Lecture 4
UNCONSTRAINED OPTIMIZATION: THE ONE-VARIABLE CASE

• Extremum of a function
• First-order conditions
• Second-order conditions
• Concavity and convexity

1. Optimization

• To find a set of set of values of the choice variables that will yield the desired extremum of the objective function
  

• The function to be maximized or minimized; y = f(x)
• The dependent variable, y, represents the object of maximization or minimization

• The independent variables that can be chosen to achieve the desired objective

2. Extremum

• An extreme value - a maximum or a minimum

• Local extremum:
  The maximum or minimum value of a function in the neighbourhood of a point
• Global extremum:
  The maximum or minimum value of a function in the range of the function
  

3. First-order condition (FOC)

• Consider the function y = f(x).
• The necessary condition for a relative extremum (maximum or minimum) is that the first-order derivative be zero, i.e.
  f'(x) = 0.

3.1 Interpretation of the FOC

• At the highest and lowest points of a curve, the tangent to the curve at such points is horizontal.
• The slope of the curve is zero.

3.2 Critical value

• Let f'(x) = 0 at x = x₀.
  • x₀: Critical value of x
  • f(x₀): Stationary value of the function
  • (x,f(x₀)): Stationary point

3.3 Question

• Why is f'(x) = 0 not a sufficient condition for a local maximum or minimum?
• Answer: Because f'(x) = 0 at some inflexion points.

Note: The first-order condition does not distinguish between a maximum and a minimum.

4. Second-order condition (SOC)

If the first-order condition is satisfied at x = x₀,
• f(x₀) is a local maximum if f''(x₀) < 0
• f(x₀) is a local minimum if f''(x₀) > 0

Maximum:
• As you move up a curve from the left, leading to a maximum, the curve gets increasingly flatter, i.e. the slope gets smaller and smaller.
• This means that f'' < 0.
• For example, if f' goes from 6 to 2, it means that f'' < 0.

Minimum:
• As you move down a curve from the left, leading to a minimum, the curve gets increasingly flatter.
• However, since the slope is negative, a flattening of the curve implies that f'' > 0.
• For example, if f' goes from -3 to -2, it means that f'' > 0.
  

5. Obtaining a local extremum

• Step 1: Obtain the first-order derivative of f(x).

• Step 2: Set f'(x) = 0.
  • Solve for x.
  • These are the critical values of x.
  • But, at this point, you do not know if they yield a maximum or a minimum.

• Step 3: Obtain the second-order derivative of f(x).

• Step 4: Determine the sign of f''(x) at the critical values of x.
  • If f'' < 0, the critical value corresponds to a maximum.
  • If f'' > 0, the critical value corresponds to a minimum.

5.1 Example

Consider the function y = f(x) = x² - 5x + 8.

1. Obtain f'(x).
2. Set f'(x) = 0. Solve for x.
3. Obtain f''(x).
4. What is the sign of f''(x) at the critical value? Does the critical value yield a maximum or a minimum?
5. Compute the stationary value of the function.
   

6. Concavity and convexity

• Refers to the curvature of a graph
• Linked to the second-order derivative

6.1 Concave curve

• A line segment joining any two points on the curve lies on or below the curve

6.2 Concave and strictly concave functions

• If f''(x) < 0 for all x, f(x) is strictly concave

6.3 Example

• f(x) = -3x² + 5x - 2
• The function is strictly concave. Why?

6.4 Convex curve

• A line segment joining any two points on the curve lies on or above the curve

6.5 Convex and strictly convex functions

• If f''(x) > 0 for all x, f(x) is strictly convex

6.6 Example

• f(x) = 3x² + 5x - 2
• The function is strictly convex. Why?

7. Profit maximization by firms

• Maximization problem:
  Firm seeks to maximize profits by choosing optimal Q.
  
• Objective function:
  The profit function
  
• Choice variable:
  Output level, Q
  
• First-order condition:
  
  • R'(Q) - C'(Q) = 0
  • MR = MC (ah, yes, the familiar condition)
  • The firm maximizes profits by producing the output level at which marginal revenue equals marginal cost.
    But how do we know that profits are maximized, not minimized, at this point?
    
• Second-order condition:
  Sufficent condition for maximum profits

8. Application: Profit maximization

• Demand curve: P(Q) = a - bQ
• Cost function: C(Q) = cQ² + eQ + F
  

Questions

1. What is the firm's fixed cost? Variable cost?
2. Obtain the firm's revenue R(Q) and marginal revenue (MR) functions.
3. Obtain the firm's marginal cost (MC) and average cost (AC) functions.
4. Write down the firm's profit function in terms of Q.
5. Obtain the first-order derivative of the profit function.
6. Obtain the second-order derivative of the profit function.
7. Under what conditions is the profit function strictly concave? Assume these conditions hold.
8. Using the first-order condition, obtain the critical value of Q. Call this Q*.
9. Using the second-order condition, establish whether the critical value corresponds to a maximum or minimum.
10. Obtain the firm's optimal price. Call this P*.
11. What is the relationship between MR and MC at Q*?
12. Sketch the demand, MR, MC and AC curves.
13. Indicate the firm's optimal output and price on the graph.
14. Use derivatives and graphs to show how the firm's price and output combination is affected by:
    1. A decrease in c.
    2. An increase in a.
    3. A decrease in F.