This is problem 12.5 from Nicholson and Snyder (2017). The textbook solution used a linear approximation but it is actually a good example of a parametric integral. Since calculus textbooks almost exclusively use physics to demonstrate applications of integration, I thought I’d work through an econ example.
Problem 12.5
Suppose that demand, the supply of entrepreneurial talent, and long-run cost for a competitive industry are given by \[\begin{gather} D(P) = 1500 - 50P \\ Q_s = .25w \\ C(q) = 0.5q^2 - 10q + w, \end{gather}\] where \(P\) is output price, \(w\) is the entrepreneur’s wage, and \(q\) is firm output.
Assume that each firm requires exactly one entrepreneur (so \(n = .25w\)).
Part a)
Find the long-run equilibrium — setting \(P = MC = AC\), \(Q = n \cdot q\), and substituting \(w = 4n\), we find \[\begin{align*} S = \sqrt{8}n^{3 / 2} = n(P + 10) \end{align*}\]
Then letting \(D = S\), we find \(n^* = 50\).
Part b)
Find the new equilibrium if demand shifts to \(D(P) = 2428 - 50P\). Setting \(D = S\) again we find \(n^* = 72\).
Part c)
The interesting one. Show that the increase in rents accruing to the entrepreneurs is identical to the change in long-run producer surplus along the market supply curve.
The change in rents is easy. We know that, \[\begin{align*} Q_s = .25w &= n \\ \implies w &= 4n \end{align*}\]
So the increase in rents is the integral, \[\begin{align*} \int_{50}^{72} 4n\ dn. \end{align*}\]
To find the producer surplus along the market supply curve treat \(S = \sqrt{8}n^{3/ 2}\) as a parametric equation for the \(x\) coordinate of the supply curve as \(n\) changes. We know that \(S = \sqrt{8}n^{3/2} = n(P + 10)\) so \(P = \sqrt{8n} - 10\): a parametric equation for the \(y\) coordinate of the supply curve in terms of \(n\).
We want to find the area to the left of the supply curve, so reversing the order of the formula for the area under a parametric curve,1 \[\begin{align} \Delta PS &= \int_{50}^{72} S \cdot P'\ dn \\ &= \int_{50}^{72} \sqrt{8}n^{3 / 2} \cdot \sqrt{\frac{2}{n}}\ dn \\ &= \int_{50}^{72} 4n\ dn. \end{align}\]
Which shows the change in rents is identical to the change in producer surplus.