Homework 4

Solution Key (Revenue-Share HHI, CR2, CR4)

👉 Download HW 4 solution file here

Definitions

Let firm \(i\) have revenue \(R_i = p_i \cdot q_i\).
Let total market revenue be \(VM\) (column VM).

Define revenue share as:
\[ sh_i = \frac{R_i}{VM} = \frac{p_i q_i}{VM} \]

  • Herfindahl–Hirschman Index (HHI, 0–10,000 scale)
    \[ HHI = \sum_{i=1}^{N} (100 \cdot sh_i)^2 \]

  • Concentration Ratio (\(CR_k\))
    \[ CR_k = \sum_{j=1}^{k} sh_{(j)} \quad \text{where } sh_{(1)} \ge sh_{(2)} \ge \cdots \]

    • CR2: sum of top two revenue shares
    • CR4: sum of top four revenue shares

Use revenue shares for all calculations:
\[sh_i = \frac{p_i q_i}{V_M}.\]

Mapping to Excel Columns

Per month (row):

Revenues

  • r1 = p1*q1 (Jeminy)

  • r2 = p2*q2 (Cousins)

  • r3 = p3*q3

  • r4 = p4*q4

  • r5 = p5*q5

  • All Others:

    r_others = MAX(0, VM - (r1 + r2 + r3 + r4 + r5))

Shares

  • sh_i = r_i / VM for each of the 5 named firms (+ Others if r_others > 0)

Excel tips (replace shares_range with the cell range of shares):

  • HHI = SUMPRODUCT((100*shares_range)^2)
  • CR2 = SUM(LARGE(shares_range,{1,2}))
  • CR4 = SUM(LARGE(shares_range,{1,2,3,4}))

Do not use QM for HHI/CR — we are using revenue shares, not quantity shares.

Graphs Over Time

Create a Year–Month column and compute monthly HHI, CR2, and CR4 using revenue shares.
Produce three line charts: HHI, CR2, and CR4 vs. date.

Interpretation Thresholds

DOJ/FTC HHI screens (0–10,000 scale):

  • Unconcentrated: HHI < 1500
  • Moderately concentrated: 1500–2500
  • Highly concentrated: > 2500

CR rules of thumb:

  • CR4 < 40% → fragmented / competitive
  • CR4 ≈ 40–60% → loose oligopoly
  • CR4 > 60% → tight oligopoly
  • High CR2 (> 40–50%) → strong duopoly tendency

Use both levels and trends to describe market structure.

Answering Management’s Questions

  • HHI over time: Report the level and trend. Compare to thresholds above.
  • CR2/CR4 over time: Highlight concentration patterns. High/stable CRs indicate persistent market power.
  • Perfect competition?
    • Only plausible if HHI is low (<1500) and CR4 < 40%.
    • Otherwise, the market is likely oligopolistic, not perfectly competitive.
  • Cousins’ position:
    Report
    \[ sh_{\text{Cousins}} = \frac{p_2 q_2}{VM} \] each month, note if it is a top-2 or top-4 firm, and whether its share is increasing or decreasing over time.

Common Pitfalls

  • Forgetting Others when VM > Σ(p_i q_i) → underestimates concentration.
  • Mixing units (quantities instead of revenues).
  • Shares must sum to 1 for accurate results.
  • HHI/CR should always be calculated monthly, then graphed over time.