Mathematical Appendix: The calculus of marginals and totals

In the text we considered several functions. When some variable Y is a function of another variable X, we can write

Y = f(X).    (A1)

For every given value of Y, this function gives us a corresponding value of X. The marginal value of Y is the rate at which Y is tending to change as X changes. Mathematically, this is the derivative of Y with respect to X. We can write

Marginal value of Y = dY/dX,    (A2)

or, in two other common alternative notations,

Marginal value of Y = f'(X),

Marginal value of Y = f_x.

If we have only the marginal relationship in (A2), we can derive the relationship of total Y to total X in (A1) up to an arbitrary constant by integrating the marginal function:

Y = ∫f'(X)dX + C,    (A3)

where C is the constant of integration.

Consider an example. Let the function be a quadratic:

Y = a + bX + cX²,   c <>a, b.    (A4)

The graph of this relationship will show Y rising up to some maximum value and then falling as X increases further.

The marginal value of Y is

dY/dX = b + 2cX.   (A5)

Since c <>b when X is zero. This tells us that the increment to Y is positive as X increased from 0 but falls steadily and eventually becomes negative. Where the marginal value is positive the graph of the original relationship is positively sloped, and where it is negative the graph of the original line is negatively sloped.

Letting the values of the parameters be a = 10, b = 2 and c = -0.1, we have

Y = 10 + 2X - 0.1X².   (A6)

The marginal value of Y is then given by

dY/dX = 2 - 0.2X,

which is zero at X = 10, positive for smaller X, and negative for larger X. This shows that increments to X up to 10 cause Y to increase while they cause Y to decrease when X exceeds 10.

If all we knew was the marginal revenue relationship of equation (A5), we could determine the total value of Y as

Y = ∫(2 - 0.2X)dX = 2X - 0.1X² + C.    (A7)

Since the original constant of a = 10 in equation (A4) disappears on differentiation, integration can only put back an arbitrary constant, C. In other words, there is no way we can discover the value of the original constant, a. But we can find the equation of the relationship between X and Y and know its shape except for an arbitrary parameter that shifts points on it upwards or downwards by a constant amount.