Marginal function in economics is defined as the change in total function due to a one unit change in the independent variable. If the total function is a continuous function and differentiable, by differentiating the total function with respect to the corresponding independent variable, the marginal function can be obtained.
Marginal cost is the change in total cost incurred from the production of an additional unit. Given that the total cost function \(TC = TC(q)\) is a continuous and differentiable function, the first-order differentiation of the total cost function is the corresponding marginal cost function:
Marginal revenue is the change in total revenue brought about by selling one extra unit of output. Given that the total revenue function \(TR = TR(q)\) is a continuous and differentiable function, the first-order differentiation of the total revenue function is the corresponding marginal revenue function:
If the total product is a function of labor (L), \(TP = TP(L)\), the marginal product of labor is \(\frac\)
If the total utility of a consumer is a function of the quantity of a good she consumes, \(TU = TU(x)\), which is a continuous and differentiable function, her marginal utility from the product is \(dTU / dx\):
With the help of the derivatives, we can find the optimum points of economic functions, if any. For example, the use of derivatives is helpful to compute the level of output at which the total revenue is the highest, the profit is the highest and (or) the lowest, marginal costs and average costs are the smallest, etc. A producer who faces a continuous and differentiable profit function, \(\pi = TR(x) - TC(x)\), can apply the techniques of optimization to find the optimum level of output and the corresponding profit. A consumer who has the utility function \(u = f(x)\) which is continuous and differentiable, can use the optimization techniques to find the level of consumption for which her utility is the highest.
Example 1: From the following economic functions, derive the corresponding marginal functions.