Application of Integrals and Differential Equations in Economics

Integrals and differential equations allow economists to study how quantities accumulate over time or across different levels, as well as to model accumulation, total change, and dynamic processes over time. Integrals are crucial for moving from marginal concepts to total quantities, such as calculating total cost from marginal cost or consumer and producer surplus in market analysis. We do this by calculating the corresponding definite integral, which can be interpreted as the area under a curve between two points. This seemingly simple geometric idea unlocks profound insights into various economic phenomena, since the area under the curves of demand functions, supply functions, cost functions, revenue functions, and so on often represents a total quantity, a cumulative value, or a measure of welfare. For instance, the area between a demand curve and a price level can represent the total benefit consumers receive beyond what they pay, the area under a marginal cost curve represents the total variable cost of production, etc. Differential equations, on the other hand, are the language of change and evolution. They enable economists to model how economic variables evolve over time, describing dynamic systems such as economic growth, market adjustments, and the accumulation of capital. In this chapter, we list some of these applications, including the ones we have already seen throughout this part, thus illustrating how integrals and differential equations provide powerful insights into economic behavior and system dynamics. We start by showing how to compute total cost (revenue) from marginal cost (revenue) function and how to compute capital stock at time T given the rate of net investment and initial capital stock K ( 0 ) $$K(0)$$ . Then we present the application of integrals to the calculation of producer surplus, consumer surplus, and total economic surplus. We continue by discussing the Lorenz curve and the Gini coefficient, which are instrumental in quantifying income or wealth inequality. Another application of integrals is the calculation of the present value of continuous future income given a continuous interest rate. As for the application of differential equations in economics, we start with the Solow-Swan growth model, which is a cornerstone of neoclassical growth theory. It explains long-run economic growth by looking at capital accumulation, labor or population growth, and increases in productivity (technological progress). We move on to the price adjustment model, which uses differential equations to model how markets adjust to equilibrium over time. Another application of differential equations deals with investment and capital stock dynamics and shows how the capital stock evolves over time. Given an initial capital stock and a path for investment, one can solve for the time path of capital stock. This equation is crucial for understanding capital accumulation and its role in production. We conclude by an application of differential equations to financial mathematics in which we derive the formula for continuous compounding.