You are currently browsing the category archive for the ‘market design’ category.

Something funny happened when I started watching Al Roth’s lecture and looked at the paper: I realized that what I always assumed is the meaning of repugnant transactions’ is not exactly the phenomena that Roth talks about. What I thought repugnant transaction’ means is a situation of two rights makes a wrong': it’s totally awesome that Xanders is willing to donate his extra kidney to Zordiac, and it’s really nice of Zordiac to donate money to Xanders, but these two nobles acts done together in exchange for each other is imoral and should be outlawed. Roth however defines repugnant transaction’ more broadly as any transaction that some people want to engage in and others don’t think they should. Consider the opening example of his paper: laws against selling horse meat in restaurants. Here what is repugnant is not the exchange but the good itself. It’s not two rights makes wrong. It’s just wrong. We outlaw the exchange simply because of constitutional reasons or because it’s impossible to enforce a ban on eating — people will simply order take away and perform the crime of eating at their homes.

In my salad days, school masters would assign boys returning from the summer hols an essay: What I did during the summer’. Yes, masters and boys. I served a portion of my youth in a misbegotten penal colony upon a wind blasted heath’. The only females present were master’s wives, matrons and the French mistress. No, not that kind, the kind that offers instruction in French. As you can see, to the lascivious minds of boys, there was no end to the double entendres. However, I digress.

Over the summer Thanh Nguyen and myself completed a paper about stable matchings. The abstract is reproduced below.

The National Resident Matching program strives for a stable matching of medical students to teaching hospitals. With the presence of couples, stable matchings need not exist. For any student preferences, we show that each instance of a stable matching problem has a nearby’ instance with a stable matching. The nearby instance is obtained by perturbing the capacities of the hospitals. Specifically, given a reported capacity $k_h$ for each hospital $h$, we find a redistribution of the slot capacities $k'_h$ satisfying $|k_h-k'_h|\le 4$ for all hospitals $h$ and $\sum_h k_h\le \sum k'_h \le \sum_h k_h+9$, such that a stable matching exists with respect to $k'$. Our approach is general and applies to other type of complementarities, as well as matchings with side constraints and contracts.

In other words, with the addition of at most 9 additional slots, one can guarantee the existence of a stable matchings. This is independent of the size of the market or doctors preferences (it does assume responsive preferences on the part of hospitals). The key tool  is Scarf’s lemma which is a wonderful device for converting results about cardinal matching problems into results about ordinal matching problems. For more on this, consult the paper by Kiralyi and Pap, who should be credited with a formulation of Scarf’s lemma that makes its usefulness evident.

Here is Al Roth’s talk in the Lindau Meeting on Economic Sciences about repugnant transactions, which I guess is the technical term for the discomfort I feel at the idea of people donating their extra kidney to those who need it in return to, you know, money.

Before he was a Nobel Laureate Roth was a Nancy L. Schwartz Memorial Lecturer. His talk was about kidney exchanges — these are exchanges between several pairs of donor+recipient involving no money but only kidneys — and he started with a survey of the audience: who is in favor of allowing selling and buying of kidneys in the free market ? (I am glad I didn’t raise my hand. The next question was about selling and buying of living hearts.) I remember noticing that there was a correlation between raised hands and seniority: For whatever reason, seniors were more likely to be in favor of the free market than juniors.

In the dinner after the talk I ended up in a table of juniors & spouses and we got to discuss our objection to the idea of letting Bob sell his Kidney to Alice, so that Bob can afford to send his daughter to college, and in doing so save Alice’s small child from orphanhood. Turned out we agreed on the policy but for different reasons. I don’t remember which was my reason. I still find both of them convincing, though less so simultaneously.

Reason I: The market price would be too low. Hungry people will compete selling their organs for a bowl of red pottage out of desperation. The slippery slope leads to poor people being harvested for their body parts.

Reason II: The market price would be too high. Only the 0.01 % will be able to afford it. The slippery slope leads to a small aristocracy who live forever by regenerating their bodies.

As I said, both (somewhat) convincing. And please don’t ask me what would be the fair price, that is neither too low nor too high.

