# Maximum price gaps of commodity between two cities

Suppose you have two cities A and B

• They are in the same country, same culture, same economical situation, etc.
• Suppose that in this country there's a single road between the two cities, where the time to pass that road with a caravan full of goods is about x days.
• The general economy is roughly equivalent to that of the late middle ages, and that salaries are more or less the same for traders,
• The transportation of goods has a fixed price p, perhaps per kg
• Assume there are no dangers at all - cities and roads are perfectly safe.

Now, given that the price of a commodity (say, rice) is p(A) in city A, what is the maximum reasonable price for the same commodity in city B? Assume also that there is no shortage in supply of the commodity.

I only ask about the maximum, as many parameters can lower the price: extra supplies, local manufacturing in city B, etc. But the maximum can - perhaps - be bounded. For example, if the cities are 1km from each other, and transport prices are very low, than it makes no sense that rice will be 10 times more expensive in B, since that gap would have been filled by wise traders.

Can you think of such a bound, in terms of p?

• Welcome Bach, made a minor edit for readability. – James Dec 23 '15 at 15:23
• Depends on what the caravan does in the other direction - it may be that all the people's salaries have already been covered. – Ian Ringrose Dec 23 '15 at 15:28
• Did you mean: Can you think of such a bound in terms of a multiple of p(A)? – realityChemist Dec 23 '15 at 15:41
• Well, yes. As a multiple of p(A)... I was thinking about bounding the ratio (p(B) / p(A)), where the bound should (probably) be a function of p. – Bach Dec 23 '15 at 15:44
• This seems almost more appropriate for the math SE than worldbuilding. – Wingman4l7 Dec 23 '15 at 23:10

I'm going to wildly swing in the general direction of this question and hope an answer falls out:

We're definitely going to see the maximum price where the product P (not p, which is the price of product P) is only manufactured in A and has to be imported to B. I'm going take your assumption about supply to mean that the manufacturing in A can cover the demand in both A and B.

To maximize the price difference between A and B, we're going to want to look for a situation in which D(A) << D(B) (these representing the demand in those cities). That is, B demands a significant amount more of P than A does. (If P was in very high demand in A, it would command a high price there. This would drive the price up in B relative to A if this demand caused a shortage in B, but we're assuming that supply can cover both cities.) We also want the supply to be very inelastic, meaning that a small increase in D will dramatically increase P. This is generally not the case for a product like rice, as you mentioned in the question. Some difficult-to-produce technology or a fine craft is more likely for this (maybe A produces particularly fine swords, but there are only so many master swordsmiths and very little room to speed up manufacture).

So, if we define E(S,P) to be the elasticity of supply of the good P, we can now say:

Now, obviously the length of the journey is also going to increase the cost almost linearly with x as long as the mode of transportation doesn't change (eg it doesn't become more economical to transport via two shorter land routes and a ship than one long land route). To figure out exactly how much, you're going to need to think about grain costs (to feed horses), wages for caravan workers and guards, etc (pluckedkiwi detailed the sort of things you should take into account here), as well as whether B has any product to ship back to A or if the caravans would need to eat the expense for the return trip. In any case, we can update our equation:

To be honest, there's definitely a lot more complexity going on than this, especially in the short term. The upper bound for that price ratio is trivially infinite (D(A) is zero), so I can't give you that, but hopefully that proportionality can serve as a rough approximation for the price difference under normal conditions.

• Could you elaborate on E(S,P)? First, what is S? Second, what is the magnitude of E(S,P)? If the supply is very elastic (say, as is the case with simple knives),and if it is very non-elastic (say, as is the case with specially crafted swords), what values would you assign to E(S,P)? – Bach Dec 29 '15 at 9:23
• I'm using S to designate supply, as opposed to demand. Sorry, I had originally defined that term elsewhere in the answer but edited it out and forgot to re-add the definition. Elasticities are defined around the value 1. So here are some supply curves: ibguides.com/images/unit-elastic-supply.png Now, S is unit elastic (has an elasticity of 1), S1 is elastic (elasticity greater than 1), and S2 is inelastic (elasticity less than 1). The bounds are [0, inf) where 0 means there is a fixed supply quantity and inf means supply is "perfectly elastic." – realityChemist Dec 29 '15 at 19:42
• So, an ancient artifact for which the technique of manufacture has been lost is going to have an elasticity of 0. Something like air would have an elasticity of supply approaching inf (but not quite there; if you need a lot of air, supply may not be able to keep up). Almost everything else is going to be somewhere in between. See en.wikipedia.org/wiki/… for some examples (and note the distinction between short run elasticity and long run elasticity; long run elasticity is more applicable to your question I think). – realityChemist Dec 29 '15 at 19:49
• Great, thanks! Your answer + explanations are very helpful. – Bach Dec 30 '15 at 9:15

