| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | 要确定完全竞争的录像带复制行业中的均衡价格，我们需要遵循以下步骤： 1. **计算每盘录像带的总成本，包括版税**。 2. **根据成本结构确定供给函数**。 3. **通过使供给和需求相等来找到均衡**。 4. **求解均衡价格**。 ### 1. 计算每盘录像带的总成本，包括版税 每家公司的平均成本为$10每盘。版税率 \( r \) 是总行业产量 \( Q \) 的递增函数，给定为 \( r = 0.002Q \)。 因此，每盘录像带的总成本 \( C \) 包括版税为： \[ C = 10 + r = 10 + 0.002Q \] ### 2. 确定供给函数 在长期均衡中，价格 \( P \) 等于每盘录像带的总成本： \[ P = 10 + 0.002Q \] ### 3. 需求函数 需求函数给定为： \[ Q = D(P) = 1,050 - 50P \] ### 4. 求解均衡价格 在长期均衡中，供给等于需求： \[ Q_s = Q_d \] 将供给函数和需求函数相等： \[ 10 + 0.002Q = P \] \[ Q = 1,050 - 50P \] 将 \( P \) 代入需求函数： \[ Q = 1,050 - 50(10 + 0.002Q) \] \[ Q = 1,050 - 500 - 0.1Q \] \[ Q + 0.1Q = 550 \] \[ 1.1Q = 550 \] \[ Q = 500 \] 然后，将 \( Q \) 代入供给函数以求得均衡价格 \( P \)： \[ P = 10 + 0.002 \times 500 \] \[ P = 10 + 1 \] \[ P = 11 \] ### 结论 在长期均衡中，录像带的均衡价格为 **$11**。 |
11 |
| deepseekr1 | Given a perfectly competitive videotape-copying industry, we need to determine the equilibrium price of copied tapes in long-run equilibrium. 1. **Cost Structure**: Each firm can copy five tapes per day at an average cost of $10 per tape. Additionally, there is a royalty cost per tape, which is an increasing function of total industry output \( Q \): \( r = 0.002Q \). 2. **Total Cost for Each Firm**: The total cost for each firm includes both the production cost and the royalty cost. For producing five tapes, the production cost is \( 5 \times 10 = 50 \) dollars. The royalty cost for five tapes is \( 5 \times 0.002Q = 0.01Q \). Therefore, the total cost for each firm is \( 50 + 0.01Q \). 3. **Average Total Cost (ATC)**: The average total cost per tape for each firm is calculated as: \[ \text{ATC} = \frac{50 + 0.01Q}{5} = 10 + 0.002Q \] 4. **Long-Run Equilibrium Condition**: In long-run equilibrium, the price \( P \) must equal the average total cost (ATC). Thus, we have: \[ P = 10 + 0.002Q \] 5. **Demand Function**: The demand function is given by: \[ Q = 1050 - 50P \] 6. **Solving the System of Equations**: - Substitute \( P = 10 + 0.002Q \) into the demand function: \[ Q = 1050 - 50(10 + 0.002Q) \] - Simplify the equation: \[ Q = 1050 - 500 - 0.1Q \] \[ Q + 0.1Q = 550 \] \[ 1.1Q = 550 \] \[ Q = \frac{550}{1.1} = 500 \] - Substitute \( Q = 500 \) back into the price equation: \[ P = 10 + 0.002 \times 500 = 10 + 1 = 11 \] Thus, the equilibrium price of copied tapes is \(\boxed{11}\). |
11 |
| gpto1 | To find the equilibrium price of copied tapes in this perfectly competitive market, we'll use the following steps: **1. Understand the Components of Total Cost per Tape:** Each firm has an average cost (excluding royalties) of **\$10** per tape. Additionally, there is a royalty cost per tape, which is a function of total industry output \( Q \): \[ \text{Royalty per tape} = r = 0.002Q \] Thus, the total cost per tape for a firm is: \[ \text{Total Cost per tape} = \text{Average Cost} + \text{Royalty per tape} = 10 + r \] **2. Long-Run Equilibrium Condition in Perfect Competition:** In the long-run equilibrium of a perfectly competitive market, firms make zero economic profit, which implies: \[ P = \text{Minimum Average Total Cost} \] Including the royalty, this becomes: \[ P = 10 + r \] **3. Relate Price to Total Quantity Demanded:** The market demand function is given by: \[ Q = D(P) = 1,050 - 50P \] **4. Express Price in Terms of Quantity:** From the equilibrium condition, we have: \[ P = 10 + 0.002Q \] This equation relates price to the total industry output \( Q \). **5. Substitute Price Back into the Demand Function:** Plugging the expression for \( P \) into the demand function: \[ Q = 1,050 - 50P = 1,050 - 50(10 + 0.002Q) \] **6. Solve for Q:** Simplify the equation: \[ Q = 1,050 - 500 - 0.1Q \\ Q + 0.1Q = 550 \\ 1.1Q = 550 \\ Q = \frac{550}{1.1} \\ Q = 500 \] **7. Solve for P:** Now use \( Q \) to find \( P \): \[ P = 10 + 0.002Q = 10 + 0.002 \times 500 = 10 + 1 = 11 \] **Answer:** The equilibrium price of copied tapes is **\$11** per tape. |