Over a lunch of burgers and envy, Mallesh Pai and discussed an odd feature of medical reidencies. This post is a summary of that discussion. It began with this question: Who should pay for the apprenticeship portion of a Doctor’s training? In the US, the apprenticeship, residency, is covered by Medicare. This was enshrined’ in the 1965 act that established Medicare:

Educational activities enhance the quality of care in an institution, and it is intended, until the community undertakes to bear such education costs in some other way, that a part of the net cost of such activities (including stipends of trainees, as well as compensation of teachers and other costs) should be borne to an appropriate extent by the hospital insurance program .

House Report, Number 213, 89th Congress, 1st session 32 (1965) and Senate Report, Number 404 Pt. 1 89th Congress 1 Session 36 (1965)).

Each year about $9.5 billion in medicare funds and another$2 billion in medicaid dollars go towards residency programs. There is also state government support (multiplied by Federal matching funds). At 100K residents a year, this translates into about about $100 K per resident. The actual amounts each program receives per resident can vary (we’ve seen figures in the range of$50K to $150K) because of the formula used to compute the subsidy. In 1997, Congress capped the amount that Medicare would provide, which results in about 30K medical school graduates competing for about 22.5K slots. Why should the costs of apprenticeship be borne by the government? Lawyers, also undertake 7 years of studies before they apprentice. The cost of their apprenticeship is borne by the organization that hires them out of law school. What makes Physicians different? Two arguments we are aware of. First, were one to rely on the market to supply physicians, it is possible that we might get to few (think of booms and busts) in some periods. Assuming sufficient risk aversion on the part of society, there will be an interest in ensuring a sufficient supply of physicians. Note similar arguments are also used to justify farm subsidies. In other words, insurance against shortfalls. Interestingly, we know of no Lawyer with the dershowitz’ to make such a claim. Perhaps, Dick the butcher (Henry VI, Part 2 Act 4) has cowed them. The second is summarized in the following from Gbadebo and Reinhardt: “Thus, it might be argued … that the complete self-financing of medical education with interest-bearing debt … would so commercialize the medical profession as to rob it of its traditional ethos to always put the interest of patients above its own. Indeed, it can be argued that even the current extent of partial financing of their education by medical students has so indebted them as to place the profession’s traditional ethos in peril.” Note, the Scottish master said as much: “We trust our health to the physician: our fortune and sometimes our life and reputation to the lawyer and attorney. Such confidence could not safely be reposed in people of a very mean or low condition. Their reward must be such, therefore, as may give them that rank in the society which so important a trust requires. The long time and the great expense which must be laid out in their education, when combined with this circumstance, necessarily enhance still further the price of their labour.” Interestingly, he includes Lawyers. If we turn the clock back to before WWII, Hospitals paid for trainees (since internships were based in hospitals, not medical schools) and recovered the costs from patient charges. Interns were inexpensive and provided cheap labor. After WWII, the GI Bill provides subsidies for graduate medical education, residency slots increased and institutions were able to pass along the costs to insurers. Medicare opened up the spigot and residencies become firmly ensconced in the system. Not only do they provide training but they allow hospitals to perform a variety of other functions such as care for the indigent at lower cost than otherwise. Ignoring the complications associated with the complementary activities that surround residency programs, who should pay for the residency? Three obvious candidates: insurers, hospitals and the doctors themselves. From Coase we know that in a world without frictions, it does not matter. With frictions, who knows? Having medicare pay makes residency slots an endowment to the institution. The slots assign to a hospital will not reflect what’s best for the intern or the healthcare system. Indeed a recent report by from the Institute of Medicine summarizes some of these distortions. However, their response to is urge for better rules governing the distribution of monies. If hospitals themselves pay, its unclear what the effect might be. For example, as residents costs less than doctors, large hospitals my bulk up of residents and reduce their reliance of doctors. However, assuming no increases in the supply of residents, wages for residents will rise etc etc. If insurers pay there might be overprovision of residents. What about doctors? To practice, a doctor must have a license. The renewal fee on a medical license is, at the top end (California), around$450 a year. In Florida it is about half that amount. There are currently about 800K active physicians in the US. To recover $10 billion (current cost of residency programs) one would have to raise the fee by a$1000 a year at least. The average annual salary for the least remunerative specialties is around $150K. At the high end about$400K. From these summary statistics, it does not appear that an extra $1K a year will break the bank, or corrupt physicians, particularly if it is pegged as a percentage rather than flat amount. The monies collected can be funneled to the program in which the physician completed his or her residency. In 1937, representatives of the Plywood trust called upon Comrade Professor Leonid Vitalievich Kantorovich with a problem. The trust produced 5 varieties of plywood using 8 different machines. How, they asked, should they allocate their limited supply of raw materials to the various machines so as to produce the maximum output of plywood in the required proportions? As problems go, it was, from this remove unremarkable. Remarkable is that the Comrade Professor agreed to take it on. The so called representatives might have been NKVD. Why? Uncle Joe’s first act upon taking power in 1929 was to purge the economists, or more precisely the Jewish ones. This was well before the purge of the communist party in 1936. Why the economists? They complained about waste in a planned economy dizzy with success.’ Yet, here were the apparatchiks of the Trust asking the Comrade Professor to reduce waste. Kantorovich writes, that at the time he was burnt out by pure mathematics. Combined with a concern at the rise of Hitler, he felt compelled to do something practical. And, so he turned his mind to the problem of the Plywood Trust. Frances Spufford, in his delightful work of faction’ called Red Plenty, imagines what Kantorovich might have been thinking. He had thought about ways to distinguish between better answers and worse answers to questions which had no right answer. He had seen a method which could do what the detective work of conventional algebra could not, in situations like the one the Plywood Trust described, and would trick impossibility into disclosing useful knowledge. The method depended on measuring each machine’s output of one plywood in terms of all the other plywoods it could have made. If he was right — and he was sure he was, in essentials — then anyone applying the new method to any production situation in the huge family of situations resembling the one at the Plywood Trust, should be able to count on a measureable percentage improvement in the quantity of product they got from a given amount of raw materials. Or you could put that the other way around: they would make a measureable percentage saving on the raw materials they needed to make a given amount of product. He didn’t know yet what sort of percentage he was talking about, but just suppose it was 3%. It might not sound like much, only a marginal gain, an abstemious eking out of a little bit more from the production process, at a time when all the newspapers showed miners ripping into fat mountains of solid metal, and the output of plants booming 50%, 75%, 150%. But it was predictable. You could count on the extra 3% year after year. Above all it was free. It would come merely by organising a little differently the tasks people were doing already. It was 3% of extra order snatched out of the grasp of entropy. In the face of the patched and mended cosmos, always crumbling of its own accord, always trying to fall down, it built; it gained 3% more of what humanity wanted, free and clear, just as a reward for thought. Moreover, he thought, its applications did not stop with individual factories, with getting 3% more plywood, or 3% more gun barrels, or 3% more wardrobes. If you could maximise, minimise, optimise the collection of machines at the Plywood Trust, why couldn’t you optimise a collection of factories, treating each of them, one level further up, as an equation? You could tune a factory, then tune a group of factories, till they hummed, till they purred. And that meant – An english description of Kantorovich’s appeared in the July 1960 issue of Management Science. The opening line of the paper is: The immense tasks laid down in the plan for the third Five Year Plan period require that we achieve the highest possible production on the basis of the optimum utilization of the existing reserves of industry: materials, labor and equipment. The paper contains a formulation of the Plywood Trust’s problem as a linear program. A recognition of the existence of an optimal solution at an extreme point as well as the hopelessness of enumerating extreme as a solution method. Kantorovich then goes on to propose his method, which he calls the method of resolving multipliers. Essentially, Kantorovich proposes that one solve the dual and then use complementary slackness to recover the primal. One might wonder how Kantorovich’s contribution differs from the contributions of Koopmans and Dantzig. That is another story and as fair a description of the issues as I know can be found in Roy Gardner’s 1990 piece in the Journal of Economic Literature. I reproduce one choice remark: Thus, the situation of Kantorovich is rather like that of the discoverer Columbus. He really never touched the American mainland, and he didn’t give its name, but he was the first one in the area. As an aside, David Gale is the one often forgotten in this discussion. If the Nobel committee has awarded the prize for Linear Programming, Dantzig and Gale would have been included. Had Gale lived long enough, he might have won it again for matching making him the third to have won the prize twice in the same subject. The others are John Bardeen and Frederick Sanger. Continuing with Spufford’s imaginings: – and that meant that you could surely apply the method to the entire Soviet economy, he thought. He could see that this would not be possible under capitalism, where all the factories had separate owners, locked in wasteful competition with one another. There, nobody was in a position to think systematically. The capitalists would not be willing to share information about their operations; what would be in it for them? That was why capitalism was blind, why it groped and blundered. It was like an organism without a brain. But here it was possible to plan for the whole system at once. The economy was a clean sheet of paper on which reason was writing. So why not optimise it? All he would have to do was to persuade the appropriate authorities to listen. Implementation of Kantorovich’s solution at the Plywood trust led to success. Inspired, Kantorovich sent a letter to Gosplan