As we are wrapping up our unit on microeconomics in our IB Economics class, I wanted to take a look at how calculus can be used to supplement the course. After all, there were many applications of calculus in physics, and I figured there would be quite a few for economics as well.
Perhaps you may only have seen linear demand and supply functions, but curvilinear curves are common as well. For instance, the shape of a unit elastic demand curve is a rectangular hyperbola. To see why, let's differentiate the rectangular hyperbola.
The price elasticity of demand (PED) is defined as the change in quantity divided by the change in price of the good, or $\frac{\frac{dQ}{Q}}{\frac{dP}{P}}$. Rearranging, $PED=\frac{dQ}{dP}\frac{P}{Q}$ Now just one more thing - according to the law of supply and demand, price is inversely related to the quantity demanded. This means that the value for PED will always be negative since $\frac{dQ}{dP}$ will be negative and $\frac{P}{Q}$. This is why PED is taken as the absolute value. Alternatively, we can simply add a negative sign in the formula to yield $$PED=-\frac{dQ}{dP}\frac{P}{Q}$$
The equation for a rectangular hyperbola can be given by $PQ=c$ where $c$ is a constant (can you see why?). Implicitly differentiating with respect to $P$, and applying the product rule, yields $$Q+P\frac{dQ}{dP}=0$$ $$P\frac{dQ}{dP}=-Q$$ $$-\frac{dQ}{dP}\frac{P}{Q}=PED=1$$.
Consumer surplus is a measure of consumer benefit which occurs when consumers pay less than the amount they were willing to pay.
For instance, suppose you went to Apple's website, looking to buy a cheap iPhone. From all the rumours, you know they're expensive, and you are only willing to spend $500. To your surprise, you find that an iPhone SE is on sale for only $400, and you gladly purchase instead. This means that you have a consumer surplus of $100, the extra amount which you were willing but did not pay.
On the other hand, producer surplus occurs when producers are able to sell a good for a higher price than the minimum price they are willing to sell them for.
Continuing with the same example as above, but from the producer's perspective, suppose you are the CEO of Apple. You currently sell the iPhone SE for $400, but you realise that because there are very few alternatives to the iPhone in this price bracket (not true), the good is inelastic and consumers are willing to pay $500 instead. Therefore, you decide to convert the consumer surplus to producer surplus by raising the price to $500. This conversion from consumer to producer surplus is the key idea of price discrimination.
It is worthy to note that producer surplus, while related, is not the same as profit. Producer surplus = $TR - TVC$, but profit = $TR - TVC - TFC$ where $TR$ = total revenue, $TVC$ = total variable costs and $TFC$ = total fixed costs. In other words, profit = producer surplus $-TFC$.
Variable costs are costs that depend on the quantity of output - things like advertisement, cost of raw materials, etc. In contrast, fixed costs must always be paid - things like rent, salaries, property tax, insurance, etc.
The supply curve represents the marginal cost of producing each additional unit, which does not take into account the fixed costs. Therefore, the blank triangular area under the supply curve represents the total variable cost. The area of the entire dotted rectangle represents total revenue, since $TR=PQ$. As stated earlier, producer surplus = $TR-TVC$, so it is equivalent to the triangular area above the supply curve and below the equilibrium.
Consumer surplus is simply the difference between the total amount that consumers are willing and able to pay (area under the demand curve) minus the total amount that consumers actually pay (area of rectangle underneath the market equilibrium).
Now that we differentiated, time to do some integration!
Firstly, let $p^*, q^*$ denote the equilibrium, and $s(q)$ and $d(q)$ denote the supply and demand functions, respectively.
In terms of calculus, the consumer surplus is given by $\int_0^{q^*}d(q)dq-\int_0^{q^*}p^*dq$. The second definite integral is the area of a rectangle given by $p^*q^*$, so the formula for consumer surplus can be simplified down to $$\bbox[yellow]{\int_0^{q^*}d(q)dq-p^*q^*}$$
Similarly, producer surplus is given by $\int_0^{q^*}p^*dq-\int_0^{q^*}s(q)dq$, and can be simplified to $$\bbox[yellow]{p^*q^*-\int_0^{q^*}s(q)dq}$$
Now, given any demand and supply function, you can use calculus to calculate the producer and consumer surplus! First solve the simultaneous equation to find the equilibrium price and quantity, $p^*$ and $q^*$. Then, simply evaluate the definite integral. There are some exercises on here and here which you may like to try.
As always, I find it extremely fascinating how mathematics is prevalent in many of the subjects we study in school, from physics to chemistry to economics. Today's finding is just another quintessential application of calculus - using derivatives to find slopes and integration to find areas under curves. While the mathematics itself is not wholly difficult in this particular case, the diverse range of its application is what I love about calculus. It is also interesting to see how our seemingly disparate topics of supply and demand and theory of the firm relate to each other.
As an extension, can you figure out how the price elasticity of demand affects the producer and consumer surplus? Tell us your findings in the comments below!
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