1. Economics 214 Lecture 12

2. Average and marginal functions The simple geometry of the relationship between marginal and average compares the slope of a ray from the origin to the value of the derivative of a function as we move out along the function. The slope of the ray from the origin will increase as we move out along the function as long as the derivative of the function is greater than the slope of the ray from the origin.

3. Marginal vs Average

4. Average and marginal functions The slope of the ray from the origin will decrease as we move out along the function as long as the derivative is less than the slope of that ray. A linear function has a constant slope, a constant derivative, but the average will varying except when the function is linear through the origin.

5. Average vs Marginal

6. Average and marginal functions The average function rises as long as the marginal function lies above it. The average function declines as long as the marginal function lies below it.

7. Marginal Cost Vs Average Cost

8. Average Product vs Marginal Product

9. Average and marginal functions

10. Basic Question How does a dependent variable change when the independent variable changes given that we have started at some initial value of the independent variable? We will discover that an easy solution to this question comes from the use of the differential.

11. Solution using Difference quotient

12. The Differential

13. Figure 6.9 Differential Approximation and Actual Change of a Function

14. Economic Example

15. Quality of Approximation The extent to which the differential serves as a good approximation to the actual change of the value of a function for discrete changes in its argument depends upon the size of the change of the argument. Table on next slide illustrates for our tax revenue function.

16. Actual an approximate changes in Revenue

17. Quality of Differential approximation

18. Quality of Differential  y (actual) dy (estimated) Difference (%) O-150 0 2-108-1101.9 4-66-706.1 6-24-3025 8181044.4 Actual and Approximate Changes with Varying Nonlinearity (x 0 =10 dx=1)