly elastc suly functons below the rce ca, the resdual demand of an arbtrary roducer s gven by (). Hence, for gven demand and rce, the roft of roducer s: π (, ) [ ] C [ ], f and <. In the tradtonal FE lterature, see e.g. [8], the KM frst-order condton s derved by smly dfferentatng () wth resect to. [ C '( )]. () Below the rce ca, all suly functons wth non-bndng caacty constrants fulfll the KM frst-order condton. Ths mles that all FE canddates are gven by - systems of dfferental equatons. The frst system has dfferental equatons and s vald for the rce nterval (, ) nterval C. The second system has - dfferental equatons and s vald for the rce, and so on. The contnuty assumton lnks the end-condtons of the systems of dfferental equatons. Includng the end-condtons, the - systems of dfferental equatons are: ( C', ) (, ) M (, ) [ C '( )] [ C '( )] ( ) [ C ' () ] ( ) [ C ' () ] [ C '( )] ( ) [ C ' () ] [ C '( )] M M ( ) () ( ) ( + ) ( ) ( ) ( + ) ( ) ( + ) M M Gven a set of values {, K, }, the - systems of dfferental equatons can be, () solved backwards. One must start wth the rce nterval (, ), for whch all the endcondtons are known,.e. and. Thus () and - () can be calculated for (, ). Ths soluton can then be used to determne the endcondtons for the rce nterval ( ),. After solvng the system of dfferental 6

9 equatons assocated wth ths rce nterval, one can roceed wth the nterval ( ) and so on., The ntegraton of the systems of ODE starts at the rce ca and roceeds n the drecton of decreasng rces. It termnates as soon as any suly functon volates the non-decreasng and non-negatve constrants. The functon Γ(,, ), K returns the termnated rce. Accordng to the conecture, all roducers wll n equlbrum offer ther frst unt of ower at C. Thus theoretcally all FE canddates must fulfll Γ[, K, ]., C Frm Frm Γ Frm Fg.. The ntegraton starts at the rce ca roceeds n the drecton of decreasng rces and s termnated as soon as any suly functon becomes nvald. Γ s defned by the termnated rce... A suffcent condton The frst-order condton and Γ[,, ] K are necessary condtons for FE,, C but not suffcent. An extremum wth vald suly functons s guaranteed, but one cannot be sure that t s a globally best resonse for a roducer to follow the FE canddate, even f the comettors follow strateges mled by the canddate. A suffcently strong second-order condton s that the market rce of the canddate globally maxmzes ( ) gven that the comettors follow the equlbrum canddate π for every,,

10 8.. The numercal algorthm For asymmetrc roducers wth general cost functons, t s very dffcult or even mossble to calculate FE analytcally. evertheless, the system of dfferental equatons n () can be solved by numercal ntegraton, gven { },,, K. By grdng the sace and by otmzaton algorthms, values { },,, K that (nearly) fulfll C Γ can be found. Consderng numercal errors, one can n ractce not rule out FE canddates that almost fulfll [ ],,, C Γ K. The second-order condton can be checked grahcally or numercally.. A EXAMPLE WITH THREE AYMMETRIC FIRM The numercal rocedure to fnd vald FE s llustrated by an examle wth three frms. Ther roducton caactes are:, and. Further, the margnal cost functon of all frms s lnear + c C u to the caacty constrant. Assume further that the rce ca s. c.. ecessary condtons The KM frst-order condtons of the FE canddates corresondng to () are gven by the followng set of systems of dfferental equatons: ,, c c c c c c c c c The varables can be normalzed such that c and. Then

