Third (to finally answer your question!), no my hypothetical model is not the same as the way they are using the term “buffer” (which seems to be more about maintaining a minimum level of excess in the system; mine is simply about the optimal tradeoff between stockouts vs excess/waste). For instance M&H say (p.25) “if there is some probability (1/n) that any given purchase will occur on a threshold, then the threshold action will trigger a reduction in production of around n units, yielding an expected impact equal to 1″ (and from the reducing suffering page: “The probability that any given chicken is the chicken that causes two cases instead of three to be purchased is 1/25”).
Sorry, I could have been more explicit in my comment. I wasn’t referring to the rest of the Reducing Suffering article, and I didn’t mean that any of that article referred to your model. M&H refer to a model similar to yours (Budolfson’s buffer model), but not in the section that I referred to (and from which you quote). What I meant is that both propose more plausible models of markets (more plausible based on observations of how grocery stores behave), and I was pointing to those alternative proposals.
M&H summarizes the main takeaway from Budolfson’s buffer model:
If a person is facing a decision with this kind of uncertainty, and they have good information about the probability of being near a threshold, this can dramatically alter the expected impact calculation. (...) Similarly, if a person knew that their purchase of a chicken was not near the threshold, they could, he argues, purchase the chicken without worry about consequences for animals.
Budolfson is correct in claiming that expected impact calculations cannot always assume that an action, on the margin, would be the same as the average effect of many such actions. The standard expected utility response given by Singer and Kagan can depend crucially on the kind of information that a person has about the location of thresholds.
This is an illustration of Budolfson’s buffer model, directly from Budolfson, 2018:
Richard makes paper T-shirts in his basement that say ‘HOORAY FOR CONSEQUENTIALISM!’, which he then sells online. The T-shirts are incredibly cheap to produce and very profitable to sell and Richard doesn’t care about waste per se, and so he produces far more T-shirts than he is likely to need each month, and then sells the excess at a nearly break-even amount at the end of each month to his hippie neighbor, who burns them in his wood-burning stove.Footnote10 For many years Richard has always sold between 14,000 and 16,000 T-shirts each month, and he’s always printed 20,000 T-shirts at the beginning of each month. Nonetheless, there is a conceivable increase in sales that would cause him to produce more T-shirts—in particular, if he sells over 18,000 this month, he’ll produce 25,000 T-shirts at the beginning of next month; otherwise he’ll produce 20,000 like he always does. So, the system is genuinely sensitive to a precise tipping point—in particular, the difference between 18,000 purchases and the ‘magic number’ of 18,001.
Presumably there could also be a conceivable decrease in sales that would cause Richard to produce fewer T-shirts, too. Richard has a historical monthly demand range that serves essentially the same purpose as your predicted demand, with thresholds for setting alternative future procurement/production decisions far enough away from the centre of the historical range, or in your case, predicted demand.
EDIT: so your last paragraph seems wrong:
I’m claiming that we don’t know, and the fact that none of these sources seems to have even considered it (or any other ones), and don’t even realize the nature of the assumptions they’re making, and nevertheless draw such strong conclusions, is again a bad sign.
Interesting—thanks for the extra info re Budolfson. I did in fact read all of M&H, and they give two interpretations of the buffer model, neither of which is related to my model, so that’s what I was referring to. [That’s also what I was referring to in my final paragraph: none of the sources you cited on that side of the causal efficacy argument seems to have considered anything like my model, which remains true given my current knowledge.] In fact if Budolfson was saying something more like my model, which does seem to be the case, then that’s an even worse sign for M&H because they must not have understood it.
The paragraph you quote from Budolfson is indeed more similar to my model, except that in my case the result follows from profit-maximizing behavior (in a competitive industry if you like!) rather than ad hoc and unusual assumptions.
Suppose that I consider a threshold (for increasing or decreasing production next cycle) right at the mean of expected sales (15,000 in the example): half the time I’ll stockout and have disappointed customers; half the time I’ll have extra stock and have to sell it on a secondary market, or give it away, or waste it. Which is worse for business? Plausibly stocking out is worse. So my threshold will be higher than the mean, reducing the probability of stocking out and increasing the prob of excess. The optimal level will be set just so that at the margin, the badness of stocking out (larger) multiplied by the prob of stocking out (smaller) will exactly offset the badness of excess times the prob of excess. Because it is above the mean, which is in fact the true best-guess state of the world (ignoring any individual consumer), and because the distribution around the mean will plausibly be Gaussian (normal), which declines exponentially from the mean—not linearly! - every individual consumer should rationally believe that their decision is less than 1/n likely to be taking place at the threshold. QED.