urging adoption of his methods. Here the fact that Kantorovich solved the dual first rather than the primal is important. Kantorovich interpreted his resolving multipliers (shadow prices today) as objectively determined prices. Kantorovich’s letter to Gosplan urged a replacement of the price system in place by his resolving multipliers. Kantorovich intended to implement optimal production plans through appropriate pieces. Gosplan, responded that reform was unecessary. Kantorovich narrowly missed a trip to the Gulag and stopped practicing Economics, for a while. Readers wanting a fuller sense of what mathematical life was like in this period should consult this piece by G. G. Lorentz. After the war, Kantorovich took up linear programming again. At Lenningrad, he headed a team to reduce scrap metal produced at the Egorov railroad-car plant. The resulting reduction in waste reduced the supply of scrap iron for steel mills disrupting their production! Kantorovich escaped punishment by the Leningrad regional party because of his work on atomic reactors. Kantorovich’s interpretation of resolving multipliers which he renamed as objectively determined valuations put him at odds with the prevailing labor theory of value. In the post Stalin era, he was criticized for being under the sway of Bohm-Bawerk, author of the notion of subjective utility. Aron Katsenelinboigen, relates a joke played by one of these critics on Kantorovich. A production problem was presented to Kantorovich where the labor supply constraint would be slack at optimality. Its objectively determined valuation’ was therefore zero, contradicting the labor theory of value. Nevertheless, Kantorovich survived. This last verse from the Ballard of L. V. Kantorvich authored by Josph Lakhman explains why: Then came a big scholar with a solution. Alas, too clever a solution. Objectively determined valuations’- That’s the panacea for each and every doubt! Truth be told, the scholar got his knukcles rapped Slightly rapped For such an unusual advice That threatened to overturn the existing order. After some thought, however, the conclusion was reached That the valuations had been undervalued This is the first of a series of posts about stability and equilibrium in trading networks. I will review and recall established results from network flows and point out how they immediately yield results about equilibria, stability and the core of matching markets with quasi-linear utility. It presumes familiarity with optimization and the recent spate of papers on matchings with contracts. The simplest trading network one might imagine would involve buyers (${B}$) and sellers (${S}$) of a homogenous good and a set of edges ${E}$ between them. No edges between sellers and no edges between buyers. The absence of an edge in ${E}$ linking ${i \in B}$ and ${j \in S}$ means that ${i}$ and ${j}$ cannot trade directly. Suppose buyer ${i \in B}$ has a constant marginal value of ${v_i}$ upto some amount ${d_i}$ and zero thereafter. Seller ${j \in S}$ has a constant marginal cost of ${c_j}$ upto some capacity ${s_j}$ and infinity thereafter. Under the quasi-linear assumption, the problem of finding the efficient set of trades to execute can be formulated as a linear program. Let ${x_{ij}}$ for ${(i,j) \in E}$ denote the amount of the good purchased by buyer ${i \in B}$ from seller ${j \in S}$. Then, the following program identifies the efficient allocation: $\displaystyle \max \sum_{(i,j) \in E} (v_i - c_j)x_{ij}$ subject to $\displaystyle \sum_{j \in S: (i,j) \in E}x_{ij} \leq d_i\,\, \forall i \in B$ $\displaystyle \sum_{i \in B:(i,j) \in E}x_{ij} \leq s_j\,\, \forall j \in S$ $\displaystyle x_{ij} \geq 0\,\, (i,j) \in E$ This is, of course, an instance of the (discrete) transportation problem. The general version of the transportation problem can be obtained by replacing each coefficient of the objective function by arbitrary numbers ${w_{ij}}$. This version of the transportation problem is credited to the mathematician F. J. Hitchcock and published in 1941. Hitchcock’s most famous student is Claude Shannon. The continuous’ version of the transportation problem was formulated by Gaspard Monge and described in his 1781 paper on the subject. His problem was to split two equally large volumes (representing the initial location and the final location of the earth to be shipped) into infinitely many small particles and then match them with each other so that the sum of the products of the lengths of the paths used by the particles and the volume of the particles is minimized. The ${w_{ij}}$‘s in Monge’s problem have a property since called the Monge property, that is the same as submodularity/supermodularity. This paper describes the property and some of its algorithmic implications. Monge’s formulation was subsequently picked up by Kantorovich and the study of it blossomed into the specialty now called optimal transport with applications to PDEs and concentration of measure. That is not the thread I will follow here. Returning to the Hitchcock, or rather discrete, formulation of the transportation problem let ${p_j}$ be the dual variables associated with the first set of constraints (the supply side) and ${\lambda_i}$ the dual variables associated with the second or demand set of constraints. The dual is $\displaystyle \min \sum_{j \in S} s_jp_j + \sum_{i \in B}d_i\lambda_i$ subject to $\displaystyle p_j + \lambda_i \geq [v_i-c_j]\,\, \forall (i,j) \in E$ $\displaystyle p_j, \lambda_i \geq 0\,\, \forall j \in S, i \in B$ We can interpret ${p_j}$ as the unit price of the good sourced from seller ${j}$ and ${\lambda_i}$ as the surplus that buyer ${i}$ will enjoy at prices ${\{p_j\}_{j \in S}}$. Three things are immediate from the duality theorem, complementary slackness and dual feasibility. 1. If ${x^*}$ is a solution to the primal and ${(p^*, \lambda^*)}$ an optimal solution to the dual, then, the pair ${(x^*, p^*)}$ form a Walrasian equilibrium. 