##### May 4, 2020

Read the Reflection, written 25 August 2021, below the following original Transmission.

Under normal circumstances, most goods and services are produced, bought, and sold through free markets. But in an emergency like a pandemic, markets may not suffice. Imagine, for example, that society suddenly needs to undertake tens or even hundreds of millions of virus tests a week (so that employers can put their employees back to work safely). To whom can we turn to produce the testing equipment? There may be many potential manufacturers, and how can we know who they all are? Even if we know their identities, how do we decide which ones should actually do the producing? How much should each produce? And what price should a producer receive to cover its costs?

If we had the luxury of time, the market might resolve all these questions: prices and quantities would adjust until supply and demand were brought into balance. But getting a new market of this size to equilibrate quickly is unrealistic. Furthermore, markets don’t work well when there are concentrations of power on either the buying or selling side, as there might well be here. Fortunately, mechanism design can be enlisted to help.1

1. Markets

Before getting to mechanism design, let’s review why markets normally work so well. Suppose that there are many buyers and producers for some good. Suppose that buyer i enjoys (gross) benefit bi(xi) from quantity xi. Similarly, each producer j incurs cost cj(yj) to produce yj. Hence, society’s net social benefit is:

$$\sum\limits_{i} b_{i}(x_{i}) - \sum\limits_{j} c_{j}(y_{j}). \tag{1}$$

At a social optimum, (1) is maximized subject to the constraint that supply equals demand:

$$\sum x_{i} = \sum y_{j} . \tag{2}$$

The solution to this constrained maximization is optimal in several senses:

(i) total production $$\sum y_{j}$$ and total consumption $$\sum x_{i}$$are optimal;

(ii) yj is optimal for each producer j; and

(iii) xi is optimal for each buyer i.

Achieving all three optimalities may seem complicated, but the market provides a simple solution. If p is the price at which the good can be bought and sold, then each buyer i maximizes

$$b_{i}(x_{i}) - px_{i} \text{ (net benefit)} \tag{3}$$

and the first-order condition for this maximization is

$$b'_{i}(x_{i}) = p (\text{ }b'\text{ denotes the derivative of } b). \tag{4}$$

Similarly, each producer j maximizes

$$p y_{j} - c_{j}(y_{j}) \text{ (profit)} \tag{5}$$

with first-order condition

$$p = c'_{j}(y_{j}). \tag{6}$$

But notice that (4) and (6) are also the first-order conditions for the problem of maximizing (1) subject to (2). And so the market outcome attains the social optimum as long as p is chosen so that (2) holds. (Mathematically, p is the Lagrange multiplier for (2).)

But how do we get the right choice of p? In a free market, p falls if supply exceeds demand and rises if demand exceeds supply. Eventually, the equilibrating price is found. But this process takes time. In the meantime, the price may be way too high, in which case, buyers who need tests are being “gouged,” or too low, in which case there may be a serious shortage of tests.

There is an additional problem with the market solution: it relies on producers and buyers being “small” so that they can’t individually affect the price. If some of these agents are big (e.g., if one of the equipment-producers supplies a significant fraction of demand), then the optimizations in (3) and (5) have to be modified and a social optimum no longer obtains. Moreover, by withholding supply, a big producer can distort the price-adjustment procedure and generate an outcome in which the price is too high and market supply is too low relative to the optimum. (A big buyer can do just the opposite.)

### 2. Mechanism Design to the Rescue2

For both reasons, we now turn to mechanism design.3 For now, let us assume that the government attaches (gross) benefit $$b(\sum y_{j})$$ to total production $$\sum y_{j}$$. (In the next section we decompose $$b(\sum y_{j})$$ into the underlying benefits $$\{b_{i}(y_{j})\}$$ of test-equipment users.)

The government is interested in maximizing the net social benefit

$$b(\sum\limits_{j} y_{j}) - \sum\limits_{j} c_{j}(y_{j})$$

but it doesn't know the cost functions $$\{c_{j}\}$$ (and may not even know the full set of potential producers). We solve this difficulty using a variant of the Vickrey–Clarke–Groves mechanism (Vickrey (1960), Clarke (1971), Groves (1973)). Specifically, the government announces a call for test-equipment production and has each potential producer j submit a cost function $$\hat{c}_{j}$$. It then computes the production levels $$\{\hat{y}_{j}\}$$ that maximize the apparent net social benefit

$$b(\sum\limits_{j} y_{j}) - \sum\limits_{j} \hat{c}_{j}(y_{j}) \tag{7}$$

and has producer k produce $$\hat{y}_{k}$$ and gives producer k a payment:

$$\left[b(\sum\limits_{j} \hat{y}_{j}) - \sum\limits_{j \neq k} \hat{c}_{j}(\hat{y}_{j})\right] - \left[b(\sum\limits_{j \neq k} \hat{y}^{*}_{j}) - \sum\limits_{j \neq k} \hat{c}_{j}(\hat{y}^{*}_{j})\right], \tag{8}$$

where the levels $$\{\hat{y}^{*}_{j}\}_{j \neq k}$$ maximize $$b(\sum\limits_{j \neq k} y_{j}) - \sum\limits_{j \neq k} \hat{c}_{j}(y_{j})$$.

