# Relevance of intersection of D and S curves - vs optimization in Calculus

Hey guys. My background is in Engineering - so Microeconomics is pretty new to me. Going through the CFA Level 1 material, and as you know the D curve slopes downwards and the S curve slopes upwards. Obviously, this is not true in all real world examples, but I suppose for the oil industry, this kind of curve would be true - probably would NOT be true for electronic devices where the supply curve will probably curve downwards as the company makes more. Obviously more many electronics, “large scale” == “cheaper”. Anyhow, the point of this question is, why is there such a huge focus on the “intersection point” of these two curves as the “price”? The “maximum profit” for the company is NOT necessarily reached at this point of intersection but at an earlier point where “more money” can be obtained for the earlier quantities. i.e. while it’s true that when you produce more you can get profits for the later quantities (until intersection) since there is a single price, you “lose” some of the potential profit for the initial quantities when the D curve is really high and you could extract more money from the early adopters if you sell the price high enough. To calculate the max profits, it should really be a maximization exercise using calculus to maximize the difference between the Price and Supply cost constrained by the Demand curve - instead of it assuming it’s at the intersection point. Am I missing the relevance of the intersection point? i.e. the purpose is not for profit maximization, but to also take the “utility” for the population as well. But no company is going to take such a “liberal” view when profits are concerned in an imperfect market and they can set the price. Thanks.

I will just zoom into your question:

the intersection point is the most efficient allocation of supply and demand, it is the point of optimal allocation of goods/resources

anything less than the quantity ( more demand, less supply) == more expensive goods

anything more than the quantity (more supply, less demand) == cheaper goods

quotas and tax will also shift the intersection, thus ending up less optimal

Howdy.

What species of engineering? Mechanical? Civil? Electrical? Aerospace? Chemical?

Yup: probably better to move discussions about economies of scale to another thread.

I believe that you may be misinterpreting the supply curve. If we say, for example, that producers will be willing to supply 100,000 widgets if they can sell them at a price of \$10 / widget, and 200,000 widgets if they can sell them at a price of \$16 / widget, those numbers – presumably – are the quantities that suppliers have determined will maximize their profits at those prices. The “maximum profit” is for the suppliers isn’t a single number; there is a maximum profit if the price is \$10 / widget, and a (probably different) maximum profit if the price is \$11 / widget, and so on. It is this continuum of quantity-that-maximizes-profit-for-a-given-price that creates the supply curve.

It should be; that’s exactly what the producers have done to create the supply curve. The good news for CFA candidates is that they don’t have to use (or even know how to use) calculus in the curriculum. (If they did, the candidate pool would be smaller. Probably considerably smaller.) And let’s not even get into Lagrange multipliers to add the constraint condition!

By this point you should realize that there’s no assumption that “the” maximum profit occurs at the intersection point; what does occur at the intersection point is that the _producers have maximized their profit given the equilibrium price _; they cannot do any better.

If you were, let’s hope that you no longer are.

Nope.

Absolutely true. Fortunately, that’s not what’s happening here.

My pleasure.

Thanks you guys - and a special thanks to S2000magician for such a detailed response!

Computer Engineering actually - both undergrad and Master’s (with focus on signal processing).

Now as you said, the supply curve is the optimal supply output at that “given price” Pi from the perspetive of the supplier. A higher price P1 (when compared to P0) would typically mean a higher quantity Q1 (when compared to Q0) - given a standard positive slope S curve. However, you also confirmed that the “max profit” depends on the price selected - and of course the corresponding Q value that corresponds to this P (e.g. the max profit at \$11/widget could be different than max profit at \$10/widget). Now my follow up question is, from all these potential “maximum profits” (also graphed as a function of P), there is going to be a single absolute max profit as a function of P. As you confirmed, this is not necessarily going to be at the “equilibrium”/intersection point, and could be a different {P, Q} value.

Don’t we need to use calculus to determine this point? Essentially, to find the “absolute max profit”, from all the “max profits” possible as a function fo the price - given the S curve and D curve, do we not need to use traditional optimizaton techniques? If so, is this part of the CFA curriculum at a later stage? It sounds like from what you said, none of the levels deal with Calculus however…

And as a related question, while the equilibrium point, as the name implies, is a good steady state “stable” point, is this actually relevant in an imperfect market where the manufacturer has complete control over the price set and the correspoding quantity? A Ferrari comes to mind - I am sure it doesn’t cost them \$250K to manufacture one - and yet the price is really high and I am guessing set purposefully high by the company at a (P, Q) point that is different from the equilibrium point to maximize the “absolute” profits - e.g. the max profit at \$250K is higher than the max profit had they set it at \$80K.

Hopefully, getting there :).

Thanks again.

My pleasure.

Years ago I had to write a nonlinear Kalman filter that formed the basis for GPS navigation. More challenging still was the nonlinear Kalman filter for navigation using deep ocean transponders (DOTs): because water density varies with depth, and sound speed varies with density, the sound paths are not straight lines.

This would be true if the maximum-profit-as-a-function-of-price weren’t a strictly monotonic function; indeed, it is (assumed to be) strictly increasing.

Profit is maximized when marginal revenue equals marginal cost. When the price increases, marginal revenue increases; therefore, going from, say, a price of \$10 / widget to a price of \$11 / widget, the maximum profit for the former will occur when marginal cost is \$10, while for the latter it will occur when marginal cost is \$11. Because costs are assumed to increase with quantity produced (remember, we’re not assuming economies of scale here), an \$11 marginal cost will occur at a higher quantity than a \$10 marginal cost: higher price implies a greater quantity to maximize profits. Furthermore – and this is the key to the whole sordid story – the maximum profit at an \$11 price will be higher than the maximum profit at a \$10 price: maximum profit is a strictly increasing function of price.