11 |
| GPT-4o | To find the equilibrium price of copied tapes in a perfectly competitive industry in long-run equilibrium, we need to understand the relationship between costs, prices, and output. ### Step-by-Step Solution: 1. **Total Cost Function**: - Each firm can copy 5 tapes per day at an average cost of $10 per tape. - The total cost for each firm to copy 5 tapes (excluding royalties) is \(5 \times 10 = \$50\). - The per-film royalty rate (r) is \(r = 0.002Q\), where Q is the total industry output. 2. **Long-Run Equilibrium Condition**: - In a perfectly competitive market, firms earn zero economic profit in the long run. - This means the price (P) must equal the long-run average cost (LAC). 3. **Calculate the Long-Run Average Cost**: - The long-run average cost per tape consists of the average production cost ($10 per tape) plus the royalty per tape. - Given that each firm produces 5 tapes and the total royalty payment is \(0.002Q\): \[ \text{Royalty per tape} = \frac{0.002Q}{Q} = 0.002Q \div Q = 0.002. \] - Therefore, the royalty per tape depends on the total industry output Q. 4. **Equilibrium Price (P)**: - In long-run equilibrium, price equals long-run average cost: \[ P = \text{LAC} = \$10 + r = \$10 + 0.002Q. \] 5. **Demand Function**: - The demand function is given by \(Q = D(P) = 1,050 - 50P\). 6. **Solve for Equilibrium Price and Quantity**: - Substitute \(P = \$10 + 0.002Q\) into the demand function: \[ Q = 1,050 - 50(\$10 + 0.002Q). \] - Simplify the equation: \[ Q = 1,050 - 500 - 0.1Q. \] - Combine like terms: \[ Q + 0.1Q = 550. \] \[ 1.1Q = 550. \] \[ Q = \frac{550}{1.1} = 500. \] 7. **Find the Equilibrium Price**: - Substitute \(Q = 500\) back into the price equation: \[ P = \$10 + 0.002Q. \] \[ P = \$10 + 0.002 \times 500. \] \[ P = \$10 + \$1 = \$11. \] ### Conclusion: The equilibrium price of copied tapes in the long-run equilibrium is **$11** per tape. |
11 |
| llama3.1-70B-Instruct | To find the equilibrium price of copied tapes in the perfectly competitive videotape-copying industry, we need to follow these steps: ## Step 1: Determine the Total Revenue (TR) and Total Cost (TC) for a Firm Given that each firm can copy 5 tapes per day at an average cost of $10 per tape, the total cost (TC) for a firm to produce Q tapes is TC = 10Q + rQ, where r is the royalty rate per tape. However, since each firm produces 5 tapes, we'll adjust our equations to reflect industry output and then solve for the price. ## Step 2: Express the Royalty Rate as a Function of Industry Output The royalty rate (r) is given as r = 0.002Q, where Q is the total industry output. ## Step 3: Determine the Industry Supply Curve In a perfectly competitive market, the supply curve is determined by the marginal cost (MC) of production. Since the average cost (AC) is $10 per tape plus the royalty rate, the marginal cost (MC) for each firm is the derivative of the total cost with respect to output, which in this case, given the linear nature of costs, is the same as the average cost. However, we need to express this in terms of industry output and price. ## Step 4: Express the Demand Curve The demand curve is given by Q = D(P) = 1,050 - 50P. ## Step 5: Find the Equilibrium Price and Quantity To find the equilibrium price and quantity, we need to set the demand curve equal to the supply curve. However, since the supply curve is not directly given, we'll use the fact that in long-run equilibrium, price equals marginal cost (P = MC). Given that MC = AC = 10 + r, and r = 0.002Q, we substitute r into the MC equation and set it equal to P. ## Step 6: Solve for Equilibrium Price Given P = MC = 10 + 0.002Q, and Q = 1,050 - 50P, we substitute Q into the MC equation to get P = 10 + 0.002(1,050 - 50P). Solving for P gives us the equilibrium price. ## Step 7: Perform the Algebra P = 10 + 0.002(1,050 - 50P) P = 10 + 2.1 - 0.1P P + 0.1P = 12.1 1.1P = 12.1 P = 12.1 / 1.1 P = 11 The final answer is: $11 |