11 9 [ ] ,., (5) Gven a set of values { },, the system reresentng the rce nterval, can be solved by numercal ntegraton 9. Ths soluton gves end-condtons for the system of dfferental equatons vald for,, whch can be solved n ts turn. The next ste s to check whether the calculated suly functons volate the non-decreasng and non-negatve requrements. The functon, Γ returns the frst rce (startng from the rce ca), for whch any suly functon volates any of the requrements. Most arameter values generate, > Γ and are accordngly not FE. Fg.. The numercal rocedure to fnd vald FE. An examle of a arameter set that generates a non-vald FE s and. Ths s the boundary condton, f one sees the rce ca as a ublc sgnal that wll coordnate the bds. Ths assumton has been suggested by Baldck & Hogan, as t gves a unque FE for 9 ee Aendx for detals of the numercal ntegraton. All suly functons are smooth u to the rce ca, where all caacty constrants bnd. Otmzaton algorthm umercal ntegraton of frst-order condton (ystem of ODE) Γ, Check valdty Otmzaton algorthm umercal ntegraton of frst-order condton (ystem of ODE) Γ, Check valdty Γ, Check valdty

12 symmetrc roducers []. They observe, however, that for asymmetrc roducers the ublc sgnal assumton often leads to nvald FE as n Fg.. In ths case, the suly functons volate several of the requrements, as suly functons should be both non-negatve and nondecreasng. To get an dea of the arameter sace for whch Γ(, ), Γ s calculated for a grd wth x onts n the sace [ ]. lot n Fg..,,, The result s resented as a contour,,9,,5, Producer Producer Producer, -, -, -,5,,5,,5,,5,,5 Fg.. The arameter set and /c generates nvald suly functons. Ths s the boundary condton when one sees the rce ca as a ublc sgnal that coordnates the bds Fg.. Contour lot of Γ (, ). There s a mnmum around and..

13 The contour lot n Fg. ndcates that (, ) Γ has a mnmum around and.. By means of an otmzaton algorthm, the estmated mnmum of Γ (, ) s located to. and.5, and the mn-value s roughly.5. Γ.5 s very close to, but stll above Γ, whch s necessary for the conectured FE. The dfference may, however, be exlaned by the numercal senstvty of the soluton. If there s a unque FE, whch one would ntutvely exect from revous FE studes [6,], then there s a unque set of {, } that gves vald suly functons. Thus there s a unque vald trle of traectores assocated wth ths set that fulflls the systems of KM frst-order condtons n (5). The slghtest devaton from ths trle, due to a small numercal error, wll lead to nvald suly functons, and Γ[, ]. nether rule out nor secure a FE. Thus Γ. 5 does, strctly seakng, > The calculated suly functons for the set {.,.5} are lotted n Fg. 5.,5,,5,,5,,5, /c Frm Frm Frm,5,,,,,,,5,6 Fg. 5. The equlbrum canddate. As shown n a revous aer, the unque asymmetrc equlbrum for constant margnal costs s ece-wse symmetrc []. Two arbtrary roducers have the same suly functon, unless the caacty constrant of one of them s bndng. Wth the strctly convex cost functons assumed n ths aer, t wll be more exensve for a smaller frm to roduce a gven suly The fmnsearch algorthm, a smlex search method of Matlab, was used n the calculaton. The estmaton deends on tolerances used n the numercal ntegraton.

14 comared to a larger frm. Thus t s exected that roducers wth more caacty sell more at every rce, as n Fg. 5. tll t s aarent that the largest frm uses hs market ower extensvely. More than % of the caacty of roducer s not offered below the rce ca. ote that frm and have a knk n ther suly functons at. A dscontnuous ncrease n the elastcty of the sules of frm and at ths ont ensures that the elastcty of ther resdual demand s contnuous. It follows from the KM frst-order condton that ths s necessary, f the suly functons of frm and are to be contnuous at... The second-order condton Does the canddate fulfll the suffcent second-order condton? Denote the suly functons of the FE canddate n Fg. 5 by X () and denote ts market rce by X (). Gven - X (), does X () globally maxmze ( ) π for every? To get an ndcaton, the so-roft lnes of all, roducers are lotted n Fg. 6-8 together wth X (). For a local extremum, a vertcal lne corresondng to a constant should have a tangency ont wth the so-roft lne at X (). Ths corresonds to the KM frst-order condton and seems to be true for every demand for all roducers wth non-bndng caacty constrants. For such frms, one can deduce from the shae of the so-roft lnes that the roft s globally maxmzed at X (). /c Equlbrum rce -... Frm cannot control the rce n ths regon.5 Vertcal lne wth constant / Fg. 6. Iso-roft lnes of frm.

