L9 notes (1): Firm Behavior and Profit Maximization

Week 9 - Firm Behavior and Profit Maximization

Admin

• Back to normal schedule: Wednesday and Friday lectures, Tuesday through Thursday tutorials.
• Project draft due in two weeks (Monday, Week 11).
• Peer reviewer assignments will be emailed in Week 10.
• Group video peer reviews will be matched with other video projects, If doing an individual report, you should be drawing from different sections of the course.
• Final exam timetable released: Thursday of Week 13, 9:00 AM - 12:15 PM (15 minutes reading time, 3 hours writing time).
• Exam structure: similar to mid-semester (20-25% multiple choice, rest short answer).
• Exam content: entire course, including pre-mid-semester and Week 12 material.
• Cheat sheet allowed: one A4 page, handwritten, both sides.

Consumption vs. Production

• Similarities:
  • Both involve inputs and outputs.
  • Consumers: inputs (money/income), output (utility from consumption).
  • Producers: inputs (costs of production), output (revenue from sales).
• Differences:
  • What people in the model want:
    • Consumers: utility from consumption.
    • Producers: revenue from sales (leading to profit).
  • Cost:
    • Consumers: fixed income (credit constrained).
    • Producers: cost of production (not typically modeled as credit constrained).
  • Value of money:
    • Consumers: money is a means to an end (buying things).
    • Firms: money (profit) is the end goal.
  • Constraints:
    • Consumers: credit constrained (fixed income).
    • Producers: typically not modeled as credit constrained (profit = revenue - costs).
  • Risk preferences:
    • Consumers: risk-averse (diminishing marginal utility of money).
      • Example: A million dollars is life-changing for an average person but not significant for someone extremely wealthy.
    • Firms: risk-neutral (linear utility in money).
      • The first $100 is just as valuable as the$1,000.

Firm as a Complicated Object (Simplified in Our Models)

• Real-world firms involve complexities:
  • Workers: minimize effort while keeping their job or seeking promotions.
  • Managers: similar to workers, but at a different level.
  • Owners: maximize profit but face principal-agent problems (motivating workers/managers).
  • Short-run vs. long-run considerations.
• Organizational economics studies these complexities.
• Our simplified model: firms are singular entities maximizing profit.

Profit Maximization

• Basic principle: maximize profit by setting marginal revenue (MR) equal to marginal cost (MC).
• Graphical representation:
  • Output (Q) vs. Dollars ($).
  • Revenue curve: increases with output.
  • Cost curve: increases with output (decreasing returns to scale).
  • Profit: difference between revenue and costs.
• Optimal quantity (Q*): maximizes the gap between revenue and cost curves.

Marginal Revenue Equals Marginal Cost

• If MC < MR: produce more to increase profit.
• If MC > MR: reduce production to increase profit.
• Optimal: MC = MR (slopes of cost and revenue curves are equal).
• Caveats:
  • Assumes convex costs.
  • If cost curves do something strange, it might be unclear what the profit maximizing level is.

Invisible Hand

• The invisible hand idea is that the market operating efficiently will lead the market operating at an equilibrium level will lead to an efficient outcome anyway.

Output vs. Dollars per Unit

• Another graphical representation:
  • Output (Q) vs. Dollars per Unit ($/Unit).
  • Marginal revenue curve (MR).
  • Marginal cost curve (MC).
  • Profit maximization: MR = MC.
• Why we like this figure: the intersection of the lines is very easy to find.

Profit Maximization - Equation

• Firm maximizes profit (π) by choosing quantity (q).
• $\pi = Revenue(q) - Costs(q)$
• No credit constraints.
• Unconstrained optimization problem.
• First-order condition: derivative of profit with respect to quantity equals zero.
• $\frac{d \pi}{dq} = \frac{dRevenue}{dq} - \frac{dCosts}{dq} = 0$
• $\frac{dRevenue}{dq} = Marginal\ Revenue$
• $\frac{dCosts}{dq} = Marginal\ Cost$
• Therefore, $MR = MC$
• Concerns:
  • Concave profit, convex cost

Marginal Revenue and Demand

• Monopolies: Firm-level demand equals market demand.
• Competitive firms: can sell as much as they like at the market price. Their demand function is just the market price.
• Oligopolies/Monopolistic competition: consider residual demand (demand after other firms' actions).
• Marginal revenue is built from firm-level demand.

Revenue and Marginal Revenue

• Revenue (R) = Price (P) * Quantity (Q)
• P is potentially a function of Q.
• Marginal revenue: extra revenue from selling an extra unit.
• $MR = \frac{dRevenue}{dQ} = \frac{d(P*Q)}{dQ}$
• Product rule:$\frac{d(uv)}{dx} = u'v + uv'$
• Applying product rule:
  • $MR = \frac{dP}{dQ} * Q + P * \frac{dQ}{dQ}$
  • $MR = \frac{dP}{dQ} * Q + P$
• If demand is constant, the marginal revenue will be equal to market price. (Perfect competition) This is not always the case

Relationship Between Marginal Revenue and Price

• For competitive firms: price does not change with quantity, so $\frac{dP}{dQ} = 0$ and MR = P.
• For non-competitive firms (monopolies): \frac{dP}{dQ} < 0 (to sell more, lower the price).
• Therefore, MR < P (or at most, MR = P).

Marginal Revenue and Elasticity

• Marginal revenue can also be expressed as:
• $MR = P * (1 + \frac{1}{PED})$ where PED is the price elasticity of demand.
• For competitive firms: PED = -∞, so MR = P.
• If demand is elastic (PED < -1): MR is negative.
• Negative marginal revenue: selling more reduces overall revenue.
• If MR < 0, the firm can increase profit by reducing production.

Generic vs. Linear Demand Functions

• Generic demand functions: marginal revenue curve is not obvious.
  • Example: if $P=Q^{-\frac{2}{3}}$, marginal revenue is $MR = \frac{1}{3} * Q^{-\frac{2}{3}}$.
• Linear demand functions: much simpler.
  • If demand is linear: $P = a - bQ$, then $MR = a - 2bQ$.
    • Marginal revenue is also linear.
    • Same Y-intercept as demand curve.
    • Twice the slope of the demand curve.