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 = PJ

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

Specific Functional Forms Linear –Q = a 0 + b 0 X + c 0 Y; b 0 = dQ/dX Log linear –Log Q = a 1 + b 1 X + c 0 Y; b 1 = %dQ/dX Double log –Log Q = a 2 + b 2 logX + c 2 logY; b 2 = (% dQ)/(%dX) Power function –Q = a 4 + b 4 X + c 4 X 2 ; dQ/dX = b 4 + 2c 4 X