Elsewhere I provide a back-of-the-envelope cost-benefit statistic to guide reopening the economy from the covid shutdowns; here I want to extend the model a bit. In that post I work in terms of the sort of model in which we use a reproduction number — in particular, a fairly homogeneous model. There I look at assessing the cost of creating opportunities for the disease to spread, and here I want to allow at least some variation in that.
For at least the past month I have been largely thinking about the epidemiology of the disease in terms of multiple populations; these could be counties or states, and it was county and state data that moved me toward this framework, but I've occasionally thought in terms of multiple populations in the same area, one of which engages in much more "social distancing" than the other. I think the following crude model will be sufficient for my purposes: suppose time is discrete, and consider a vector v at each time, with each component indicating infectious cases in a particular area, and vt+1=Rvt, where the reproduction number R is now a matrix instead of a number. I don't suppose that it's constant with time, but I am going to consider changing one element Rij at a single moment in time with R unchanged at all other times; to be clear, it may be changing over time, but the counterfactual follows the same path as the baseline scenario except for a single element of the matrix at a single time.
For T>t+1 let MT be the product of the transition matrices from time t+1 to time T-1, such that vT=MTRvt. The change in vT,l due to a change in Rij
is the amount of that change times MT,li vt,j. If you can place a cost on each exposure — a cost that may be different at different times and different for different populations — and multiply each row of each MT by the relevant cost and then add up the M, you get a matrix C; the associated marginal cost of an increase in Rij is Clivt,j, i.e. the relevant cost from the C matrix times the current prevalence in the population from which we're increasing the spread.[1] The real question, now, is how to get any kind of bead on C. In parallel with the earlier post, I'll note that the sum over rows of CRv is the total cost of all future infections; this will allow us to make contact with other attempts to do a cost benefit analysis on the entire crisis.
I'm not sure there's benefit in producing more formulas by imposing more structure on C; I could note that, if R were constant (and all of its eigenvalues below 1) and the costs were constant or exponentially declining, then we would basically have C=(1-R)-1, and even with varying R we can maybe read something off of that. It's worse to transmit infections to places that tend to transmit more infections; it's worse to transmit infections to places that do so, and so on.
[1] This may seem on some level obvious; I would kind of hope it does. Note an implication, though: there is substantial benefit in cutting transmission from a hotspot, and that benefit is largely independent of whether it's to a hotspot or seeding a new location. In conversation I sometimes get the impression that people think that, well, if we're trading people between hotspots, that doesn't matter, but if each infected person in each place transmits the disease, on average, to 0.8 people in their hotspot and 0.4 in the other, you will get exponential growth that could be eliminated by stopping the interchange.
Monday, May 11, 2020
Saturday, May 9, 2020
batch auctions for foreign exchange
The foreign exchange market is quite decentralized, and I've been thinking recently that it might be convenient for some players in the market for there to be a daily batch auction, perhaps early in the morning in NYC, late morning in London, and evening in Tokyo. Academics often seem to like batch auctions for the thickness (liquidity) they offer, but one of my motivations was the existence of currency derivatives and indexes; there are traders who may be trying to hedge in the spot market against an "end-of-day" risk, and I thought having a batch auction that provided fixes for derivatives would make it easier to avoid some basis risk.
Foreign exchange markets can feel a bit like barter in some ways; if I'm buying pounds from a trader in London, we'll both think of ourselves as the buyer, and if someone in Europe wants to trade dollars for yen, it's not clear whether that's a purchase or a sale.[1] In some ways, a centralized auction alleviates the double-coincidence-of-wants problem and makes barter feasible and even makes money (or, in this case, a vehicle currency) redundant, but there is a complication: consider a trader who enters, into the auction, an order expressing a desire to exchange 10 euros for 900 yen, and suppose the auction determines that the clearing price is 100 yen per euro. A European entering an order to buy 900 yen expects to end up exchanging 9 euros for 900 yen, while a Japanese person entering an order to sell 10 euros expects to exchange 10 euros for 1000 yen. An American may wish to hand over 10 euros and receive 900 yen and to receive the gains from trade in US dollars. An order now should specify the currency of the order, which may be the currency the trader wants to buy, or sell, or something else. In more generality, the trader may express three baskets of currencies, expecting that, if the trade is executed, the trader will give up basket A and receive basket B and some multiple of basket C such that the currency received and the currency provided have the same value at the clearing prices.
Actually determining how to clear the markets turns out to be equivalent to solving a convex optimization problem, at least if currencies are arbitrarily divisible and orders can be partially executed, at least provided that, for each order that executes, the currency associated with that order is provided by some combination of other orders that execute. The solution technique is likely to involve iteratively finding the excess demand for different currencies given different price vectors, and it seems likely that in a practical distributed setting you would want to group orders by currency, where you would first figure out, given the proposed price, which orders execute, the total gains from trade those orders, and how much currency that adds to the demand vector. One potential complication here is created by orders that clear exactly, with no gains from trade, which may end up partially executing; when excess demand of different groups of orders is being aggregated it may be necessary, especially late in the process of finding the execution price, to retain a lot of possible excess demand vectors to aggregate the partial calculations.
[1] This is the source of some confusion regarding options at times; the terms "put" and "call" are similarly poorly defined. In some ways, though, this is clarifying, though I think I should let go of this particular tangent at this point.
Foreign exchange markets can feel a bit like barter in some ways; if I'm buying pounds from a trader in London, we'll both think of ourselves as the buyer, and if someone in Europe wants to trade dollars for yen, it's not clear whether that's a purchase or a sale.[1] In some ways, a centralized auction alleviates the double-coincidence-of-wants problem and makes barter feasible and even makes money (or, in this case, a vehicle currency) redundant, but there is a complication: consider a trader who enters, into the auction, an order expressing a desire to exchange 10 euros for 900 yen, and suppose the auction determines that the clearing price is 100 yen per euro. A European entering an order to buy 900 yen expects to end up exchanging 9 euros for 900 yen, while a Japanese person entering an order to sell 10 euros expects to exchange 10 euros for 1000 yen. An American may wish to hand over 10 euros and receive 900 yen and to receive the gains from trade in US dollars. An order now should specify the currency of the order, which may be the currency the trader wants to buy, or sell, or something else. In more generality, the trader may express three baskets of currencies, expecting that, if the trade is executed, the trader will give up basket A and receive basket B and some multiple of basket C such that the currency received and the currency provided have the same value at the clearing prices.
Actually determining how to clear the markets turns out to be equivalent to solving a convex optimization problem, at least if currencies are arbitrarily divisible and orders can be partially executed, at least provided that, for each order that executes, the currency associated with that order is provided by some combination of other orders that execute. The solution technique is likely to involve iteratively finding the excess demand for different currencies given different price vectors, and it seems likely that in a practical distributed setting you would want to group orders by currency, where you would first figure out, given the proposed price, which orders execute, the total gains from trade those orders, and how much currency that adds to the demand vector. One potential complication here is created by orders that clear exactly, with no gains from trade, which may end up partially executing; when excess demand of different groups of orders is being aggregated it may be necessary, especially late in the process of finding the execution price, to retain a lot of possible excess demand vectors to aggregate the partial calculations.
[1] This is the source of some confusion regarding options at times; the terms "put" and "call" are similarly poorly defined. In some ways, though, this is clarifying, though I think I should let go of this particular tangent at this point.
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