Mathematical Framework of Economic Analysis: Linear Models and Differential Calculus

1. Economics 214 Lecture 2 Mathematical Framework of Economic Analysis Continued

2. Linear Models The equations of our model are in the linear equation form.

3. Linear Equations • Systems of linear equations can be solved by linear algebra or matrix algebra. • Matrix algebra can tells us if our system has an unique solution. • Matrix algebra can be used to do comparative statics. • How does an equilibrium value change when one of the exogenous variables change. i.e. how does equilibrium national income change when investment changes.

4. Differential Calculus Suppose the events of the past several months cause consumers in our economy to decide to increase their rate of saving. i.e. reduce β in our consumption function. We may ask ourselves how this change in behavior affects the equilibrium national income in our economy. To analyze this problem we will have to make use of differential calculus. In Economics 111, we called this event the Paradox of Thrift.

5. Extreme Values

6. Extreme Value • From the previous graph, we might ask ourselves at what level of employment is output maximum? • This is an example of an extreme value problem. • We need differential calculus to solve this problem.

7. Production Example

8. Constrained Optimization • In microeconomics, we learned we could produce given level of output with various mixtures of the inputs. • We asked what combination minimized the cost of producing a given level of output. • This is a constrained optimization problem. • We need differential calculus to get a solution.

9. Dynamic Analysis • Dynamic Analysis focuses on models in which time and the time path of variables are explicitly included. • Difference equations focuses on models in which time is treated as a series of distinct periods.