15 The tangency condton s not necessarly fulflled n regons where roducers cannot control the rce due to a bndng caacty constrant or a bndng rce ca. For examle, frm cannot, due to hs caacty constrant, unlaterally ush the rce below X () for >. By ncreasng hs mark-us, he s stll able to ncrease the market rce. However, accordng to Fg. 6, such devatons decrease hs roft. ether frm nor can control the rce for > X. Ther caacty constrants revent them from reducng the rce and the rce ca revents them from ncreasng the rce. Frm could reduce the rce for > X accordng to Fg. 8 t would not be roftable. Thus t seems that X () globally maxmzes ( ) π for every, f the aggregate suly of the comettors s gven by - X ()., X, but.5 /c Frm cannot control the rce n ths regon / Fg.. Iso-roft lnes of frm. ote that X does not nclude, as suly functons are left-hand contnuous.

16 .5 /c / Fg. 8. Iso-roft lnes of frm... Welfare loss Wth symmetrc cost functons, as n [6], or asymmetrc caactes and dentcal constant margnal costs, as n [], there s no neffcency, as demand s nelastc and all frms oerate at the same margnal cost. But there s a welfare loss, f margnal costs are ncreasng and large frms have larger mark-us for every margnal cost, as n Fg. 5. The roducton s neffcent, as some unts wth a hgh margnal cost wll be acceted from small frms nstead of cheaer roducton from larger frms. For the examle wth three frms, the welfare loss s llustrated n Fg. 9. In the examle wth three frms, the welfare loss s, relatvely seakng, largest for the demand outcome X, where the caacty constrant of frm starts to bnd. For hgher demand, roducton from frm and that s cheaer than the most exensve generator of frm s acceted, and the cost rato decreases. There s another knk n the rato, when the caacty constrant of frm bnds. The roducton s otmal when the whole caacty s needed,.e..

17 ,5 Cost/Otmal cost,,,,,,,,,6,8, Fg. 9. The total roducton cost relatve the otmal cost for the examle wth three frms. The roblem of neffcent roducton has lead von der Fehr & Harbord [] to suggest that electrc ower markets should consder Vckrey auctons nstead of unform-rce auctons. The advantage wth the Vckrey aucton s that t s otmal for roducers to bd ther true margnal costs, as they are offered an nformaton rent. 5. COCLUIO Frms tycally have non-constant margnal costs and asymmetrc roducton caactes. In ths general case, the frst-order condtons of a uly Functon Equlbrum (FE) consttute a system of non-autonomous ordnary dfferental equatons. olvng such a system analytcally s very dffcult and robably mossble. evertheless, t can be solved by numercal ntegraton. There s one roblem, however, electrcty auctons normally requre nondecreasng suly functons and t has been observed by Baldck & Hogan that numercally calculated asymmetrc FE:s tend to volate ths restrcton []. The three excetons are: symmetrc frms wth dentcal cost functons, cases wth affne solutons.e. affne margnal costs and no caacty constrants and when there are small varatons n the demand. In ths aer a numercal rocedure s suggested that can solve the roblem of nvald asymmetrc FE. It s conectured that the general asymmetrc FE has roertes smlar to those found n the case of constant margnal costs, whch has been analyzed n []. The caacty constrants of small frms bnd at lower rces comared to frms wth a hgher 5