2. The set of optimal dual prices, i.e., Walrasian prices live in a lattice. 3. The dual is a (compact) representation of the TU (transferable utility) core of the co-operative game associated with this economy. 4. Suppose the only bilateral contracts we allow between buyer ${i}$ and seller ${j}$ are when ${(i,j) \in E}$. Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair ${(i,j) \in E}$. 5. Because the constraint matrix of the transportation problem is totally unimodular, the previous statements hold even if the goods are indivisible. As these are standard, I will not reprove them here. Note also, that none of these conclusions depend upon the particular form of the coefficients in the objective function of the primal. We could replace ${[v_i - c_j]}$ by ${w_{ij}}$ where we interpret ${w_{ij}}$ to be the joint gain gains from trade (per unit) to be shared by buyer ${i}$ and seller ${j}$. Now, suppose we replace constant marginal values by increasing concave utility functions, ${\{U_i(\cdot)\}_{i \in B}}$ and constant marginal costs by ${\{C_j (\cdot)\}_{j \in S}}$? The problem of finding the efficient allocation becomes: $\displaystyle \max \sum_{i \in B}U_i(\sum_{j: (i,j) \in E}x_{ij}) - \sum_{j \in S}C_j(\sum_{i: (i,j) \in E}x_{ij})$ subject to $\displaystyle \sum_{j \in S: (i,j) \in E}x_{ij} \leq d_i\,\, \forall i \in B$ $\displaystyle \sum_{i \in B:(i,j) \in E}x_{ij} \leq s_j\,\, \forall j \in S$ $\displaystyle x_{ij} \geq 0\,\, (i,j) \in E$ This is an instance of a concave flow problem. The Kuhn-Tucker-Karush conditions yield the following: 1. If ${x^*}$ is a solution to the primal and ${(p^*, \lambda^*)}$ an optimal Lagrangean, then, the pair ${(x^*, p^*)}$ form a Walrasian equilibrium. 2. The set of optimal Lagrange prices, i.e., Walrasian prices live in a lattice. 3. Suppose the only bilateral contracts we allow between buyer ${i}$ and seller ${j}$ are when ${(i,j) \in E}$. Furthermore, a contract can specify only a quantity to be shipped and price to be paid. Then, we can interpret the set of optimal primal and dual solutions to be the set of contracts that cannot be blocked (suitably defined) by any buyer seller pair ${(i,j) \in E}$. Notice, we lose the extension to indivisibility. As the objective function in the primal is now concave, an optimal solution to the primal may occur in the interior of the feasible region rather than at an extreme point. To recover integrality’ we need to impose a stronger condition on ${\{U_i\}_{i \in B}}$ and ${\{C_j\}_{j \in S}}$, specifically, they be ${M}$-concave and convex respectively. This is a condition tied closely to the gross substitutes condition. More on this in a subsequent post. On many campuses one will find notices offering modest sums to undergraduates to participate in experiments. When the experimenter does not attract sufficiently many subjects to participate at the posted rate, does she raise it? Do undergraduates make counter offers? If not, why not? An interesting contrast is medical research where there has arisen a class of human professional guinea pigs. They have a jobzine and the anthropologist Roberto Abadie has book on the subject. Prices paid to healthy subjects to participate in trials vary and increase with the potential hazards. The jobzine I mention earlier provides ratings of various research organizations who carry out such studies. A number of questions come to mind immediately: how are prices determined, are subjects in a position to offer informed consent, should such contracts be forbidden and does relying on such subjects induce a selection bias? In the March 23rd edition of the NY Times Mankiw proposes a do no harm’ test for policy makers: …when people have voluntarily agreed upon an economic arrangement to their mutual benefit, that arrangement should be respected. There is a qualifier for negative externalities, and he goes on to say: As a result, when a policy is complex , hard to evaluate and disruptive of private transactions, there is good reason to be skeptical of it. Minimum wage legislation is offered as an example of a policy that fails the do no harm test. The association with the Hippocratic oath gives it an immediate appeal. I think the test to be more Panglossian (or should I say Leibnizian) than Hippocratic. There is an immediate heart strings’ argument against the test, because indentured servitude passes the do no harm’ test. However, indentured servitude contracts are illegal in many jurisdictions ( repugnant contracts?). This argument raises only more questions, like why would we rule out such contracts? I want to focus instead on two other aspects of the do no harm’ principle contained in the words voluntarily’ and benefit’. What is voluntary and benefit compared to what? To fix ideas imagine two parties, who if they work together and expend equal effort can jointly produce a good worth$1. How should they split the surplus produced? How will they split the surplus produced? An immediate answer to the should’ question is 50-50. A deeper answer would suggest that they each receive their marginal product (or added value) of $1, but this impossible without an injection of money from the outside. There is no immediate answer to the will’ question as it will depend on the outside options of each of the agents and their relative patience. Suppose for example, the outside option of each party is$0, one agent is infinitely patient and the other has a high discount rate. It isn’t hard to construct a model of bargaining where the lions share of the gains from trade go to the patient agent. Thus, what will’ happen will be very different from what should’ happen. What will’ happen depends on the relative patience and outside options of the agents at the time of bargaining. In my extreme example of a very impatient agent, one might ask why is it that one agent is so impatient? Is the patient agent exploiting the impatience of the other agent coercion?

