Random Sales, and the elasticity of supply for the option of doing nothing

This post is not about Boxing Day sales. It's about those seemingly random sales, where a good is heavily discounted for no obvious reason.

A couple of years ago I went to Canadian Tire planning to buy a $200 socket set. And was surprised to find the one I wanted on special that day at $80. I was pleased of course; but why would Canadian Tire do that?

I don't normally do micro. But this question kept bugging me. Here's my theory. It's a real option theory of price discrimination.

1. Sometimes I go to Canadian Tire and buy something I need.

2. Sometimes I go to Canadian Tire and buy something I don't need, but it's on special, and I might need it in the future.

3. Sometimes, but only very rarely, I go to Canadian Tire and buy something I need, and find it's on special.

My demand function when I'm doing 1 is less elastic than when I'm doing 2. By holding short random specials, Canadian Tire can ensure that 3 happens very rarely, so it can more or less segregate the two markets 1 and 2. From then on it's the standard price discrimination story, where Canadian Tire maximises profit by selling exactly the same good for a lower price in the more elastic market 2 than in the less elastic market 1.

But why is the demand curve more elastic in market 2 than in market 1?

The more alternatives you have to buying a good, the greater is the elasticity of demand. If they raise the price when I know I need a socket set (market 1), my alternative is to use my existing set of wrenches. If they raise the price when I don't know whether I need a socket set (market 2), I have two alternatives: I can use my existing set of wrenches; or, I can simply wait and see if I need a socket set. I always have (at least) one more alternative when I'm buying something that I don't know if I will need. I have the option to simply wait and see.

If Canadian Tire want me to part with my money when I don't yet know if I need a socket set, they are selling me a socket set, but they are also buying my option of doing nothing. Obviously I won't sell my option of doing nothing, except at a positive price. I won't buy the socket set until I need it unless they offer me a discount.

If Canadian Tire raises the regular price, it only loses those customers who have a lower value on a socket set. If Canadian Tire raises the special price, it loses those customers who have a lower value on a socket set, and it also loses those customers who have a higher value on their option of doing nothing. The elasticity of demand for socket sets on special is the regular elasticity of demand, plus the elasticity of supply of customers' option of doing nothing.

Here's a simple story:

There are 100 consumers who might need a socket set. Consumer i will need a socket set from Canadian Tire with probability Pi. And if it turns out he does need it, will be willing to pay Wi to buy it from Canadian Tire. All consumers have different Pi's and Wi's.

Each consumer can buy the socket set early at the special price S, before knowing whether he needs it. Or can wait, and if he needs it, buy it at the regular price R, provided the price is less than his willingness to pay W.

The marginal shopper who waits until he needs it, and who is just indifferent to buying the socket set at the regular price R, has Wi=R. The benefit of having a socket set equals the regular price. The elasticity of demand for late shoppers is the same as the elasticity of W. If Canadian Tire raises the regular price, the only customers it loses are the customers who don't need a socket set quite that much.

The marginal shopper who buys before he knows whether he needs it, and who is just indifferent to buying the socket set at the special price S, has PiWi=S. The expected benefit of a socket set equals the special price. The elasticity of demand for early shoppers is the same as the elasticity of PW. If Canadian Tire raises the special price, it loses the customers who don't need a socket set quite that much if they do need one, and it also loses the customers whose probability of needing a socket set is not that high. Unless Pi and Wi are strongly negatively correlated, the elasticity of PW will always be greater than the elasticity of W.

Now, if Canadian Tire raises the special price S, some fraction of the customers it loses will find they do need a socket set, and will buy one at the regular price when they need it. So that demand isn't lost to Canadian Tire, merely postponed. But as long as P is less than one, it won't recover all the lost customers that way. Some will find they don't need a socket set after all. Or maybe they find they do need one, but buy it from another store.

Comments

[Adam P]

yeah, I think this basically makes sense Nick.

[Nick Rowe]

Adam: thanks. I wasn't sure. And couldn't quite do the math to prove it. Though anybody with halfway decent math could take my model, assume a distribution F(P,W), and solve for the elasticities in the two markets.