18 caacty. The caacty constrant of the second largest frm starts to bnd at the rce ca. The largest frm has a erfectly elastc suly at the rce ca. Excet for the two largest frms, the rces at whch the caacty constrants of frms bnd are unknown constants, and so s. The frst-order condtons of ths assumed equlbrum yeld - systems of nonautonomous ordnary dfferental equatons. Gven {, K, }, the set of systems can, be solved by means of numercal ntegraton. One starts at the rce ca and roceeds n the drecton of a decreasng. When any of the suly functons volates the restrctons a suly functon s ncreasng and non-negatve the ntegraton s termnated. The functon ( ) Γ, K, returns the rce at whch the ntegraton termnates. For a vald FE canddate, Γ must, n theory, return the margnal cost of the cheaest unt. Based on the results for asymmetrc roducers wth constant margnal costs one would ntutvely exect a unque FE. Then the equlbrum can be found by an otmzaton algorthm mnmzng Γ. The rocedure for fndng asymmetrc FE canddates s llustrated by an examle wth three frms and lnear margnal costs. Contour lots of Γ ndcate that t has a unque mnmum ust above the margnal cost of the cheaest unt. Ths s exected as a numercal error would force the robably unque trle of FE traectores slghtly off ther track. Further, numercally calculated so-roft lnes ndcate that no roducer wll fnd t roftable to unlaterally devate from the FE canddate. Thus the second-order condton seems to be fulflled. At the rce, for whch the caacty constrant of the smallest frm starts to bnd, the elastcty of the suly of the two larger frms wll ncrease dscontnuously. Ths ensures that the elastcty of the resdual demand of the two frms s contnuous at. Thus n equlbrum, all frms, but the smallest, wll have knks n ther suly functons below ther caacty constrant. The numercal rocedure could be generalzed to any ncreasng and convex cost functon and, wth enough comuter ower, any number of frms. The rocedure s more lkely to generate vald FE wth the conectured roertes, f the larger of any two frms has weakly larger margnal cost for any ercentage of the caacty and f all frms have the same margnal cost at zero suly. Wth adustments n the conectured roertes, e.g. the order n whch frms caactes bnd, t should be ossble to aly the method to even more general cost functons. For asymmetrc frms wth ncreasng margnal costs, asymmetrc mark-us mly neffcent roducton. The reason s that large frms have more market ower. For every 6

19 margnal cost, small frms have lower mark-us comared to large frms. Hence, some costly generators of small frms wll be acceted nstead of cheaer roducton from larger frms. 6. REFERECE [] R. Baldck and W. Hogan, Caacty constraned suly functon equlbrum models for electrcty markets: tablty, on-decreasng constrants, and functon sace teratons, Unversty of Calforna Energy Insttute POWER Paer PWP-89, August. [] F. Bolle, uly functon equlbra and the danger of tact colluson. The case of sot markets for electrcty, Energy Economcs,,. 9-, 99. [].H. von der Fehr and D. Harbord, ot Market Cometton n the UK Electrcty Industry, The Economc Journal,,. 5-56, 99. [] T. Genc &. Reynolds, uly Functon Equlbra wth Pvotal Electrcty ulers, htt:// March. [5] R.J. Green and D.M. ewbery, Cometton n the Brtsh Electrcty ot Market, Journal of Poltcal Economy, vol., no. 5, , 99. [6] P. Holmberg, Unque suly functon equlbrum wth caacty constrants, Workng Paer :, Usala Unversty. [] P. Holmberg, Asymmetrc suly functon equlbrum wth constant margnal costs, Usala Unversty, 5. [8] P.D. Klemerer and M.A. Meyer, uly functon equlbra n an olgooly under uncertanty, Econometrca, 5,. -, 989. [9] V. Krshna, Aucton theory, Academc Press, London,.

20 [] D. M. ewbery, Caacty-constraned uly Functon Equlbra: Cometton and Entry n the Electrcty ot Market, Manuscrt. Cambrdge: Unv. Cambrdge, Det. Al. Econm., 99. APPEDIX The numercal ntegraton s erformed n Matlab. It has been observed by ewbery that the couled dfferental equatons assocated wth FE are stff and hghly senstve to the startng ont chosen for the numercal ntegraton []. The examle studed n ths aer has the same roblem. Thus a robust solver s used, the ode5s of Matlab wth the backward dfferentaton oton. When usng numercal ntegraton algorthms, t s often necessary to rewrte the system of dfferental equatons on the form x ( t) f ( x). of dfferental equatons below. The frst-order condton s: [ C ( )] [ C '( )] M ' Ths transformaton s llustrated for the system The system can be rewrtten on the followng form: C C ' '( ) ( ) M (6) ummng over all equaltes yelds: C ' As, ( ) the system n (6) can now be rewrtten: C C whch has the form x ( t) f ( x).. ' M ' C C ' '( ) ( ) A more general exresson, whch also consders elastc demand, has been derved by Baldck & Hogan []., 8

