### Krugman is Wrong

**Matthew Martin**6/23/2013 01:42:00 PM

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I almost missed this when I read his column, but there is actually an analytical error:

"To a large extent, the price you pay for an iWhatever is disconnected from the cost of producing the gadget"Matt Yglesias offered a somewhat better critique of Krugman's claim by pointing out that Apple is not a monopoly, even if it is monopolistically competitive.

Ok, for the record, Krugman does use the weasel words "to a large extent," nor is it completely clear that he meant to suggest that all of the "rents" he mentioned were specifically monopoly rents. Mostly, I just want to use this as a teachable moment, to demonstrate that costs matter, a lot, even if you are a monopolist. Warning, I will use math to make this point (not real math, mind you, but the kind of math even Matt Yglesias could do).

Typically, we use $Y$ to denote the number of goods produced, but this is Apple, so we will use the symbol $iY$ instead. Furthermore, we will denote the total cost to Apple to produce $iY$ number of gadgets by $c\left(iY\right)$. Suppose they sell the gadget at a price $P$--we're supposing, for the moment, that Apple is a monopoly, so they can choose whatever price they want. So we can calculate Apples total profits from selling $iY$ number of gadgets at price $P$: $$profit=PiY-c\left(iY\right)$$

This is something you can find in any introductory microeconomics text, so I will skip ahead to the result. However, my belief is that all academic knowledge should be totally open, free, and accessible to the non-expert public, so non-economists unfamiliar with the result can read a longer explanation after the jump, including the relevant assumptions about $P$ and $c\left(iY\right)$. For now, we solve $$\max_{iY} PiY-c\left(iY\right)$$ to maximize Apple's profits, which gives us the result that $$\frac{dP}{diY}iY+P-\frac{dc\left(iY\right)}{diY}=0.$$ Rearranging this result gives us the cannonical textbook model of monopoly profit-maximizing behavior: $$\underbrace{\frac{dP}{diY}iY+P}_{marginal~revenue}=\underbrace{\frac{dc\left(iY\right)}{diY}}_{marginal~cost}$$

What we see, then, is that Apple's price strategy, quantity sold, and yes, profits, do actually depend on costs, which appear in the right-hand-side of the profit-maximization condition above. So Krugman is completely out of line to say that they don't.

An explanation of the calculus methods below:

Appendix (read only if you have no idea how I reached the result above):

Intuitively, the number of gadgets Apple can sell depends on the price they charge--more people will buy them at lower prices than at higher prices. Hence, $P$ is actually a function of $iY$. In competitive markets, firms are forced to accept whatever the market price is for a good, but in this case Apple is a monopoly, and therefore free to choose any point along the demand function $P=P\left(iY\right)$. For illustration, we suppose that both the demand function $P$ and the cost function $c\left(iY\right)$ are continuously differentiable.This need not be the case in the real world--Apple cannot sell half an iPod--but given the scale of Apple's operations, this assumption offers a reasonable approximation of the real world.

Now, we will need to use a little calculus here--nothing more than you should have learned in high school. But all you really need to know is that the notation $\frac{dP}{diY}$ (called the derivative of $P$ with respect to $iY$ refers to the rate that the price consumers are willing to pay, $P$, changes in response to an increase in the quantity sold. Similarly $\frac{dc\left(iY\right)}{diY}$ denotes the rate at which Apple's costs, $c\left(iY\right)$ changes in response to an increase in the quantity $iY$ sold. As mentioned earlier, fewer people will buy the product at higher prices, so $\frac{dP}{diY}<0$ for all $iY>0$. We will further assume that Apple's production costs increase at an increasing rate, a concept known as diminishing returns. This implies that $\frac{dc\left(iY\right)}{diY}\geq 0$, but it also means something a bit more than that--in particular, that the function $c\left(iY\right)$ is concave. By Fermat's theorem, a necessary condition for a continuously differentiable function to achieve a maximum at a particular point is that the first derivative must equal zero. The additional assumption that the cost function is strictly concave ensures that any point where the derivative is zero is the unique maximum.

We are now ready to maximize Apple's profits: $$\max_{iY} PiY-c\left(iY\right).$$ A concave increasing function achieves its maximum where the first derivative equals zero: $$\frac{dP}{diY}iY+P-\frac{dc\left(iY\right)}{diY}=0$$ Rearranging that gives us: $$\underbrace{\frac{dP}{diY}iY+P}_{marginal~revenue}=\underbrace{\frac{dc\left(iY\right)}{diY}}_{marginal~cost}$$

If you aren't sure how I arrived at that equation, all I did was take the first derivative of $profit$ with respect to quantity, $iY$. Consult a calculus textbook for the rules on taking derivatives and finding extrema.