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Economics Terms A-Z

Derivatives of Function

Derivatives of functions are computed using differential calculus. They are widely applied in economic modelling to measure the effects and rates of change in economic variables, as well as to determine maximum and minimum values of functions.

Consider the following production cost function for a good\[C\left(q\right)=k+aq^2\] where \(k\) is a fixed cost, \(q\) the number of units produced and \(a\) a variable-cost parameter. \(C\left(q\right)\) represents the total cost of production for \(q\) units of the good. When making the decision about how much of the good to produce a profit-maximising firm will be interested in the cost of producing the last unit of the good (marginal cost) and aim to set this equal to the extra revenue from selling that unit (marginal revenue). Suppose that (total) revenue from selling this good is simply the unit price \(p\) times the quantity sold\[R\left(q\right)=pq.\] Then profit \(\pi\) will be revenue minus cost\[\pi\left(q\right)=R\left(q\right)-C\left(q\right).\]

Maximum profit is found by taking the first derivative of the profit function and setting it equal to zero\[\begin{aligned} \frac{\partial\pi\left(q\right)}{\partial q}=\frac{\partial R\left(q\right)}{\partial q}-\frac{\partial C\left(q\right)}{\partial q}&=0\\ \Leftrightarrow\ \ \ \frac{\partial R\left(q\right)}{\partial q}&=\frac{\partial C\left(q\right)}{\partial q}\\ \Leftrightarrow\ \ \ p&=2aq\\ \Rightarrow\ \ \ q^{*}&=\frac{p}{2a}.\end{aligned}\] Since the first derivative of the profit function is zero in this point, profits are neither growing nor falling. It can be verified that the point is a maximum (and not a minimum) by taking the second derivative of the profit function\[\begin{aligned} \frac{\partial^{2}\pi\left(q\right)}{\partial q^{2}}&=\frac{\partial^{2}\left(pq-\left(k+aq^{2}\right)\right)}{\partial q^{2}}\\ &=-2a,\end{aligned}\] which is negative for positive \(a\), i.e. the gradient of the profit function is decreasing in \(q\). This means that profits are still growing at \(q<q^{*}\) but falling for \(q>q^{*}\) and therefore profits are at a maximum at \(q=q^{*}\).

This example is represented in the following graph where it is shown how profits \(\pi\) are maximised at point \(q=q^{*}\) where the first derivative of the revenue function (marginal revenue) \(\frac{\partial R}{\partial q}\) equals the first derivative of the cost function (marginal cost) \(\frac{\partial C}{\partial q}\).

Derivatives of Function

The above example involves simple, direct, partial relationships between output quantity \(q\) and revenue/ cost/ profit. In most real-world situations revenue and cost will also depend on other variables, for example sales effort \(e\) and marketing expenditure \(m\) for revenue, and the price of raw materials \(r\) and wages \(w\) for cost. Insofar as these other variables interact with the production quantity, it may be necessary to calculate the total derivatives of the revenue and cost functions with respect to \(q\) in order to determine the optimal level of production,\[\begin{aligned} \frac{dR\left(q,e\left(q\right),m\left(q\right)\right)}{dq}&=\frac{\partial R}{\partial q}+\frac{\partial R}{\partial e}\frac{de}{dq}+\frac{\partial R}{\partial m}\frac{dm}{dq};\\ \frac{dC\left(q,r\left(q\right),w\left(q\right)\right)}{dq}&=\frac{\partial C}{\partial q}+\frac{\partial C}{\partial r}\frac{dr}{dq}+\frac{\partial C}{\partial w}\frac{dw}{dq}.\end{aligned}\] Calculating these total derivatives takes into account any indirect effects of changing the production quantity on revenue and cost that arise due to potential differences in sales effort, marketing expenditure, the price of raw materials and wages brought about by changing the production quantity.

Further reading

Hal Varian offers a simple introduction for economists on how to calculate derivatives of functions, both total and partial, in the mathematical appendix to his popular text book Intermediate Microeconomics: A Modern Approach (see sections A10 to A13 of the ninth edition).

Good to know

More often than not, when economists compute derivatives of functions they are performing ceteris-paribus analysis. Taking the partial derivative of a function \(y=f\left(x_{1},\cdot\right)\) with respect to \(x_{1}\) is essentially to estimate the effect of \(x_{1}\) on \(y\) while assuming that all else remains equal.