I’m not sure what you mean by M&H not understanding Budolfson. They give a brief overview of the model, but the section from M&H I referred to (“Efficient Responsive Supply Chains and Causal Efficacy”) describes the market as they understand it, in a way that’s not consistent with Budolfson. The implicit reply is that Budolfson’s model does not match their observations of how the market actually works.
I think how they’d respond to your model is:
stores do use explicit demand predictions to decide procurement,
they are constantly making new predictions,
these predictions are in fact very sensitive to recent individual purchase decisions, and actually directly so.
Suppose the store makes stocking decisions weekly. If demand is lower one week than it would have otherwise been, their predictions for the next week will be lower than they would have otherwise been. Of course, there’s still a question of how sensitive: maybe they give little weight to their actual recent recorded purchases[1] relative to other things, like others’ market forecasts or sales the same time in past years.[2] But M&H would contend that actually they are very sensitive to recent purchases, and I would guess that’s the case, too, because it probably is one of the most predictive pieces of information they can use, and plausibly the most predictive. They don’t provide direct estimates of the sensitivity based on empirical data and maybe they don’t back these claims with strong enough evidence at all (i.e. maybe stores don’t actually usually work this way), and it’s fair to point out these kinds of holes in their arguments if someone wants to use their paper to make a strong case.
Here are relevant quotes:
For example, modern grocery stores have check-out procedures that track the sale of each product and automatically order replacements from the parent companies. Even in industries that are not vertically integrated, standard information technology allows firms to track sales in great detail, down to individual transactions (Salin 2000). In addition, these companies track the rates of orders to optimize shipping and refrigeration times and to minimize waste. (...) In this kind of system, the large distributors that contract with farms actually do know the rate at which chickens are being purchased throughout their network.
(...)
Given this description of the way these markets function, we can now describe the causal chain that connects an individual’s purchase to a farmer’s production decision. When a person decides to stop purchasing chickens, the result is that their local grocery store automatically starts ordering chickens more slowly, to reflect the decreased rate of sale. The distributor (perhaps Chickens R Us) will automatically adjust their shipments of chickens to that store. Since some shipments will require preset bundles of chickens, there will be a threshold at which a delivery of meat comes a day later, to reflect the slower demand. This “threshold” does not mean, however that the information going down the supply chain is less precise. As Chickens R Us is managing their supply of chickens in the distribution network, they are also managing the rate at which they send contracts of birds to their “growers” and the number of growers that get contracts.
I would correct the one sentence to “When a person decides to stop purchasing chickens, the result is that their local grocery store automatically starts ordering chickens more slowly than they otherwise would have, to reflect the lower than otherwise rate of sale.”
I still haven’t read Budolfson, so I’m not claiming that M&H misinterpret him. As I said, I did read their entire paper, and in the section specifically about him they describe two interpretations of “buffer”, neither of which matches my model. So if his model is similar to mine, they got it wrong. If his model is different than mine, then they don’t seem to have ever considered a model like mine. Either way a bad sign.
Everything you write about how you think they might respond to me (i.e. your three bullet points and the subsequent paragraph) is 100% consistent with my model and doesn’t change any of its implications. In my model stores use predicted demand and can update it as often as they want. The point is that purchasing is in bulk (at least at some level in the supply chain); therefore there is a threshold; and the optimal threshold (every single time) will be chosen to be away from the mean prediction. This can still be extremely sensitive, and may well be. [Apologies if my brief descriptions were unclear, but please do take another look at it before responding if you don’t see why all this is the case.]
To the final point, yes of course if someone decides to stop purchasing then the store [probabilistically] starts ordering fewer chickens [than otherwise]; I didn’t disagree with that sentence of theirs, and it is also 100% consistent with my model. The question is the magnitude of that change and whether it is linear or not, crucial points to which they have nothing to contribute.
EDIT: I did misunderstand at this point, as you pointed out in your reply.
Ok, I think I get your model, but I don’t really see why a grocery store in particular would follow it, and it seems like a generally worse way to make order decisions for one. I think it’s more plausible for earlier parts of the supply chain, where businesses may prefer to produce consistent volumes, because there are relevant thresholds (in revenue) for shutting down, downsizing, expanding and entering the market, and it’s costly to make such a decision (selling/buying capital, hiring/firing staff) only to regret it later or even flip-flop.[1] It takes work to hire someone, so hiring and firing (in either order) is costly. Capital assets lose value once you purchase or use them, so buying and selling (in either order) is costly. If changes in a business’ production levels often require such a decision, that business has reason to try to keep production more consistent or stick with their plans to avoid accumulating such costs. But not all changes to production levels require such decisions.