[Frances Woolley]

Nick, what's interesting, though, is that even though the sale might appear random to you, it probably isn't. School supplies predictably come on sale in August/September when demand is very high - which seems at first glance to be counter-intuitive. I stocked up on baking supplies about 3 or 4 weeks ago when they were all on sale - just at the start of the pre-Christmas baking binge. I'd be willing to bet that a fairly small fraction of sales events are, in fact, truly random sales.

This doesn't mean that your model is wrong - in fact, putting baking supplies on sale a few weeks before Christmas might be the optimal way of getting a type 2 shopper like me to buy stuff. I'd be reluctant to buy, say, bulk chopped pecans in February, because they don't keep perfectly, and there's limited room in my freezer. But if they come on sale 8 weeks before Christmas, and I know I'll be making a big double-batch of nanaimo bars for holiday parties some time in the next couple of months.

[Stephen Gordon]

Mmmm.... Nanaimo bars....

[Frances Woolley]

Stephen, the best part is the recipe, which requires one to "chill" on three different occasions - done most effectively on the sofa with an injection of "liquidity"

[Nick Rowe]

Frances: Why do turkeys always seem to be cheaper just before Christmas and Thansgiving? Yep, it's counter-intuitive, if you start from the standard competitive equilibrium demand and supply curve. I think (not sure) there's an existing literature on this. One theory is economies of scale in production and/or distribution, so the MC curve slopes down. The second theory is that each store's elasticity of demand is higher when a lot of people are buying turkeys, and are all looking for the store with the lowest price, so the markup of price over MC is smaller (since MR is close to P when elasticity is high). Both these theories assume monopolistic competition (as does mine).

But none of the standard theories seemed to work for a lot of durable goods (e.g. tools) that seem to go on special at Canadian Tire. Go into Canadian Tire on any day, and you can always find some tools that are deeply discounted, while other tools are at the regular price. Go back a week later, and it's different tools. Socket sets and impact wrenches aren't deeply seasonal items (OK, maybe impact wrenches when it's time to put on Winter Tires, but that doesn't explain last week's special). And you often see one socket set on special, and a different socket set at the regular price.

What I can't decide is how much this works for grocery store specials as well. There are things you *know* you will need, but don't need quite yet. Like your examples. So the elasticity of demand should be the same early as late. But maybe early shoppers will be visiting other stores before they need them, while last minute shoppers won't have time, so you get the different demand elasticities that way. But that's not really an option value, because there's no uncertainty.

[Adam P]

Wait, I don't think anything in Nick's model implied that sales are *only* put on randomly.

Surely there are many that are set for other reasons. Nonetheless I thought Nick had a point in that a certain amount of randomness can facilitate price discrimination.

The idea is to optimize the arrival rate/duration of the sales so as to capture case 2 without the frequency of case 3 getting to high. Like always there's a trade-off, more case 2 sales at the cost of more occurances of 3, so you try to find the optimum.

[Fmb]

How much of increased money demand is due to a similar real option argument? Roughly volatility is up, so options worth more so don't squander them.

[Nick Rowe]

Fmb: My guess is "quite a lot". But I can't really put any sort of number on it. Here's an old post I did on the topic: http://worthwhile.typepad.com/worthwhile_canadian_initi/2010/07/cash-as-the-real-real-option-to-do-anything.html

[Fmb]

Thanks, just read that post.

Now I'm wondering if there's a possible "Increasing vol bound" where increasing the money supply increases Vol which influences money demand enough to sop up the new supply. If so, loosening might have no net impact on equilibrium.

Considering the current spread of opinions about the effects of QE, this seems at least superficially plausible.

[reason]

This may not be relevant to spanners, but often sales are related to stock management upstream from the retailer at the manufacturer. If the manufacturer has old stock it wants to clear out, it can offer that to a retailer on a heavily discounted basis. Watch for a new model in the near future.

[Hugh]

Random positive stimulation causes bargain hunters to search Can Tire for deals. This gets them in store more often. More often than stores which have regular prices that are lower on average but not lower on sale. And they (we) buy things we'll never use.

On the other hand, many people are not bargain hunters and like to shop at stores that are reliably reasonable (Walmart? at least it matches the image they project). Bargain hunting is a lot of work and may well not pay off.

It surely matters whether a put-off sale is likely to go to the same store or not. Unless the store is short-sighted and cares way more about a sale this month rather than next month.

Another effect: an ad for a sale price is more interesting than an ad for a regular price.

On balance, the effects seem more psychological than economic.