Claim: Given that the government chooses $$\{\hat{y}_{j}\}$$ to maximize (7) and pays producer k the amount (8), it is optimal for producer k to report its costs truthfully, i.e., it will take $$\hat{c}_{k} = c_{k}$$.

Proof: The second expression in square brackets in (8) doesn't depend on $$\hat{c}_{k}$$ and so doesn't affect producer k's maximization. In effect, producer k maximizes

$$b(\sum\limits_{j} \hat{y}_{j}) - \sum\limits_{j \neq k} \hat{c}_{j}(\hat{y}_{j}) - c_{k}(\hat{y}_{k}). \tag{9}$$

But (9) is just net social benefit with cost functions $$c_{k}$$ and $$\{\hat{c}_{j}\}_{j \neq k}$$, i.e., producer k's objective is the same as society's. Thus, the optimal choice of $$\hat{c}_{k}$$ is indeed $$c_{k}$$. Q.E.D.

Let us now decompose $$b(\cdot)$$ into $$\sum\limits_{i} b_{i}(\cdot)$$.

Because government doesn't know the benefit functions $$\{b_{i}\}$$, it will have buyers report $$\{\hat{b}_{i}\}$$ (as well as having producers report $$\{\hat{c}_{j}\}$$) and, instead of maximizing (7) it will choose $$\{\hat{x}_{i}\}$$ and $$\{\hat{y}_{j}\}$$ to maximize

$$\sum\limits_{i} \hat{b}_{i}(\hat{x}_{i}) - \sum\limits{j} \hat{c}_{j}(\hat{y}_{j}) \text{ subject to } \sum\limits_{i} \hat{x}_{i} - \sum\limits_{j} \hat{y}_{j} \tag {10}$$

Buyer h then receives $$\hat{x}_{h}$$ and pays

$$\left[\sum\limits_{j} \hat{c}_{j}(\hat{y}_{j}) - \sum\limits_{i \neq h} \hat{b}_{i}(\hat{x}_{i})\right] - \left[\sum\limits_{j} \hat{c}_{j}(\hat{y}^{*}_{j}) - \sum\limits_{i \neq h} \hat{b}_{i}(\hat{x}^{*}_{i})\right],$$

where $$\{\hat{x}^{*}_{i}\}$$ and $$\{\hat{y}^{*}_{j}\}$$ maximize

$$\sum\limits_{i \neq h} \hat{b}_{i}(x_{i}) - \sum\limits_{j} \hat{c}_{j}(y_{j}). \tag{11}$$

By analogy with producer k's problem in section 2, it is optimal for buyer h in these circumstances to set $$\hat{b}_{h} = b_{h}$$.

### 4. Simple Example

Imagine that there is just a single buyer with benefit function $$b(\cdot)$$ and a single producer with cost function $$c(\cdot)$$. In that case, the government

(i) has the buyer report $$\hat{b}(\cdot)$$ and the producer report $$\hat{c}(\cdot)$$;

(ii) calculates $$z^{*}$$ to maximize $$\hat{b}(z) - \hat{c}(z)$$;r

(iii) has the producer produce $$z^{*}$$ and deliver this to the buyer; and

(iv) pays the producer $$\hat{b}(z^{*})$$ and taxes the buyer $$\hat{c}(z^{*})$$.

Notice that the buyer's objective function is

$$b(z) - \hat{c}(z)$$

and the producer's is

$$\hat{b}(z) - c(z)$$

and so it is optimal for the buyer to report $$\hat{b} = b$$ and for the producer to report $$\hat{c} = c$$.

As usual in the mechanism design literature, the way to align social and individual goals is to give individual producers and buyers monetary transfers (either positive or negative) that transform their personal objective functions into the social objective function.

Footnotes

1. This piece is based on a Santa Fe Institute webinar talk on the Complexity of COVID-19, April 14, 2020.
2. This section and the next are a bit math-heavy. For a simple explanation, see section 4.
3. An alternative to markets or mechanism design would be for government to simply order some company or companies to produce all the equipment. But this might be an extraordinarily inefficient outcome if these companies aren't up to the task or if there are other companies who could produce it much more cheaply (which the government is not likely to know in advance). Moreover, how does the government know equipment level is  "right"?

Harvard University
Santa Fe Institute

REFERENCES

1. Clarke, E. 1971. “Multipart Pricing of Public Goods.” Public Choice. 11: 17-33.
2. Groves, T. 1973. “Incentives in Teams.” Econometrica. 41(4 ): 617-631.
3. Vickrey, W. 1961. “Counterspeculation, Auctions, and Competitive Sealed Tenders.” Journal of Finance. 16 (1): 8-37.

Listen to SFI President David Krakauer discuss this Transmission in episode 32 of our Complexity Podcast.