(I don’t mean to imply you don’t understand all of the above; this is just me thinking through it, checking my understanding and showing others interested.)
I don’t think a grocery store has to adjust its capital or staff to order more or less, or at least not for the vast majority of marginal changes in order size. Same for distributors/wholesalers.
I’m not sure about broiler farms. They’d sometimes just have to wait longer for a contract (or never get one again), or maybe they’d get a smaller contract and raise fewer broilers (the market is contract-based in the US, and the farms don’t own the broilers[2]), so it often just wouldn’t be their decision. But on something like your model, if a farm was planning to enter the market or expand, and contracts or revenues (or market reports) come only slightly worse than expected (still above the threshold in your model, and which is far more likely than coming below the threshold), they’d enter/expand anyway. For farms not planning to expand/enter the market, maybe they’d even take on a contract they don’t expect to pay for its variable costs, just to get more favour from the companies contracting them in the future or to push out competitors. Or, just generally, the contracts would very disproportionately be above their thresholds for shutdown, as they expect them to be. Also, many individual farmers are probably subject to the sunk cost fallacy.
Then there are the integrator/processor companies like Tyson that contract the farms. A small number of companies control a large shares of this part of the supply chain, and they’ve been caught price-fixing (see here and here), which undermines the efficiency (and of course competitiveness) of the market. Below their predictions, maybe they’d want to keep giving farms contracts in order to keep them from shutting down or to keep them from switching to competitors, because it’ll be harder/slower to replace them if demand recovers, or just to hurt competitors. Or, if they were already planning to expand production, but sales come in below expectation, they’d do it anyway for similar reasons.
Here’s an example for a grocery store:
Suppose, to avoid stockouts (like you propose they should), as a rule, they order 7 more units than (the expected value of) their predicted sales.
Suppose they would have predicted 123 sales for the next period had you not abstained. Because you abstained, they instead predict 122. So, as a result of your abstention, they order 129 instead of 130, and you make a difference, at least at this level.
Now, maybe they need to order in specific multiples of units. Say they need to order in multiples of 10, and they order the minimum multiple of 10 that’s at least 7 over what they predict.
In the above case, your abstention makes no difference, and they would order 130 either way, but that’s just one case. The threshold to order 10 fewer is when the prediction modulo 10 would have been 4 and your abstention drops it below that.[3] If you look at a randomly sampled period where they need to order, there’s not really any reason to believe that their prediction modulo 10 will be especially unlikely to be 4 compared to any of the other digits.[4]
Broiler production contracts add another risk aside from the way in which compensation is determined. Traditionally, broiler contracts have not required strong commitments by integrators. In 2006, about half of broiler contracts were “flock to flock”; that is, the integrator made no specific commitment to provide birds beyond the current flock’s placement. Those contracts that specified a longer duration (usually 1 to 5 years) rarely committed the integrator to a specified number of birds or flocks in a year.
I guess one way would be if they have sufficiently consistent purchases and choose a supplier based on the multiple to get their prediction modulo the multiple away from the threshold. I think it’s very unlikely they’d switch suppliers just to get their predictions in a better spot with respect to multiples.
Hi—thanks again for taking more time with this, but I don’t think you do understand my model. It has nothing to do with capital assets, hiring/firing workers, or switching suppliers. All that it requires is that some decisions are made in bulk, i.e. at a level of granularity larger than the impact of any one individual consumer. I agree this is less likely for retail stores (possibly some of them order in units of 1? wouldn’t it be nice if someone actually cared enough to look into this rather than us all arguing hypothetically...), but it will clearly happen somewhere back up the supply chain, which is all that my model requires.
Your mistake is when you write “Say they need to order in multiples of 10, and they order the minimum multiple of 10 that’s at least 7 over what they predict.” That’s not what my model predicts (I think it’s closer to M&H’s first interpretation of buffers?), nor does it make economic sense, and it builds in linearity. What a profit-maximizing store will do is to balance the marginal benefit and marginal cost. Thus if they would ideally order 7 extra, but they have to order in multiples of 10 and x=4 mod10, they’ll order x+6 not x+16 (small chance of one extra stock-out vs large chance of 10 wasted items). They may not always pick the multiple-of-10 closest to 7 extra, but they will balance the expected gains and losses rather than using a minimum. From there everything that I’m suggesting (namely the exponential decline in probability, which is the key point where this differs from all the others) follows.