When parties negotiate to their mutual benefit, it is to their benefit relative to the status quo. When the status quo presents one agent an outside option that is untenable, say starvation, is bargaining voluntary, even if the other agent is not directly threatening starvation? The difficulty with the do no harm’ principle in policy matters is the assumption that the status quo does less harm than a change in it would. This is not clear to me at all. Let me illustrate this  with two examples to be found in any standard microeconomic text book.

Assuming a perfectly competitive market, imposing a minimum wage constraint above the equilibrium wage would reduce total welfare. What if the labor market were not perfectly competitive? In particular, suppose it was a monopsony employer constrained to offer the same wage to everyone employed. Then, imposing a minimum wage above the monopsonist’s optimal wage would increase total welfare.

Penn state runs auctions to license its intellectual property. For each license on the block there is a brief description of what the relevant technology is and an opening bid which I interpret as a reserve price. It also notes whether the license is exclusive or not. Thus, the license is sold for a single upfront fee. No royalties or other form of contingent payment. As far as I can tell the design is an open ascending auction.

What is the institutional detail that makes electricity special? Its in the physics that I will summarize with a model of DC current in a resistive network. Note that other sources, like Wikipedia give other reasons, for why electricity is special:

Electricity is by its nature difficult to store and has to be available on demand. Consequently, unlike other products, it is not possible, under normal operating conditions, to keep it in stock, ration it or have customers queue for it. Furthermore, demand and supply vary continuously. There is therefore a physical requirement for a controlling agency, the transmission system operator, to coordinate the dispatch of generating units to meet the expected demand of the system across the transmission grid.

I’m skeptical. To see why, replace electricity by air travel.