## Reflection

August 25, 2021

### FURTHER REFLECTIONS ON MECHANISM DESIGN AND THE PANDEMIC

In May 2020, I wrote a Transmission based on my participation in an April SFI panel on complexity and the COVID-19 pandemic. My particular angle was to offer some suggestions about how mechanism design—the “reverse engineering” part of economics—might be useful in dealing with the pandemic.

I noted then that, in normal circumstances, ordinary markets do an excellent job of ensuring that the goods and services people want and need are produced and distributed. If, for example, there is currently greater demand for potatoes than available supply, we can expect the price of potatoes to rise. This will have two effects: (1) demand will be curtailed, but, more importantly, (2) potato growers will be induced to sell more potatoes, thus ameliorating the initial shortage.

However, as I discussed, there are at least two reasons why a laissez-faire approach is not likely to work very well for certain critical goods during a pandemic: (1) there may be no existing market for a good, yet it is needed right away; and (2) the good is, at least in part, a public good (its benefits go not just to the person using the good but to everyone else as well).

Suppose, for example, that a country needs to acquire millions of SARS-CoV-2 test kits quickly. This exact good hasn’t been produced before, since the virus is new. Thus, there is no existing market (although there do exist companies producing similar products).

In principle, the country’s government could leave matters to the market: suppliers that wish to produce kits would produce them and sell them to citizens (or hospitals or employers) who wish to buy.

But there are several problems with this solution. In particular, how is a supplier to know (at least at first) how many test kits to produce? After all, this is a new good and demand for it is uncertain. Furthermore, the supplier doesn’t yet know who else will be producing test kits and how much they will produce. Under such circumstances, the supplier may be reluctant to incur the significant setup costs entailed in production until the uncertainties are resolved. Given time, the market would ultimately resolve them through the equilibration of supply and demand. But that process isn’t instantaneous, and test kits are needed quickly.

Furthermore, given that supply can’t be ramped up immediately, prices are likely to be high at first, which will disproportionately hurt poorer citizens and businesses (the very groups that are worst hit by the pandemic).

And, finally, the market approach ignores the public good aspect of test kits. If I buy and use a test kit, I will get some benefit—I will know whether or not I have the virus and can take proper precautions and seek treatment if I do. But most of the benefit goes to other people, who will be protected from infection if I quarantine as a result of testing positive. Since I have little incentive to take into account those other benefits, I am likely to underpurchase test kits. And so the market system will result in too few kits being supplied and used.

At the opposite extreme, an alternative solution would be for the government to step in, pick some potential suppliers, and order them to produce test kits—that is, a command economy approach. Indeed, this approach was actually used to some extent in the United States for ventilators.

But it gives rise to some difficult questions. Which suppliers should the government choose? Clearly, it would like to choose the ones with the lowest production costs, but it doesn’t know which ones those are. Indeed, the government might not even know the identities of all potential suppliers. Moreover, how many test kits should the government order? And how do the suppliers’ costs get covered (if, in fact, they do)?

For all these reasons, I proposed a mechanism-design solution in my May 2020 Transmission. In this mechanism, the government first announces its intention to buy test kits and invites potential suppliers to furnish information about their costs. Then, on the basis of this information, the government gives each supplier a target output level (possibly zero if the supplier costs are too high) and a corresponding price that it is willing to pay for this output. After the kits are delivered, it then turns around and resells them to society for a very low or zero price. I showed that it is possible for the government to design the mechanism so that suppliers are induced to provide accurate cost information and the production targets maximize the net social benefit from test kits.

To what extent did the US government actually use a mechanism like this for critical pandemic goods? In the case of virus test kits and personal protective equipment, the answer appears to be: almost not at all. And, as a result, there were dangerous shortages of both, especially in the first year of the pandemic. Our country has endured a staggering death toll—over 600,000 people lost already—due in large part to government mismanagement. And when the final reckoning is done, the absence of test kits and protective equipment is almost certain to be an important part of the story.

The one bright spot for mechanism design was vaccine development. There, instead of leaving everything to the market, the Trump administration created Operation Warp Speed. In particular, the administration picked a number of pharma companies on the basis of their reputation or promise and covered a lot of their upfront development costs. It also offered them futures contracts: if they successfully developed a vaccine and got it approved (at least on an emergency basis), the government promised to buy a large number of doses at a specified price. And, indeed, we ended up with several vaccines in record time.

But Operation Warp Speed didn’t go far enough, at least as far as the developing world is concerned. Although there have been enough vaccine doses for almost everyone who wants them in the US and Europe, only about 2% of Africans have been vaccinated so far. Furthermore, the fact that successful pharma companies retain patent rights over their vaccines is proving to be a major stumbling block to getting enough doses to the Third World.

A far preferable solution would have been for pharma companies to give up their intellectual property in exchange for a hefty buyout and for the vaccines to have been put in the public domain. That would have allowed doses to be manufactured on a far more massive scale.

Had it been used properly, mechanism design could have saved hundreds of thousands—perhaps millions—of lives. As it was, I would give the United States and the developed world a grade of D for their response to the pandemic. Not a complete failure, but not something to be proud of either.

Read more thoughts on the COVID-19 pandemic from complex-systems researchers in The Complex Alternative, published by SFI Press.