And a quick reminder: I’m not claiming that my model is the right one or the best one, however it is literally the first one that I thought of and yet no one else in this literature seems to have considered it. Hence my conclusion that they’re making far stronger claims than are possibly warranted.
(I’ve edited this comment, but the main argument about grocery stores hasn’t changed, only some small additions/corrections to it, and changes to the rest of my response.)
Thanks for clarifying again. You’re right that I misunderstood. The point as I now understand is that they expect the purchases (or whatever they’d ideally order, if they could order by individual units) to fall disproportionately in one order size and away from each threshold for lower and higher order sizes, i.e. much more towards the middle, and they’ve arranged for their order sizes to ensure this.
I’ll abandon the specific procedure I suggested for the store, and make my argument more general. For large grocery stores, I think my argument at the end of my last comment is still basically right, though, and so you should expect sensitivity, as I will elaborate further here. In particular, this would rule out your model applying to large grocery stores, even if they have to order in large multiples, assuming a fixed order frequency.
Let’s consider a grocery store. Suppose they make purchase predictions p (point estimates or probability distributions), and they have to order in multiples of K, but I’ll relax this assumption later. We can represent this with a function f from predictions to order sizes so that f(p)=K∗g(p), where g is an integer-valued function.f can be the solution to an optimization problem, like yours. I’m ignoring any remaining stock they could carry forward for simplicity, but they could just subtract it from p and put that stock out first. I’m also assuming a fixed order frequency, but M&H mention the possibility of “a threshold at which a delivery of meat comes a day later”. I think your model is a special case of this, ignoring what I’m ignoring and with the appropriate relaxations below.
I claim the following:
Assuming the store is not horrible at optimizing, f should be nondecreasing and scale roughly linearly with p. What I mean by “roughly linearly with p” is that for (the vast majority of possible values of) p, we can assume that f(p+K)=f(p)+K, and that values of p where f(p+1)=f(p)+K, i.e. the thresholds, are spaced roughly K apart. Even if different order sizes didn’t differ in multiples of some fixed number, something similar should hold, with spacing between thresholds roughly reflecting order size differences.
A specific store might have reason to believe their predictions are on a threshold much less than 1/K of the time across order decisions, but only for one of a few pretty specific reasons:
They were able to choose K the first time to ensure this, intentionally or not, and stick with it and f regardless of how demand shifts.
The same supplier for the store offers different values of K (or the store gets the supplier to offer another value of K), and the store switches K or uses multiple values of K simultaneously in a way that avoids the thresholds. (So f defined above isn’t general enough.)
They switch suppliers or products as necessary to choose K in a way that avoids the thresholds. Maybe they don’t stop offering a product or stop ordering from the same supplier altogether, but optimize the order(s) for it and a close substitute (or multiple substitutes) or multiple suppliers in such a way that the thresholds are avoided for each. (So f defined above isn’t general enough.)
If none of these specific reasons hold, then you shouldn’t expect to be on the threshold much less than 1/K of the time,[1] and you should believe E[f(p−1)]≈E[f(p)]−1, where the expectation is taken over your probability distribution for the store’s prediction p.
How likely are any of these reasons to hold, and what difference should they make to your expectations even if they did?
The first reason wouldn’t give you far less than 1/K if the interquartile range of their predictions across orders over time isn’t much smaller than K, but they prefer or have to keep offering the product anyway. This is because the thresholds are spaced roughly K apart, p will have to cross thresholds often with such a large interquartile range, and if p has to cross thresholds often, it can’t very disproportionately avoid them.[2]
Most importantly, however, if K is chosen (roughly) independently of p, your probability distribution for pmodK for a given order should be (roughly) uniform over 0,..., K−1,[3] so p should hit the threshold with probability (roughly) 1/K. It seems to me that K is generally chosen (roughly) independently of p. In deciding between suppliers, the specific value of K seems less important than the cost per unit, shipping time, reliability and a lower value of K.[4] In some cases, especially likely for stores belonging to large store chains, there isn’t a choice, e.g. Walmart stores order from Walmart-owned distributors, or chain stores will have agreements with the same supplier company across stores. Then, having chosen a supplier, a store could try to arrange for a different value of K to avoid thresholds, but I doubt they’d actually try this, and even if they did try, suppliers seem likely to refuse without a significant increase in the cost per unit for the store, because suppliers have multiple stores to ship to and don’t want to adjust K by the store.
