1. Calculus-Based Optimization AGEC 317
Economic Analysis for Agribusiness and Management

2. Readings
Optimization Techniques

3. Why Consider Optimization?
Consumers maximize utility or satisfaction.
Producers or firms maximize profit or minimize costs.
Optimization is inherent to economists.

4. Setting-Up Optimization Problems
Define the agent’s goal: objective function and identify the agent’s choice (control) variables
Identify restrictions (if any) on the agent’s 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)

5. Mathematically, Optimize y = f(x1, x2, . . . ,xn)
subject to (s.t.)
gj (x1, x2, ,xn) ≤ bj or
= bj j = 1, 2, . . ., m.
≥ bj
y = f(x1, x2, ,xn) → objective function
x1, x2, ,xn → set of decision variables (n)
optimize → either maximize or minimize
gi(x1, x2, ,xn) → constraints (m)

6. Constraints refer to restrictions on resources
legal constraints
environmental constraints
behavioral constraints

7. 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:

8. 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 maximum.

9. Example PROFIT = -40 + 140Q – 10Q2 Find Q that maximizes profit
140 – 20Q = set 0
Q = 7
- 20 < 0
max profit occurs at Q = 7
max profit = (7) – 10(7)2
max profit = $450

10. Minimization Problems: Min 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.

11. Example COST = 15 - .04Q + .00008Q2 Find Q that minimizes cost
Q = set 0
Q = 250
> 0
Minimize cost at Q = 250
min cost = $10

12. Unconstrained Multivariate Optimization
Max
FOC:
SOC:
Min
FOC:
SOC:

13. Example PROFIT is a function of the output of two products
(e.g.heating oil and gasoline)
Q1 Q2
so,
Solve Simultaneously Q1 = 5.77 units
Q2 = 4.08 units

14. Second-Order Conditions
(-20)(-16) – (-6)2 > 0
320 – 36 > 0
we have maximized profit.

15. Constrained Optimization
Solution: Lagrangian Multiplier Method
Maximize y = f(x1, x2, x3, …, xn)
s.t. g(x1, x2, x3, …, xn) = b
Solution:
Set up Lagrangian:
FOC:

16. Lagrangian Multiplier
Interpretation of Lagrangian Multiplier λ: the shadow value of the constrained resource.
If the constrained resource increases by 1 unit, the objective function will change by λ units.

17. Example Maximize Profit = subject to (s.t.) 20Q1 + 40Q2 = 200
Could solve by direct substitution
Note that 20Q1 = 200 – 40Q2
or Q1 = 10 – 2Q2
Combine like terms.

18. Lagrangian Multiplier Method