Let ${V}$ be the set of vertices and ${E^*}$ the set of edges a the network. It will be convenient in what follows to assign (arbitrarily) an orientation to each edge in ${E^*}$. Let ${E}$ be the set of directed arcs that result. Hence, ${(i,j) \in E}$ mens that the edge ${(i,j)}$ is directed from ${i}$ to ${j}$. Notice, if ${(i,j) \in E}$, then ${(i,j) \not \in E}$.

Associated with each ${(i,j) \in E}$ is a number ${x_{ij}}$ that we interpret as a flow of electricity. If ${x_{ij} > 0}$ we interpret this to be a flow from ${i}$ to ${j}$. If ${x_{ij} < 0}$ we interpret this as a flow from ${j}$ to ${i}$.

1. Let ${\rho_{ij}}$ is the resistance on link ${(i,j)}$.
2. ${c_i}$ unit cost of injecting current into node ${i}$.
3. ${v_i}$ marginal value of current consumed at node ${i}$.
4. ${d_i}$ amount of current consumed at node ${i}$.
5. ${s_i}$ amount of current injected at node ${i}$.
6. ${K_{ij}}$ capacity of link ${(i,j)}$.

Current must satisfy two conditions. The first is conservation of flow at each node:

$\displaystyle s_i + \sum_{k: (k,i) \in E}x_{ji} = \sum_{j: (i,j) \in E}x_{ij} + d_i\,\, \forall i \in V$

The second is Ohm’s law. There exist node potentials ${\{\phi_i\}_{i \in V}}$ such that

$\displaystyle \rho_{ij}x_{ij} = \phi_i - \phi_j\,\, \forall (i,j) \in E.$

Using this systems equations one can derive the school boy rules for computing the resistance of a network (add them in series, add the reciprocals in parallel). At the end of this post is a digression that shows how to formulate the problem of finding a flow that satisfies Ohm’s law as an optimization problem. Its not relevant for the economics, but charming nonetheless.

At each node ${i \in V}$ there is a power supplier with constant marginal cost of production of ${c_i}$ upto ${S_i}$ units. At each ${i \in V}$ there is a consumer with constant marginal value of ${v_i}$ upto ${D_i}$ units. A natural optimization problem to consider is

$\displaystyle \max \sum_{i \in V}[v_id_i - c_is_i]$

subject to

$\displaystyle \sum_{j: (i,j) \in E}x_{ij} -\sum_{j: (j,i) \in E}x_{ji} - s_i + d_i= 0\,\, \forall i \in V$

$\displaystyle \rho_{ij}x_{ij} = \mu_i - \mu_j\,\, \forall (i,j) \in E$

$\displaystyle -K_{ij} \leq x_{ij} \leq K_{ij}\,\, \forall (i,j) \in E$

$\displaystyle 0 \leq s_i \leq S_i\,\, \forall i \in V$

$\displaystyle 0 \leq d_i \leq D_i\,\, \forall i \in V$

This is the problem of finding a flow that maximizes surplus.

Let ${{\cal C}}$ be the set of cycles in ${(V, E^*)}$. Observe that each ${C \in {\cal C}}$ corresponds to a cycle in ${(V, E)}$ if we ignore the orientation of the edges. For each cycle ${C \in {\cal C}}$, let ${C^+}$ denote the edges in ${E}$ that are traversed in accordance with their orientation. Let ${C^-}$ be the set of edges in ${C}$ that are traversed in the opposing orientation.

We can project out the ${\phi}$ variables and reformulate as

$\displaystyle \max \sum_{i \in V}[v_id_i - c_is_i]$

subject to

$\displaystyle \sum_{j: (i,j) \in E}x_{ij} -\sum_{j: (j,i) \in E}x_{ji} - s_i + d_i= 0\,\, \forall i \in V$

$\displaystyle \sum_{(i,j) \in C^+}\rho_{ij}x_{ij} - \sum_{(i,j) \in C^-}\rho_{ij}x_{ij} = 0\,\, \forall \,\, C \in {\cal C}$