Stores similarly probably wouldn’t follow the strategies in the second and third reasons because they wouldn’t be allowed to, or even if they could, other considerations like cost per unit, shipping time, reliability and stocking the same product would be more important. Also, if the order quantities vary often enough between orders based on such strategies, you’d actually be more likely to make a difference, although smaller when you do.
So, I maintain that for large stores, you should believe E[f(p−1)]≈E[f(p)]−1.
And a quick reminder: I’m not claiming that my model is the right one or the best one, however it is literally the first one that I thought of and yet no one else in this literature seems to have considered it. Hence my conclusion that they’re making far stronger claims than are possibly warranted.
Fair. I don’t think they should necessarily have considered it, though, in case observations they make would have ruled it out, but it seems like they didn’t make such observations.
but it will clearly happen somewhere back up the supply chain, which is all that my model requires.
I don’t think this is obvious either way. This seems to be a stronger claim than you’ve been making elsewhere about your model. I think you’d need to show that it’s possible and worth it for those at one step of the supply chain to choose K or suppliers like in a way I ruled out for grocery stores and without making order sizes too sensitive to predictions. Or something where my model wasn’t general enough, e.g. I assumed a fixed order frequency.
It could be more than 1/K, because we’ve ruled out being disproportionately away from the threshold by assumption, but still allowed the possibility of disproportionately hitting it.
I would in fact expect lower numbers within 0, …, K−1 to be slightly more likely, all else equal. Basically Benford’s law and generalizations to different digit positions. Since these are predictions and people like round numbers, if K is even or a multiple of 5, I wouldn’t be surprised if even numbers and multiples of 5 were more likely, respectively.
Except maybe if the minimum K across suppliers is only a few times less than p, closer to p or even greater, and they can’t carry stock forward past the next time they would otherwise receive a new shipment.
Sorry, I could have been more explicit in my comment. I wasn’t referring to the rest of the Reducing Suffering article, and I didn’t mean that any of that article referred to your model. M&H refer to a model similar to yours (Budolfson’s buffer model), but not in the section that I referred to (and from which you quote). What I meant is that both propose more plausible models of markets (more plausible based on observations of how grocery stores behave), and I was pointing to those alternative proposals.
M&H summarizes the main takeaway from Budolfson’s buffer model:
This is an illustration of Budolfson’s buffer model, directly from Budolfson, 2018:
Presumably there could also be a conceivable decrease in sales that would cause Richard to produce fewer T-shirts, too. Richard has a historical monthly demand range that serves essentially the same purpose as your predicted demand, with thresholds for setting alternative future procurement/production decisions far enough away from the centre of the historical range, or in your case, predicted demand.
EDIT: so your last paragraph seems wrong:
Interesting—thanks for the extra info re Budolfson. I did in fact read all of M&H, and they give two interpretations of the buffer model, neither of which is related to my model, so that’s what I was referring to. [That’s also what I was referring to in my final paragraph: none of the sources you cited on that side of the causal efficacy argument seems to have considered anything like my model, which remains true given my current knowledge.] In fact if Budolfson was saying something more like my model, which does seem to be the case, then that’s an even worse sign for M&H because they must not have understood it.
The paragraph you quote from Budolfson is indeed more similar to my model, except that in my case the result follows from profit-maximizing behavior (in a competitive industry if you like!) rather than ad hoc and unusual assumptions.
Suppose that I consider a threshold (for increasing or decreasing production next cycle) right at the mean of expected sales (15,000 in the example): half the time I’ll stockout and have disappointed customers; half the time I’ll have extra stock and have to sell it on a secondary market, or give it away, or waste it. Which is worse for business? Plausibly stocking out is worse. So my threshold will be higher than the mean, reducing the probability of stocking out and increasing the prob of excess. The optimal level will be set just so that at the margin, the badness of stocking out (larger) multiplied by the prob of stocking out (smaller) will exactly offset the badness of excess times the prob of excess. Because it is above the mean, which is in fact the true best-guess state of the world (ignoring any individual consumer), and because the distribution around the mean will plausibly be Gaussian (normal), which declines exponentially from the mean—not linearly! - every individual consumer should rationally believe that their decision is less than 1/n likely to be taking place at the threshold. QED.
I’m not sure what you mean by M&H not understanding Budolfson. They give a brief overview of the model, but the section from M&H I referred to (“Efficient Responsive Supply Chains and Causal Efficacy”) describes the market as they understand it, in a way that’s not consistent with Budolfson. The implicit reply is that Budolfson’s model does not match their observations of how the market actually works.