Therefore (finally!), there is no global maximum profit: as the price increases, the maximum profit increases.

As the function is strictly increasing, you’re not going to calculate the derivative wrt price, set it equal to zero, and solve for the price: the derivative’s never zero; it’s always positive.

Ferraris come to my mind frequently as well: the license plate on my (red, obviously) S2000 is “SUDO 4RE”.

Your example is not one of a perfectly competitive market; Ferraris sell in a market probably best described as an oligopoly (cars in general sell in a market of monopolistic competition, but the extreme high-end you chose is more of an oligopoly). As you read further in economics, you’ll see that Ferrari doesn’t price their cars at the (pure competition) equilibrium point; they price their cars at the profit-maximizing point and sell fewer than they would in a purely competitive market.

You can certainly think of other examples where the supplier doesn’t have absolute control over price and quantity: surely you’ve seen half-off (or more) sales on merchandise that didn’t sell at the manufacturer’s desired price.

Good to hear.

My pleasure.

Ah the Kalman filter. :). I did some work with that as well. Nothing as cool as what you did though. If you don’t mind me asking, how many years were you in Engineering before you switched to finance? I’ve been in Engineering for a few years now - but want to make the switch :).

That is certainly true. But, wouldn’t that be constrained by the demand curve? In an extreme case using your numbers, lets say that the demand curve is a downward sloping line where it intersects the y-axis at 10.5 (i.e. even at very low quantities, no one in the market will pay \$11 for the widget). So in this case, the price line at \$11 (horizontal line) is completely above the demand curve and they won’t sell anything at that price. As you move the price line lower, at \$10/widget, it may be that it intersects with the demand curve at a quantity that is less than the quantity where MC == MR == \$10). This way the “maximum” profit that is possible at \$10 will not be realized. Whereas, if you lower the price a bit more lets say to \$9/widget, while the “potential” profit may be less than at \$10/widget (as you described above), the “actual” profit may be more since the demand curve “doesn’t get in the way” as much at \$9/widget. i.e. for the first N widgets, you will make less profit with \$9/widget than when compared to the \$10/widget case. However, with \$9/widget you will “sell more” since the demand curve allows more widgets to be sold at the lower price of \$9/widget. This whole paragraph assumes that the price line is completely horizontal and every widget (including the first widget) was sold at the stated price.

So in this sense, don’t we have to do some optimization to calculate the “trade-off” between the “extra profit” (\$10/widget vs \$9/widget) for the first N widgets versus the extra profits you will get with the extra widgets sold at \$9/widget?

Also, as I mentioned, in my example, no one will buy widgets at \$11/widget, since the demand curve is completely below the \$11/widget horizontal line price point. Is my approach wrong or am I complicating things unnecessarily?

Of course. Red is the only colour that makes sense! :).

Thanks S2000magician. I appreciate you taking the time to help me with this.

ε²–

I started working full-time in the late '70s, and joined PIMCO (my first job in “finance”) in 1995; so, perhaps 18 years.

I started writing software to run numerical control systems (milling machines, lathes, punch presses, coördinate measuring machines, and so on), then moved to the job where I got to write the Kalman filters (amongst other things: analyzing GPS receiver algorithm design, developing GPS simulators). After that I was a warhead designer for 10 years (single and multiple explosively formed penetrator (EFP) warheads), wrote software to estimate the cost for constructing petroleum processing plants, wrote software to monitor freeway traffic . . . all the while teaching university mathematics part-time. After six years at PIMCO (I was hired because I knew graphics programming – they needed someone to finish the graphic output of their mortgage-backed security analysis software (the original programmer had resigned) – but that grew into rewriting most of the analysis software to improve accuracy and efficiency (e.g., redesigning it to run on threaded machines for parallel processing), then developing prepayment models for mortgages), I returned to “engineering” as a project risk manager at (eventually) Northrop, then moved into consulting in project risk management, and teaching CFA review courses, project risk management, cost management, quality management, cash flow analysis, problem solving and decision making, financial mathematics, and financial modeling.

It’s constrained by the demand curve only in the sense that, although the suppliers would be happy to sell (i.e., maximize their profits by selling), say, 120,000 widgets at \$11 / widget (which is what the supply curve shows), customers won’t buy 120,000 widgets at \$11 / widget; they’ll buy only, say, 80,000.

The supply curve is the continuum of quantity/price pairs that suppliers would be willing to sell, without regard to what consumers would be willing to buy. Similarly, the demand curve is the continuum of quantity/price pairs that consumers would be willing to buy, irrespective of what suppliers would be willing (or able) to supply. That’s why the equilibrium point is important: it’s the only point where suppliers are willing and able to supply exactly the number of widgets that consumers are willing to buy: no more, no fewer.

I think that your error is in thinking that the supply curve is created with the demand curve in mind (as a constraint to the number of widgets that suppliers can sell at a given price). The supply curve is created in isolation: if demand were infinite (or, more reasonably, always sufficient to cover whatever suppliers are willing to produce and sell), then how many widgets will maximize profits at each possible price? Similarly, the demand curve is created in isolation, assuming that suppliers will be willing to supply whatever quantity is demanded at a given price. Only when those two curves are plotted against each other do we have the constraint: excess demand below the equilibrium quantity/price will spur suppliers to produce more, while excess supply above the equilibrium quantity/price will constrain suppliers (who aren’t maximizing profit because they’re not selling everything they produce) and force them to reduce production.

You’re quite welcome. I hope that it’s helping, if just a bit.