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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