I think how they’d respond to your model is:
stores do use explicit demand predictions to decide procurement,
they are constantly making new predictions,
these predictions are in fact very sensitive to recent individual purchase decisions, and actually directly so.
Suppose the store makes stocking decisions weekly. If demand is lower one week than it would have otherwise been, their predictions for the next week will be lower than they would have otherwise been. Of course, there’s still a question of how sensitive: maybe they give little weight to their actual recent recorded purchases[1] relative to other things, like others’ market forecasts or sales the same time in past years.[2] But M&H would contend that actually they are very sensitive to recent purchases, and I would guess that’s the case, too, because it probably is one of the most predictive pieces of information they can use, and plausibly the most predictive. They don’t provide direct estimates of the sensitivity based on empirical data and maybe they don’t back these claims with strong enough evidence at all (i.e. maybe stores don’t actually usually work this way), and it’s fair to point out these kinds of holes in their arguments if someone wants to use their paper to make a strong case.
Here are relevant quotes:
I would correct the one sentence to “When a person decides to stop purchasing chickens, the result is that their local grocery store automatically starts ordering chickens more slowly than they otherwise would have, to reflect the lower than otherwise rate of sale.”
Or, indirectly, through leftover stocks or stockouts.
Although eventually that should get picked up.
I still haven’t read Budolfson, so I’m not claiming that M&H misinterpret him. As I said, I did read their entire paper, and in the section specifically about him they describe two interpretations of “buffer”, neither of which matches my model. So if his model is similar to mine, they got it wrong. If his model is different than mine, then they don’t seem to have ever considered a model like mine. Either way a bad sign.
Everything you write about how you think they might respond to me (i.e. your three bullet points and the subsequent paragraph) is 100% consistent with my model and doesn’t change any of its implications. In my model stores use predicted demand and can update it as often as they want. The point is that purchasing is in bulk (at least at some level in the supply chain); therefore there is a threshold; and the optimal threshold (every single time) will be chosen to be away from the mean prediction. This can still be extremely sensitive, and may well be. [Apologies if my brief descriptions were unclear, but please do take another look at it before responding if you don’t see why all this is the case.]
To the final point, yes of course if someone decides to stop purchasing then the store [probabilistically] starts ordering fewer chickens [than otherwise]; I didn’t disagree with that sentence of theirs, and it is also 100% consistent with my model. The question is the magnitude of that change and whether it is linear or not, crucial points to which they have nothing to contribute.
EDIT: I did misunderstand at this point, as you pointed out in your reply.
Ok, I think I get your model, but I don’t really see why a grocery store in particular would follow it, and it seems like a generally worse way to make order decisions for one. I think it’s more plausible for earlier parts of the supply chain, where businesses may prefer to produce consistent volumes, because there are relevant thresholds (in revenue) for shutting down, downsizing, expanding and entering the market, and it’s costly to make such a decision (selling/buying capital, hiring/firing staff) only to regret it later or even flip-flop.[1] It takes work to hire someone, so hiring and firing (in either order) is costly. Capital assets lose value once you purchase or use them, so buying and selling (in either order) is costly. If changes in a business’ production levels often require such a decision, that business has reason to try to keep production more consistent or stick with their plans to avoid accumulating such costs. But not all changes to production levels require such decisions.
(I don’t mean to imply you don’t understand all of the above; this is just me thinking through it, checking my understanding and showing others interested.)
I don’t think a grocery store has to adjust its capital or staff to order more or less, or at least not for the vast majority of marginal changes in order size. Same for distributors/wholesalers.
I’m not sure about broiler farms. They’d sometimes just have to wait longer for a contract (or never get one again), or maybe they’d get a smaller contract and raise fewer broilers (the market is contract-based in the US, and the farms don’t own the broilers[2]), so it often just wouldn’t be their decision. But on something like your model, if a farm was planning to enter the market or expand, and contracts or revenues (or market reports) come only slightly worse than expected (still above the threshold in your model, and which is far more likely than coming below the threshold), they’d enter/expand anyway. For farms not planning to expand/enter the market, maybe they’d even take on a contract they don’t expect to pay for its variable costs, just to get more favour from the companies contracting them in the future or to push out competitors. Or, just generally, the contracts would very disproportionately be above their thresholds for shutdown, as they expect them to be. Also, many individual farmers are probably subject to the sunk cost fallacy.
