Using Mathematics to Learn Economics Short-hand skills Equilibrium (static) analysis Comparative...

Using Mathematics to Learn Economics

• Short-hand skills

• Equilibrium (static) analysis

• Comparative statics analysis– Differentiation– Partial derivatives

• Optimization– Use in decision making

Rules of Differential Calculus

• Constant rule

• Power-function rule

• Sum-difference rule

• Partial derivatives

Optimization Techniques

• Unconstrained optimization

• Constrained optimization– Substitution method– Lagrangian multiplier method

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, .

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

Typical Mathematical Functions

• Demand and supply curves

• Total revenue functions

• Production function

• Cost functions

• Profit functions

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