THE SIMPLE MATHEMATICS OF MICRO-ECONOMICS Production theory. Begin with a production function: Q = 21a + 9a2 - a3, showing output, Q, as a function of the variable input, labor, a. Marginal product of labor is a) the addition to total product resulting from the use of one more unit of a. b) the slope of the total product curve, and c) the first derivative of a total product, TP, function, thus, dQ/da = MPa = 21 + 18a - 3a2* _______________________ *You will remember about first derivatives that: to get the derivative of a constant, dQ/da = 0. For example, in the expression Q = 21 + 7a - a2 + 3a3, the number 21 is considered a constant, for which dQ/da = 0. To get the derivative of a variable “a” preceded by a coefficient m, or ma, dQ/da = m. For example, in the expression Q = 21 + 7a - a2 + 3a3, the term 7a consists of the coefficient 7 and the variable a, for which dQ/da = 7. To get the derivative of a variable with an exponent, an, dQ/da = nan-1. For example, in the expression Q = 21 + 7a - a2 + 3a3, the term a2 consists of the variable a and the exponent 2, for which dQ/da = 2a. (By the formula, this would read 2a1, but any number or variable taken to the first power, I.e., having an exponent of 1, is not changed. So we need not write the 1.) To get the derivative of a variable with both coefficient m and exponent n, such as ma n, we have dQ/da = n(m)an-1. For example, in the expression Q = 21 + 7a - a2 + 3a3, 3 the term 3a consists of the variable a with the coefficient 3 and exponent 3, for which dQ/da = 9a2. For the entire expression, Q = 21 + 7a - a2 + 3a3, we get the derivative dQ/da = 7 - 2a + 9a2. For average product, AP, we have APa = TP/a - (21a + 9a2 - a3)/a = 21 + 9a - a2

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Cost Analysis Now consider costs, using the same kind of analysis. We recall the simple short-run relationships, specifically that total cost, TC, equals total fixed costs,