$\displaystyle -K_{ij} \leq x_{ij} \leq K_{ij}\,\, \forall (i,j) \in E$

$\displaystyle 0 \leq s_i \leq S_i\,\, \forall i \in V$

$\displaystyle 0 \leq d_i \leq D_i\,\, \forall i \in V$

Recall the scenario we ended with in part 1. Let ${V = \{1, 2, 3\}}$, ${E = \{(1,3), (1,2), (2,3)\}}$ and in addition suppose ${\rho_{ij} =1}$ for all ${(i,j)}$. Only ${(1,3)}$ has a capacity constraint of 600. Let ${D_1 = D_2 = 0}$ and ${S_3 = 0}$. Also ${c_1 = 20}$ and ${c_2 = 40}$ and each have unlimited capacity. At node 3, the marginal value is ${V > 40}$ upto 1500 units and zero thereafter. The optimization problem is

$\displaystyle \max Vd_3 - 20s_1 - 40 s_2$

subject to

$\displaystyle x_{12} + x_{13} - s_1 = 0$

$\displaystyle x_{23} - s_2 - x_{12} = 0$

$\displaystyle d_3 - x_{13} - x_{23} = 0$

$\displaystyle x_{13} - x_{23} - x_{12} = 0$

$\displaystyle -600 \leq x_{13} \leq 600$

$\displaystyle 0 \leq d_3 \leq 1500$

Notice, for every unit of flow sent along ${(1,3)}$, half a unit of flow must be sent along ${(1,2)}$ and ${(2,3)}$ as well to satisfy the cycle flow constraint.

The solution to this problem is ${x_{13} = 600}$, ${x_{12} = -300}$, ${x_{23} = 900}$, ${s_1 = 300}$, ${s_2 = 1200}$ and ${d_3 = 1500}$. What is remarkable about this not all of customer 3’s demand is met by the lowest cost producer even though that producer has unlimited capacity. Why is this? The intuitive solution would have been send 600 units along ${(1,3)}$ and 900 units along ${(1,2) \rightarrow (2,3)}$. This flow violates the cycle constraint.

In this example, when generator 1 injects electricity into the network to serve customer 3’s demand, a positive amount of that electricity must flow along every path from 1 to 3 in specific proportions. The same is true for generator 2. Thus, generator 1 is unable to supply all of customer 3’s demands. However, to accommodate generator 2, it must actually reduce its flow! Hence, customer 3 cannot contract with generators 1 and 2 independently to supply power. The shared infrastructure requires that they co-ordinate what they inject into the system. This need for coordination is the argument for a clearing house not just to manage the network but to match supply with demand. This is the argument for why electricity markets must be designed.

The externalities caused by electricity flows is not a proof that a clearing house is needed. After all, we know that if we price the externalities properly we should be able to implement the efficient outcome. Let us examine what prices might be needed by looking at the dual to the surplus maximization problem.

Let ${y_i}$ be the dual variable associated with the flow balance constraint. Let ${\lambda_C}$ be associated with the cycle constraints. Let ${\nu_i}$ and ${\theta_i}$ be associated with link capacity constraints. Let ${\mu_i}$ and ${\sigma_i}$ be associated with the remaining tow constraints. These can be interpreted as the profit of supplier ${i}$ and the surplus of customer ${i}$ respectively. For completeness the dual would be:

$\displaystyle \min \sum_{(i,j) \in E}[\nu_{ij} + \theta_{ij}]K_{ij} + \sum_{i \in V}[S_i \mu_i + D_i \sigma_i]$

subject to

$\displaystyle -\theta_{ij} + \nu_{ij} + \rho_{ij}\sum_{C^+ \ni (i,j)}\lambda_C - \rho_{ij}\sum_{C^- \ni (i,j)}\lambda_C + y_i - y_j = 0\,\, \forall (i,j) \in E$

$\displaystyle \mu_i - y_i \geq -c_i\,\, \forall i \in V$

$\displaystyle \sigma_i + y_i \geq v_i\,\, \forall i \in V$

$\displaystyle \nu_{ij}, \theta_{ij}, \mu_i, \sigma_i \geq 0\,\, \forall i \in V,\,\,\forall (i,j) \in E$

Now ${y_i}$ has a natural interpretation as a price to be paid for consumption at node ${i}$ for supply injected at node ${i}$. ${\mu_i}$ and ${\nu_i}$ can be interpreted as the price of capacity. However, ${\lambda_C}$ is trickier, price for flow around a cycle? It would seem that one would have to assign ownership of each link as well as ownership of cycles in order to have a market to generate these prices.