Generally under a stable situation for goods which flow from A to B we would expect

p(b) = p(A) + trading overheads + profit margin for the traders.

Trading overheads include the cost of actually transporting the goods, the "cost of capital" for purchasing and reselling the goods. The risk of loss/spoilage of goods during transport.

profit margin for the traders will also vary. If there is a shortage of traders then the traders may make a lot of profit. If there is a glut of traders then margins may be very thin. A large profit margin for the traders will encourage an increase in the number of traders drving the margins back down but that increase may take time to realise.

And in the short term there will likely be instability in the price because of the time delay between deciding to bring goods in and actually doing so.

Determining the maximum bound of the price of a commodity is trivial but unsatisfying - it is simply the total wealth of the purchaser. In this scenario, that information is unstated.

To manufacture a scenario where this is the case, let's assume that the entirety of City B is poisoned, and there is only one dose of antidote. In this scenario, the only rational outcome would be the wealthiest resident of City B would use all the wealth they could acquire to purchase the antidote.

Otherwise, assuming a rational market and perfect competition, the maximum price would fall on the intersection of the supply and demand curves or equilibrium. The distance between the cities, the safety of the roads, or the cost of transport would shift the absolute cost (as would changes in demand) but can't be used to determine an arbitrary maximum bound of cost.

Basically, there's not enough information in this scenario to determine an upper bound to the cost beyond 'all the money City B can acquire'.

• That neglects the fact that the transport between the cities also affects the supply and demand (someone transporting goods from A to B increases the demand at A and the supply at B, thus in an ideal market raising the prices at A and lowering them at B. I believe if enough transport capacity is available (another point not addressed by the question), in a perfectly rational and competitive market this should limit the price difference to at most the transport cost. – celtschk Dec 25 '15 at 20:37

Just as there are many parameters which could lower the price, there are many which can raise the price.

The long-run average cost, presuming no temporary shocks to supply or demand, is going to be the price in the other city plus transportation cost. That transport cost would need to include the cost of financing if relevant (getting the money to fund the merchant's initial purchase), any tariffs or tolls, loss of product during transport (damage from weather or pests), transaction costs (license fees to be permitted to sell?), etc.

Just add up the cost needed to transport the good, plus the cost to purchase the good, and adjust for opportunity costs (are there better uses of those funds and merchant's time? probably not a concern unless transport is restricted as someone would find it worth the trip for just enough to feed themselves) to find the typical price. The only way the price (pB = pA + costs) would be expressed as a multiple of the purchase price in the other city would be if the overwhelming cost of the enterprise is in financing the trip, and the interest charged being a percentage of the amount borrowed.

To find a maximum, you need to consider the elasticity of demand and the ability to defer consumption. Something like foodstuffs may have a very low elasticity of demand (everyone needs to eat), and relatively low ability to defer consumption (you can put off eating for a few days, but it becomes increasingly unpleasant). Supply imbalances would only last roughly the round-trip time (for price signals to be sent to the other city, and goods to return, presuming available transport capacity) - people would be aware of this and adjust price expectations accordingly.

As many have noted, a reasonable starting point for analysis is to consider the cost of transporting the product from A to B.

But that's only one of many factors. There are many, many others beyond what you mention.