Then there are the integrator/processor companies like Tyson that contract the farms. A small number of companies control a large shares of this part of the supply chain, and they’ve been caught price-fixing (see here and here), which undermines the efficiency (and of course competitiveness) of the market. Below their predictions, maybe they’d want to keep giving farms contracts in order to keep them from shutting down or to keep them from switching to competitors, because it’ll be harder/slower to replace them if demand recovers, or just to hurt competitors. Or, if they were already planning to expand production, but sales come in below expectation, they’d do it anyway for similar reasons.
Here’s an example for a grocery store:
Suppose, to avoid stockouts (like you propose they should), as a rule, they order 7 more units than (the expected value of) their predicted sales.
Suppose they would have predicted 123 sales for the next period had you not abstained. Because you abstained, they instead predict 122. So, as a result of your abstention, they order 129 instead of 130, and you make a difference, at least at this level.
Now, maybe they need to order in specific multiples of units. Say they need to order in multiples of 10, and they order the minimum multiple of 10 that’s at least 7 over what they predict.
In the above case, your abstention makes no difference, and they would order 130 either way, but that’s just one case. The threshold to order 10 fewer is when the prediction modulo 10 would have been 4 and your abstention drops it below that.[3] If you look at a randomly sampled period where they need to order, there’s not really any reason to believe that their prediction modulo 10 will be especially unlikely to be 4 compared to any of the other digits.[4]
I see papers on sunk-cost hysteresis and entry and exist decisions under uncertainty, like Baldwin, 1989, Dixit, 1989, Gschwandtner and Lambson, 2002.
Also:
For their prediction x, if x mod10=4, then they order x+16. If x mod10=3, then they order x+7.
I guess one way would be if they have sufficiently consistent purchases and choose a supplier based on the multiple to get their prediction modulo the multiple away from the threshold. I think it’s very unlikely they’d switch suppliers just to get their predictions in a better spot with respect to multiples.
Hi—thanks again for taking more time with this, but I don’t think you do understand my model. It has nothing to do with capital assets, hiring/firing workers, or switching suppliers. All that it requires is that some decisions are made in bulk, i.e. at a level of granularity larger than the impact of any one individual consumer. I agree this is less likely for retail stores (possibly some of them order in units of 1? wouldn’t it be nice if someone actually cared enough to look into this rather than us all arguing hypothetically...), but it will clearly happen somewhere back up the supply chain, which is all that my model requires.
Your mistake is when you write “Say they need to order in multiples of 10, and they order the minimum multiple of 10 that’s at least 7 over what they predict.” That’s not what my model predicts (I think it’s closer to M&H’s first interpretation of buffers?), nor does it make economic sense, and it builds in linearity. What a profit-maximizing store will do is to balance the marginal benefit and marginal cost. Thus if they would ideally order 7 extra, but they have to order in multiples of 10 and x=4 mod10, they’ll order x+6 not x+16 (small chance of one extra stock-out vs large chance of 10 wasted items). They may not always pick the multiple-of-10 closest to 7 extra, but they will balance the expected gains and losses rather than using a minimum. From there everything that I’m suggesting (namely the exponential decline in probability, which is the key point where this differs from all the others) follows.
And a quick reminder: I’m not claiming that my model is the right one or the best one, however it is literally the first one that I thought of and yet no one else in this literature seems to have considered it. Hence my conclusion that they’re making far stronger claims than are possibly warranted.
(I’ve edited this comment, but the main argument about grocery stores hasn’t changed, only some small additions/corrections to it, and changes to the rest of my response.)
Thanks for clarifying again. You’re right that I misunderstood. The point as I now understand is that they expect the purchases (or whatever they’d ideally order, if they could order by individual units) to fall disproportionately in one order size and away from each threshold for lower and higher order sizes, i.e. much more towards the middle, and they’ve arranged for their order sizes to ensure this.
I’ll abandon the specific procedure I suggested for the store, and make my argument more general. For large grocery stores, I think my argument at the end of my last comment is still basically right, though, and so you should expect sensitivity, as I will elaborate further here. In particular, this would rule out your model applying to large grocery stores, even if they have to order in large multiples, assuming a fixed order frequency.
Let’s consider a grocery store. Suppose they make purchase predictions p (point estimates or probability distributions), and they have to order in multiples of K, but I’ll relax this assumption later. We can represent this with a function f from predictions to order sizes so that f(p)=K∗g(p), where g is an integer-valued function.f can be the solution to an optimization problem, like yours. I’m ignoring any remaining stock they could carry forward for simplicity, but they could just subtract it from p and put that stock out first. I’m also assuming a fixed order frequency, but M&H mention the possibility of “a threshold at which a delivery of meat comes a day later”. I think your model is a special case of this, ignoring what I’m ignoring and with the appropriate relaxations below.
