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rewrite same which is given below step wise in organised manner 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: $$ R_{\text{no-bundle}}= \begin{cases} 62 & \text{if } x\le 15,\\[4pt] 2x+32 & \text{if } 1522$** $\Rightarrow x+24>46$. Now the smaller bundle valuation is 46 (Talker). Setting $P=46$ sells the bundle to **both** and yields revenue $2\cdot46=92$. (If $x+24$ is so large that selling only to the gamer at $P=x+24$ yields more than 92, i.e. $x+24>92$ or $x>68$, then the firm would instead sell only to the gamer at price $x+24$.) So bundling revenue: $$ R_{\text{bundle}}= \begin{cases} 2x+48 & \text{if } x\le 22,\\[4pt] 92 & \text{if } 2268. \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

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rewrite same which is given below step wise in organised manner 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: $$ R_{\text{no-bundle}}= \begin{cases} 62 & \text{if } x\le 15,\\[4pt] 2x+32 & \text{if } 1522$** $\Rightarrow x+24>46$. Now the smaller bundle valuation is 46 (Talker). Setting $P=46$ sells the bundle to **both** and yields revenue $2\cdot46=92$. (If $x+24$ is so large that selling only to the gamer at $P=x+24$ yields more than 92, i.e. $x+24>92$ or $x>68$, then the firm would instead sell only to the gamer at price $x+24$.) So bundling revenue: $$ R_{\text{bundle}}= \begin{cases} 2x+48 & \text{if } x\le 22,\\[4pt] 92 & \text{if } 2268. \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

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Asked by: prem
Created at: 2025-08-24 23:11:20

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Here’s a stepwise, organized rewrite of your solution, making all logic and transitions clear: --- # Step-by-Step Solution ## **Step 1: Problem Setup** - **Products:** Phone and Internet. - **Consumers:** Two types. - **Talker:** Phone = \$30, Internet = \$16 ⇒ Bundle = \$46. - **Gamer:** Phone = \$x, Internet = \$24 ⇒ Bundle = \$x + \$24. - **Assumptions:** 1 unit of each consumer. The firm can - (A) Sell each product separately (set prices for Phone and Internet individually). - (B) Use **pure bundling** (sell only the bundle at one price). - **Goal:** For each pricing strategy, **find the firm’s maximum revenue**. Then, **compare which is better** as $x$ varies. --- ## **Step 2: Revenue from Separate Sales (No Bundling)** ### **A. Internet Pricing** - Values: Talker = \$16, Gamer = \$24. - Price = \$24 ⇒ Only Gamer buys ⇒ Revenue = \$24. - Price = \$16 ⇒ Both buy ⇒ Revenue = \$16 × 2 = \$32. - **Optimal:** Set price at \$16 ⇒ **Internet revenue = \$32**. ### **B. Phone Pricing** - Values: Talker = \$30, Gamer = \$x. - Consider cases: 1. **If $x \le 15$:** - Set price = \$30 ⇒ Only Talker buys ⇒ **Revenue = \$30**. 2. **If $15 < x < 30$:** - Set price = \$x ⇒ Both buy ⇒ **Revenue = \$2x** (since $2x > 30$ for $x>15$). 3. **If $30 \le x \le 60$:** - Set price = \$30 ⇒ Both buy ⇒ **Revenue = \$60**. 4. **If $x > 60$:** - Set price = \$x ⇒ Only Gamer buys ⇒ **Revenue = \$x**. ### **C. Total Revenue (Phone + Internet)** \[ R_{\text{no-bundle}} = \begin{cases} 62 & \text{if } x \le 15 \\ 2x + 32 & \text{if } 15 < x < 30 \\ 92 & \text{if } 30 \le x \le 60 \\ x + 32 & \text{if } x > 60 \end{cases} \] --- ## **Step 3: Revenue from Pure Bundling** ### **A. Bundle Values** - Talker: \$46 - Gamer: \$x + \$24 ### **B. Possible Bundle Prices** - To sell to **both**: Bundle price $P \le \min(46, x+24)$. **Case 1:** $x+24 \le 46$ (i.e., $x \le 22$) - Set $P = x+24$, both buy. - **Revenue = $2(x+24) = 2x + 48$** **Case 2:** $x+24 > 46$ (i.e., $x > 22$) - Set $P = 46$, both buy. - **Revenue = $2 \times 46 = 92$** - Exception: If $x+24 > 92$ ($x>68$), better to sell only to Gamer at $P=x+24$, revenue = $x+24$. ### **C. Bundling Revenue Summary** \[ R_{\text{bundle}} = \begin{cases} 2x + 48 & \text{if } x \le 22 \\ 92 & \text{if } 22 < x \le 68 \\ x + 24 & \text{if } x > 68 \end{cases} \] --- ## **Step 4: Compare Bundling vs. No Bundling** ### **Case-by-case Comparison** 1. **$x \le 15$:** - $R_{\text{bundle}} = 2x + 48$ - $R_{\text{no-bundle}} = 62$ - $2x + 48 > 62 \implies x > 7$ - **Bundling is better for $7 < x \le 15$** 2. **$15 < x < 22$:** - $R_{\text{bundle}} = 2x + 48$ - $R_{\text{no-bundle}} = 2x + 32$ - $2x + 48 > 2x + 32$ always - **Bundling is always better here** 3. **$22 < x < 30$:** - $R_{\text{bundle}} = 92$ - $R_{\text{no-bundle}} = 2x + 32$ - $2x + 32 < 92$ for $x < 30$ - **Bundling is better for $22 < x < 30$** 4. **$30 \le x \le 60$:** - Both revenues are $92$ - **Equal** 5. **$60 < x \le 68$:** - $R_{\text{bundle}} = 92$ - $R_{\text{no-bundle}} = x + 32$ - $x + 32 > 92$ for $x > 60$ - **No-bundling is better** 6. **$x > 68$:** - $R_{\text{bundle}} = x + 24$ - $R_{\text{no-bundle}} = x + 32$ - No-bundling is better by \$8 --- ## **Step 5: Final Answer (When Is Bundling Better?)** Pure bundling is **more profitable** than selling separately **exactly when:** \[ \boxed{7 < x < 30} \] - At $x = 7$ and $x = 30$, revenues are equal. - For $x \le 7$ or $x \ge 30$, pure bundling is not strictly better.

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