21 WORKIG PAPER * Edtor: ls Gottfres : Thomas Aronsson and ören Blomqust, Redstrbuton and Provson of Publc Goods n an Economc Federaton.. :5 Matas Eklöf and Danel Hallberg, Prvate Alternatves and Early Retrement Programs.. :6 Bertl Holmlund, ckness Absence and earch Unemloyment. 8. : Magnus Lundn, ls Gottfres and Tomas Lndström, Prce and Investment Dynamcs: An Emrcal Analyss of Plant Level Data.. :8 Mara Vredn Johansson, Allocaton and Ex Ante Cost Effcency of a wedsh ubsdy for Envronmental ustanablty: The Local Investment Program. 6. :9 ören Blomqust and Vdar Chrstansen, Taxaton and Heterogeneous Preferences. 9. : Magnus Gustavsson, Changes n Educatonal Wage Premums n weden: : Magnus Gustavsson, Trends n the Transtory Varance of Earnngs: Evdence from weden and a Comarson wth the Unted tates. 6. : Annka Alexus, Far Out on the Yeld Curve.. : Pär Österholm, Estmatng the Relatonsh between Age tructure and GDP n the OECD Usng Panel Contegraton Methods.. : Per-Anders Edn and Magnus Gustavsson, Tme Out of Work and kll Derecaton. 9. :5 ören Blomqust and Luca Mcheletto, Redstrbuton, In-Knd Transfers and Matchng Grants when the Federal Government Lacks Informaton on Local Costs.. :6 Ida Häkknen, Do Unversty Entrance Exams Predct Academc Achevement? 8. : Mkael Carlsson, Investment and Uncertanty: A Theory-Based Emrcal Aroach.. :8. Anders Klevmarken, Towards an Alcable True Cost-of-Lvng Index that Incororates Housng. 8. * A lst of aers n ths seres from earler years wll be sent on request by the deartment.

22 :9 Matz Dahlberg and Karn Edmark, Is there a Race-to-the-Bottom n the ettng of Welfare Beneft Levels? Evdence from a Polcy Interventon.. : Pär Holmberg, Unque uly Functon Equlbrum wth Caacty Constrants.. 5: Mkael Bengtsson, clas Berggren and Henrk Jordahl, Trust and Growth n the 99s A Robustness Analyss.. 5: clas Berggren and Henrk Jordahl, Free to Trust? Economc Freedom and ocal Catal.. 5: Matz Dahlberg and Eva Mörk, Publc Emloyment and the Double Role of Bureaucrats. 6. 5: Matz Dahlberg and Douglas Lundn, Antderessants and the ucde Rate: Is There Really a Connecton?. 5:5 Mara Vredn Johansson, Tobas Heldt and Per Johansson, Latent Varables n a Travel Mode Choce Model: Atttudnal and Behavoural Indcator Varables.. 5:6 Katarna ordblom and Henry Ohlsson, Tax Avodance and Intra-Famly Transfers. 5. 5: ören Blomqust and Luca Mcheletto, Otmal Redstrbutve Taxaton when Government s and Agents Preferences Dffer.. 5:8 Ruth-Aïda ahum, Income Inequalty and Growth: A Panel tudy of wedsh Countes :9 Olof Åslund and Peter Fredrksson, Ethnc Enclaves and Welfare Cultures Quas-exermental Evdence.. 5: Annka Alexus and Erk Post, Exchange Rates and Asymmetrc hocks n mall Oen Economes.. 5: Martn Ågren, Myoc Loss Averson, the Equty Premum Puzzle, and GARCH.. 5: Pär Holmberg, umercal Calculaton of an Asymmetrc uly Functon Equlbrum wth Caacty Constrants. 8. ee also workng aers ublshed by the Offce of Labour Market Polcy Evaluaton htt:// I 8-9