# What tools are available to understand situations where the supply curve is clearly a function of the demand curve?

Consider the "Matrix Regurgitated: Coming Only To Cinemas" market:

• Buyers. People who would potentially be willing to spend money to watch a 4th Matrix film in cinemas, if ever such a film were released.

• Sellers. Warner Brothers (or whoever owns the rights to these films).

Lets make the simplifying assumption that this film will cost exactly 500 million dollars to make.

Assume the demand curve looks something like this: Okay, now lets think about supply. Let $p$ denote very high price; say, $\$30$per ticket. Assume, in particular, that$p$is far toward the right-hand-side of our graphs, where the quantity demanded is very low. Then if tickets are sold at price$p$, quantity demanded will be very low, so revenue from the film will be very low, so it won't be worth spending the 500 million to make the movie, so quantity supplied will be exactly$0$units. Now consider a demand-side shift. Perhaps some well-known film critic publishes her paradigm-shattering thesis on why the Matrix Sequels were, in point of fact, the best sequels of all time. This causes the demand curve to shift up: Once again, let$p$denote very high price; say,$\$30$ per ticket. Then if tickets are sold at price $p$, quantity demanded will still be pretty high (because the whole curve has shifted up), so revenue will be pretty high, maybe revenue will equal one billion dollars, so therefore it probably makes sense to spend the initial 500 million to make the movie, so quantity supplied will be strictly more than $0$ units.

This is kind of weird, right? Its as if every time demand changes, we have to change the supply curve, too. Looking at it another way, supply isn't purely a function of price-per-unit; its also a function of the demand curve.

Question. What tools are available to help think about situations like this, where the supply curve is clearly a function of the demand curve?

The equilibrium quantity supplied is dependent on where the demand curve intersects the supply curve, but that doesn't make supply itself a function of demand.

To understand this better, it would be helpful to draw a supply and demand graph with both of the curves, rather than just the one curve you've got in your drawings. You're getting into some weird hand-waving that's making things overly complicated. Before making these abstract hypotheticals, you have to make sure you understand the fundamentals of what is happening.

• You haven't understood the example at all. Try drawing a reasonable supply curve for this particular example and you'll see what I mean. – goblin May 16 '16 at 1:53
• @goblin If you were dealing with a monopolist with perfectly inelastic supply curve, then when the demand curve shifts, equilibrium quantity doesn't change, but the profitability of it does since the equilibrium price does. That would completely explain your scenario. Also I'm not sure why you draw the graph with the axes switched like that. – Kitsune Cavalry May 16 '16 at 2:01
• Also if you're mixing the two markets together (the one for making the movie and the one for selling seats) for some unjustified reason, even then you could probably justify your profitability argument with a normal supply curve and seeing the shifts along the curve. – Kitsune Cavalry May 16 '16 at 2:03

I think confusion has arisen because "quantity" is used here to refer to two things: the quantity of films made (a binary 0/1 with LRMC=\$500m); and quantity of viewings (which you've assume has SRMC=0 and can be any non-negative integer).

Other than that, it's an unremarkable case of a monopoly supplier who will pick the price of viewings to maximise profit, and then only produce the film if that's positive. In this case, that's the same as maximising revenue, because for viewings, SRMC=0.

So your tools are simply: differentiate the total profit function of viewings (which in this case is identical to the total revenue function) to find the turning point, verify it's a maximum, round it either up or down to an integer (pick the direction that gives higher profit), and compare the revenue at that point with the cost of producing the film. Each time the demand-curve shifts, you have to do the differentiation again.

The question is not clear to me but relates to the canonical example of instrumental variables, which involved attempts to estimate specifically demand and supply curves. Here, the OP considers an exogenous demand-side shift so an ordinary least squares (OLS) regression of quantities on prices is a possible tool to "think about situations like this". However, if the demand and supply curves shift over time, the observed data on quantities and prices reflect a set of equilibrium points on both curves. Consequently, a tool such an ordinary least squares regression of quantities on prices fails to identify either the supply or demand relationship.

Wright (1928) confronted this issue in the seminal application of instrumental variables (IVs). He suggested that certain curve shifters, called IVs, can be used to address the problem:

Such additional factors may be factors which (A) affect demand conditions without affecting cost conditions or which (B) affect cost conditions without affecting demand conditions.

A variable he used for the demand curve shifter was the price of substitute goods, such as cottonseed, while a variable he used for the supply curve shifter was yield per acre, which can be thought of as primarily determined by the weather.

However, IVs may have potential pitfalls: an instrument that is correlated with the error term, or weakly correlated with the endogenous variable, or not every observation's behavior is affected by the instrument.

To overcome omitted variables bias, instruments are sometimes derived from natural experiments. Recent years have seen a resurgence in the use of instrumental variables in this way — that is, to exploit situations where the forces of nature or government policy have conspired to produce an environment somewhat akin to a randomized experiment.

This answer is based on Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments by Joshua D. Angrist and Alan B. Krueger.

• +1, It's definitely not immediately obvious to tell whether a price change is due to supply or demand shift. This probably answers a more fundamental (and clear) idea of what the user is asking. – Kitsune Cavalry May 16 '16 at 0:28
• Could it come from the fact that the question is unclear? – emeryville May 16 '16 at 2:54