Calculus-Based Optimization AGEC 317 Economic Analysis for Agribusiness and Management Readings Optimization Techniques Why Consider Optimization? Consumers maximize utility or satisfaction. Producers or firms maximize profit or minimize costs. Optimization is inherent to economists. Setting-Up Optimization Problems Define the agents goal: objective function and identify the agents choice (control) variables Identify restrictions (if any) on the agents choices (constraints). If no constraints exist, then we have unconstrained minimization or maximization problems. If constraints exist, what type? Equality Constraints (Lagrangian) Inequality Constraints (Linear Programming) Mathematically, Optimize y = f(x 1, x 2,...,x n ) subject to (s.t.) g j (x 1, x 2,...,x n ) b j or = b j j = 1, 2,..., m. or b j y =f(x 1, x 2,...,x n ) objective function x 1, x 2,...,x n set of decision variables (n) optimize either maximize or minimize g i (x 1, x 2,...,x n ) constraints (m) Constraints refer to restrictions on resources legal constraints environmental constraints behavioral constraints Review of Derivatives y=f(x): First-order condition: Second-order condition: Constant function: Power function: Sum of functions: Product rule: Quotient rule: Chain rule: Unconstrainted Univariate Maximization Problems: max f(x) Solution: Derive First Order Condition (FOC): f(x)=0 Check Second Order Condition (SOC): f(x)0 Local vs. global: If more than one point satisfy both FOC and SOC, evaluate the objective function at each point to identify the minimum. Example COST = Q Q 2 Find Q that minimizes cost Q = set 0 Q = > 0 Minimize cost at Q = 250 min cost = $10 Min FOC: SOC: Unconstrained Multivariate Optimization Max FOC: SOC: Example PROFIT is a function of the output of two products (e.g.heating oil and gasoline) Q 1 Q 2 so, Solve Simultaneously Q 1 = 5.77 units Q 2 = 4.08 units Second-Order Conditions (-20)(-16) (-6) 2 > 36 > 0 we have maximized profit. Constrained Optimization Solution: Lagrangian Multiplier Method Maximize y = f(x 1, x 2, x 3, , x n ) s.t. g(x 1, x 2, x 3, , x n ) = b Solution: Set up Lagrangian: FOC: Lagrangian Multiplier Interpretation of Lagrangian Multiplier : the shadow value of the constrained resource. If the constrained resource increases by 1 unit, the objective function will change by units. Example Maximize Profit = subject to (s.t.) 20Q Q 2 = 200 Could solve by direct substitution Note that 20Q 1 = 200 40Q 2 or Q 1 = 10 2Q 2 Maximize Profit = Combine like terms. Maximize Profit = Lagrangian Multiplier Method