Let's solve this carefully and step-by-step. **Setup / interpretation.** There are two products (Phone and Internet) and two consumer types (one Talker and one Gamer). Each consumer will pay up to the amounts in the table: * Talker: Phone = \$30, Internet = \$16 ⇒ bundle value = \$46. * Gamer: Phone = \$x, Internet = \$24 ⇒ bundle value = \$x+24. We assume one unit of each type (so total firm revenue is the sum over the two consumers). The firm can either (A) sell the two goods separately (no bundling) and choose each product’s price, or (B) use **pure bundling** and sell only the bundle at a single price. We compare the firm’s maximum revenue under the two options. --- ## 1) Revenue with **no bundling** (optimal separate prices) Treat each product separately and pick the price that maximizes revenue from the two consumers. **Internet:** valuations 16 (Talker) and 24 (Gamer). * If price = 24 → sell only to Gamer → revenue = 24. * If price = 16 → sell to both → revenue = 16×2 = 32. Best choice: **price = 16**, revenue = **\$32**. **Phone:** valuations 30 (Talker) and x (Gamer). Consider cases. * If $x \le 15$: best is to price at 30 and sell only to Talker → phone revenue = \$30. * If $15 < x < 30$: best is to set price = x and sell to both → phone revenue = \$2x. (Because 2x > 30 for x>15.) * If $30 \le x \le 60$: best is to set price = 30 and sell to both → phone revenue = \$60. * If $x > 60$: best is to set price = x and sell only to Gamer → phone revenue = \$x. So total no-bundling revenue (phone + internet = phone\_rev + 32) is: $$ R_{\text{no-bundle}}= \begin{cases} 62 & \text{if } x\le 15,\\[4pt] 2x+32 & \text{if } 15<x<30,\\[4pt] 92 & \text{if } 30\le x\le 60,\\[4pt] x+32 & \text{if } x>60. \end{cases} $$ (At the boundary $x=15$ or $x=30$ the formulas coincide appropriately.) --- ## 2) Revenue with **pure bundling** Bundle values: * Talker: 46 * Gamer: $x+24$ To sell to both, bundle price $P$ must be ≤ min(46, x+24). The firm will choose the highest feasible price that reaches the desired set (one or both consumers). Two main subcases (depending on which consumer has the smaller bundle valuation): **(i) If $x \le 22$** $\Rightarrow x+24 \le 46$. Then both consumers’ bundle valuations ≥ $x+24$ only if $P\le x+24$. The firm’s best revenue is to sell the bundle to **both** at $P=x+24$: revenue $=2(x+24)=2x+48$. (That dominates trying to sell to one only.) **(ii) If $x>22$** $\Rightarrow x+24>46$. Now the smaller bundle valuation is 46 (Talker). Setting $P=46$ sells the bundle to **both** and yields revenue $2\cdot46=92$. (If $x+24$ is so large that selling only to the gamer at $P=x+24$ yields more than 92, i.e. $x+24>92$ or $x>68$, then the firm would instead sell only to the gamer at price $x+24$.) So bundling revenue: $$ R_{\text{bundle}}= \begin{cases} 2x+48 & \text{if } x\le 22,\\[4pt] 92 & \text{if } 22<x\le 68,\\[4pt] x+24 & \text{if } x>68. \end{cases} $$ --- ## 3) Compare bundling vs no bundling We want $R_{\text{bundle}} > R_{\text{no-bundle}}$. Work through ranges: 1. $x\le 15$: $R_{\text{bundle}}=2x+48,\; R_{\text{no}}=62.$ $2x+48>62 \iff 2x>14\iff x>7.$ So for $7<x\le 15$ bundling is better; for $x\le 7$ no-bundle is better. At $x=7$ they tie (both 62). 2. $15<x<22$: $R_{\text{bundle}}=2x+48,\; R_{\text{no}}=2x+32.$ $2x+48>2x+32$ always → **bundling strictly better** for all $x\in(15,22)$. 3. $22\le x<30$: For $x=22$ bund=92 and no-bundle $=2x+32=76$ (bundling better). For $22<x<30$, bund=92 and no-bundle $=2x+32<92$ (since $2x+32<2\cdot30+32=92$). So **bundling better** for all $x\in(22,30)$. At $x=30$ both give 92. 4. $30\le x\le 60$: $R_{\text{bundle}}=92,\; R_{\text{no}}=92$. They are **equal** (neither strictly better). 5. $60<x\le 68$: $R_{\text{bundle}}=92,\; R_{\text{no}}=x+32>92$. So **no-bundling better**. 6. $x>68$: $R_{\text{bundle}}=x+24,\; R_{\text{no}}=x+32$. So **no-bundling better** (by \$8). --- ## **Final answer** Pure bundling is **more profitable** than not bundling **exactly when** $$ \boxed{7 < x < 30.} $$ (At $x=7$ and $x=30$ revenues are equal; for $x\le 7$ or $x\ge 30$ bundling is not strictly better.)