What alternative products are available in B that people might use as a substitute? Perhaps people in B see potatoes as an alternative to rice, and they are just as happy to eat potatoes as to eat rice. In that case, if the price of rice goes about the price of potatoes, they'll just eat potatoes instead. No matter what the price of rice in A or the shipping costs to get it to B, they won't pay more for rice than they pay for potatoes. If they like potatoes but they like rice more, or if they like to have variety, they may be willing to pay somewhat more for rice, but the price of potatoes will put downward pressure on the price of rice.

How much do people in B want the product, compared to people in A? If the people in A love rice but the people in B hate rice and will only eat it if the alternative is starving to death, the price of rice in B might well be LESS than the price in A despite the shipping costs. (Merchants may only carry rice to B when, say, their caravans would otherwise be under capacity, and they may as well carry a product on which they make a tiny profit rather than have wasted capacity and make nothing.)

Also, bear in mind that there's no such thing as "there is no shortage". Well, there are a few goods that are so abundant that people can get all they want for free. Like air. That may be the only one. Besides that, the law of supply and demand kicks in. If the supply goes up, the price goes down until demand matches supply, and vice versa. There can be temporary surpluses or shortages until the market can adjust, but eventually an equilibrium is reached. There is almost no limit to how much of a product people will consume if the price is low enough.

In this case, suppose people in B are willing to pay a higher price for rice than the price in A plus the shipping costs. Then merchants will have an incentive to ship more rice from A to B to make a higher profit. Either people in A make due with less rice, or the price in A and/or B has to go up until there is an equilibrium again.

Also note that you can't calculate an expected price by "adding in the profit for the merchant". How much profit do you assume? There is no universal law that says merchants always get 17.3% of the selling price or any such formula. Merchants set a price that maximizes their profit, that is, where the number of units they sell at a given price times the profit per unit is the largest for all possible prices. If the maximum is negative, they'll quit selling this product. (Once they are convinced this is a long-term situation and not a temporary fluke.) If they can make more money selling other products, they'll switch to other products. If they can make more money by selling their camels to the butcher shop, that's what they'll do. (Well, lots of caveats on that. Sometimes people will continue in a profession because they enjoy doing it, even though they could make more money doing something else. Or they're tradition-bound, this is what my father did and this is what my grandfather did, etc, and so they stick to it. Etc.)

I think perhaps your traders won't bother with rice - if it costs p to transport 1kg of goods, no matter what the goods are, surely they will transport things that maximise the selling price of the kilogram. Rice is basically bulky and cheap. 1kg of it costs very little. Jewellery is really expensive when you have 1kg of it - a gold ring, for example, weighs maybe 10 grams as an upper bound. Therefore, they can bring 100 gold rings from A to B for the same transportation cost, but each gold ring probably sells for the same as a 25kg bag of rice.

In reality, the cost of transportation is only loosely reflected in the price paid - if city B has a famine (maybe due to the local warehouses being destroyed in a fire - something local to B), the price of rice will rise well above the costs of transporting it, and the price of gold rings will probably fall (who wants gold rings when they need food). This bears no relationship to the cost of getting the goods there - it applies to both traders who were already there, but not yet at the warehouses, and to traders who immediately set off from city A.

Essentially, the maximum bound is whatever someone is willing to pay for an item, which isn't really subject to a formula.

• A note on your first paragraph - It's not just about maximizing the value of the cargo. In fact, that might be a very bad idea if there is any decent risk of loss, such as from thieves. Liquidity is also a concern, which you do sort-of cover in the second paragraph, but its not just a concern during famine. If you could take a full cargo of rice, sell it, buy whatever cargo for the return trip and return - that may well earn you more money than if you were to have your money tied up in jewelry the whole time, trying to sell it. – DoubleDouble Dec 23 '15 at 16:51
• @DoubleDouble -- furthermore, having unsellable cargo in a transport vehicle can turn into a financial liability due to demurrage charges -- this may not be a big deal with a horse-drawn cart, but is a major problem if it's a ship, intermodal container or railcar you've tied up! – Shalvenay Dec 24 '15 at 6:49
• Thanks for your answer. It seems correct ("the maximum bound is whatever someone is willing to pay"), still not very helpful, as the question is, perhaps, exactly about that quantity, and naively I believe that there are formulae governing these "willing to pay" quantities... – Bach Dec 29 '15 at 8:27