Intermediate Microeconomics

1. Intermediate Microeconomics Math Review

2. Functions and Graphs • Functions are used to describe the relationship between two variables. • Ex: Suppose y = f(x), where f(x) = 10 - 2x • This means • if x is 2, y must be 10 – 2(2) = 6 • if x is 4, y must be 10 – 2(4) = 2 * This relationship can also be described via a graph.

3. Rate-of-Change and Slope • We are often interested in rate-of-change of one variable relative to the other. • For example, how do profits (y) change as a firm increases quantity supplied (x)? • This is captured by the slope of a graph.

4. Rate-of-Change and Slope • For linear functions, this is constant and equal to “rise/run” or Δy/Δx. y 4 2 y 6 2 2 -4 1 -2 2 4 x 3 4 x slope = rise/run =(change in y)/(change in x) = -4/2 = -2 slope = rise/run = -2/1 = -2

5. Non-linear Relationships • Things gets slightly more complicated when relationships are “non-linear”. • Consider the functional relationship y = f(x), where f(x) = 3x2 + 1 • For non-linear relationships, rise/run is just an approximation of the slope at any given point. • This approximation is better, the smaller the change in x we consider. y 13 4 y 28 4 slope = 9/1 = 9 slope = 24/2 = 12 24 9 2 1 1 3 1 2 x x

6. Non-linear Relationships • Analytically: • Consider again the relationship y = f(x), where f(x) = 3x2 + 1 • Starting at x = 1, if we increase x by 2 what will be the corresponding change in y? • Similarly, starting at x = 1, if we increase x by 1 what will be the corresponding change in y? • So this functional relationship between x and y means that how much y changes due to a change in x depends on how big of a change in x and where you evaluate this ratio.

7. The Derivative • As discussed before, we get a better approximation to the relative rate-of-change the smaller the change in x we consider. • In particular, given a relationship between x and y such that y = f(x) for some function f(x), we have been considering the question of “if x increases by Δx, what will be the relative change in y?”, or • The derivative is just the limit of this expression as Δx goes to zero, or • We will also sometimes express the derivative of f(x) as f’(x)

8. The Derivative • Given y = f(x), where f(x) = 3x2 + 1, • f’(x) = (2)3x2-1 = 6x • So at x = 1, f’(1) = 6(1) = 6 • This means the slope of f(x) = 3x2 + 1 at x = 1 equals 6. • Equivalently, this means that at x = 1, y is increasing at a rate 6 times faster than x. • Alternatively, at x = 3, f’(3) = 6(3) = 18 • This means the slope of f(x) = 3x2 + 1 at x = 3 equals 18. • Equivalently, this means that at x = 3, y is increasing at a rate 18 times faster than x. y 28 4 slope = 18 slope = 6 1 3 x

9. The Derivative • This shows that rise/run method with non-linear functions will give an approximation of the slope at any given x*, where this approximation is essentially the average slope between x* and x* + Δx. • Obviously, the smaller the Δx, the better the approximation, or the closer the rise/run calculation will be to the derivative. y 28 13 4 slope = 12 slope = 9 slope = 6 1 2 3 x

10. Rules for Taking Derivatives • Basic functions: • f(x) = axb + c f’(x) = (b)axb-1 • Log functions: • f(x) = a log x f’(x) = a/x • Product Rule: • f(x) = g(x)h(x) f’(x) = g’(x)h(x) + g(x)h’(x) • Quotient Rule: • f(x) = g(x)/h(x) f’(x) = [g’(x)h(x) – g(x)h’(x)]/h(x)2 • Chain Rule: • f(x) = g(h(x)) f’(x) = g’(h(x))h(x)

11. Second Derivatives • A Second Derivative is just taking the derivative of the derivative. • Going back to y = f(x), where f(x) = 3x2 + 1, • f ’(x) = 6x > 0 • f ”(x) = 6 > 0 • Intuitively, if the first derivative give you the slope of a function at a given point, the second derivative gives you the slope of the slope of a function at a given point. • For y = f(x), where f(x) = 3x2 + 1, • The positive first derivative tells us that y increases as x increases, • The positive second derivative tells us that the slope of f(x) increases as x increases, meaning y increases more quickly as x increases.

12. Thinking about Derivatives Graphically y f(x) = 3x2 + 1 x

13. Thinking about Derivatives Graphically y f(x) = 3x2 + 1 f’(x) = 6x x x

14. Thinking about Derivatives Graphically y f(x) = 3x2 + 1 f’(x) = 6x f”(x) = 6 x x x

15. Thinking about Derivatives Graphically Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5. y f(x) = 10 – 2x0.5 x

16. Thinking about Derivatives Graphically Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5. y x f(x) = 10 – 2x0.5 f’(x) = -x-0.5 x

17. Thinking about Derivatives Graphically Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5. y x f(x) = 10 – 2x0.5 f’(x) = -x-0.5 f”(x) = x-1.5 x x

18. Partial Derivatives • Often we will want to consider functions of more than one variable. • For example: y = f(x, z), where f(x, z) = 5x2z + 2 • We will often want to consider how the value of such function changes when only one of its arguments changes. • This is called a Partial derivative.

19. Partial Derivatives • The Partial derivative of f(x, z) with respect to x, is simply the derivative of f(x, z) taken with respect to x, treating z as just a constant. • Examples: • What is the partial derivative of f(x, z) = 5x2z3 + 2 with respect to x? With respect to z? • What is the partial derivative of f(x, z) = 5x2z3 + 2z with respect to x? With respect to z?