Using Mathematics to Learn Economics

1. Using Mathematics to Learn Economics • Short-hand skills • Equilibrium (static) analysis • Comparative statics analysis • Differentiation • Partial derivatives • Optimization • Use in decision making

2. Rules of Differential Calculus • Constant rule • Power-function rule • Sum-difference rule • Partial derivatives

3. Optimization Techniques • Unconstrained optimization • Constrained optimization • Substitution method • Lagrangian multiplier method

4. Lagrangian Method • Objective functions are often constrained by one or more “constraints” (time, capacity, or money) • Max L = (objective fn) -{constraint = 0} • Min L = (objective fn) +{constraint = 0} • An artificial variable is created for each constraint, traditionally called lambda, .

5. Example using Lagrangian Function • Minimize Crime in your town • Police, P, costs $15,000 each. • Jail, J, costs $10,000 each. • Budget is $900,000. • Crime function is estimated: C = 5600 - 4PJ

6. Typical Mathematical Functions • Demand and supply curves • Total revenue functions • Production function • Cost functions • Profit functions

7. Specific Functional Forms • Linear • Q = a0 + b0X + c0Y; b0 =dQ/dX • Log linear • Log Q = a1 + b1X + c0Y; b1 =%dQ/dX • Double log • Log Q = a2 + b2 logX + c2 logY; b2 = (%dQ)/(%dX) • Power function • Q = a4 + b4X + c4X2; dQ/dX = b4 + 2c4X