I claim the following:
Assuming the store is not horrible at optimizing, f should be nondecreasing and scale roughly linearly with p. What I mean by “roughly linearly with p” is that for (the vast majority of possible values of) p, we can assume that f(p+K)=f(p)+K, and that values of p where f(p+1)=f(p)+K, i.e. the thresholds, are spaced roughly K apart. Even if different order sizes didn’t differ in multiples of some fixed number, something similar should hold, with spacing between thresholds roughly reflecting order size differences.
A specific store might have reason to believe their predictions are on a threshold much less than 1/K of the time across order decisions, but only for one of a few pretty specific reasons:
They were able to choose K the first time to ensure this, intentionally or not, and stick with it and f regardless of how demand shifts.
The same supplier for the store offers different values of K (or the store gets the supplier to offer another value of K), and the store switches K or uses multiple values of K simultaneously in a way that avoids the thresholds. (So f defined above isn’t general enough.)
They switch suppliers or products as necessary to choose K in a way that avoids the thresholds. Maybe they don’t stop offering a product or stop ordering from the same supplier altogether, but optimize the order(s) for it and a close substitute (or multiple substitutes) or multiple suppliers in such a way that the thresholds are avoided for each. (So f defined above isn’t general enough.)
If none of these specific reasons hold, then you shouldn’t expect to be on the threshold much less than 1/K of the time,[1] and you should believe E[f(p−1)]≈E[f(p)]−1, where the expectation is taken over your probability distribution for the store’s prediction p.
How likely are any of these reasons to hold, and what difference should they make to your expectations even if they did?
The first reason wouldn’t give you far less than 1/K if the interquartile range of their predictions across orders over time isn’t much smaller than K, but they prefer or have to keep offering the product anyway. This is because the thresholds are spaced roughly K apart, p will have to cross thresholds often with such a large interquartile range, and if p has to cross thresholds often, it can’t very disproportionately avoid them.[2]
Most importantly, however, if K is chosen (roughly) independently of p, your probability distribution for p mod K for a given order should be (roughly) uniform over 0,..., K−1,[3] so p should hit the threshold with probability (roughly) 1/K. It seems to me that K is generally chosen (roughly) independently of p. In deciding between suppliers, the specific value of K seems less important than the cost per unit, shipping time, reliability and a lower value of K.[4] In some cases, especially likely for stores belonging to large store chains, there isn’t a choice, e.g. Walmart stores order from Walmart-owned distributors, or chain stores will have agreements with the same supplier company across stores. Then, having chosen a supplier, a store could try to arrange for a different value of K to avoid thresholds, but I doubt they’d actually try this, and even if they did try, suppliers seem likely to refuse without a significant increase in the cost per unit for the store, because suppliers have multiple stores to ship to and don’t want to adjust K by the store.
Stores similarly probably wouldn’t follow the strategies in the second and third reasons because they wouldn’t be allowed to, or even if they could, other considerations like cost per unit, shipping time, reliability and stocking the same product would be more important. Also, if the order quantities vary often enough between orders based on such strategies, you’d actually be more likely to make a difference, although smaller when you do.
So, I maintain that for large stores, you should believe E[f(p−1)]≈E[f(p)]−1.
Fair. I don’t think they should necessarily have considered it, though, in case observations they make would have ruled it out, but it seems like they didn’t make such observations.
I don’t think this is obvious either way. This seems to be a stronger claim than you’ve been making elsewhere about your model. I think you’d need to show that it’s possible and worth it for those at one step of the supply chain to choose K or suppliers like in a way I ruled out for grocery stores and without making order sizes too sensitive to predictions. Or something where my model wasn’t general enough, e.g. I assumed a fixed order frequency.
It could be more than 1/K, because we’ve ruled out being disproportionately away from the threshold by assumption, but still allowed the possibility of disproportionately hitting it.
For realistic distributions of p across orders over time.
I would in fact expect lower numbers within 0, …, K−1 to be slightly more likely, all else equal. Basically Benford’s law and generalizations to different digit positions. Since these are predictions and people like round numbers, if K is even or a multiple of 5, I wouldn’t be surprised if even numbers and multiples of 5 were more likely, respectively.
Except maybe if the minimum K across suppliers is only a few times less than p, closer to p or even greater, and they can’t carry stock forward past the next time they would otherwise receive a new shipment.