11 |
| llama3.1-405B-Instruct | To find the equilibrium price of copied tapes in the perfectly competitive videotape-copying industry, we need to follow these steps: 1. **Determine the total cost function for a firm**: The average cost per tape is $10, and each firm can copy 5 tapes per day. Therefore, the total cost (TC) for a firm is the sum of the cost of copying tapes and the royalty payment. The cost of copying tapes per day is $10 * 5 = $50. However, to express this in terms of the industry's total output (Q), we need to consider the royalty payment as well. 2. **Express the royalty payment in terms of Q**: The royalty rate (r) is given by r = 0.002Q. This rate is per tape, so the total royalty payment for the industry is r * Q = 0.002Q * Q = 0.002Q^2. 3. **Find the total revenue (TR) for the industry**: The demand function is Q = 1,050 - 50P. To find the total revenue, we express P in terms of Q and then multiply by Q. Rearranging the demand function gives P = (1,050 - Q) / 50. Therefore, TR = P * Q = ((1,050 - Q) / 50) * Q. 4. **Set the marginal revenue (MR) equal to the marginal cost (MC) to find the equilibrium output (Q)**: In a perfectly competitive market, the price (P) equals the marginal cost (MC). The MC here includes the cost of copying tapes plus the marginal increase in the royalty payment. The MC of copying tapes is $10 (since the cost is constant), and the marginal increase in the royalty payment is the derivative of the total royalty payment with respect to Q, which is d(0.002Q^2)/dQ = 0.004Q. 5. **Solve for Q**: Setting MR = MC, we equate the price (from the demand function) to the MC (including the royalty). So, (1,050 - Q) / 50 = 10 + 0.004Q. Solving this equation for Q gives us the equilibrium output. 6. **Calculate the equilibrium price (P)**: Once we have Q, we can substitute it back into the demand function to find the equilibrium price. Let's solve the equation from step 5: (1,050 - Q) / 50 = 10 + 0.004Q Multiplying both sides by 50 to get rid of the fraction: 1,050 - Q = 500 + 0.2Q Bringing terms involving Q to one side and constants to the other: 1,050 - 500 = 0.2Q + Q 550 = 1.2Q Dividing both sides by 1.2: Q = 550 / 1.2 Q = 458.33 Now, substituting Q back into the demand function to find P: P = (1,050 - 458.33) / 50 P = 591.67 / 50 P = 11.8334 Therefore, the equilibrium price of copied tapes is approximately $11.83. |
11 |
| DeepSeek-Coder-V2 | To find the equilibrium price of copied tapes in the perfectly competitive videotape-copying industry, we need to follow these steps: 1. **Determine the total cost per tape for each firm:** - The average cost of copying a tape is $10. - The royalty rate (r) is an increasing function of total industry output (Q): r = 0.002Q. - Therefore, the total cost per tape for each firm is the sum of the average cost and the royalty rate: $10 + 0.002Q. 2. **Set up the profit maximization condition:** - In a perfectly competitive market, firms will produce at the point where the price (P) equals the marginal cost (MC). - The marginal cost in this case is the total cost per tape: MC = $10 + 0.002Q. - Therefore, the equilibrium condition is P = $10 + 0.002Q. 3. **Use the demand function to find the equilibrium price:** - The demand function is given by Q = D(P) = 1,050 - 50P. - Substitute the demand function into the equilibrium condition: P = $10 + 0.002(1,050 - 50P). 4. **Solve for the equilibrium price (P):** - Start by simplifying the equation: P = $10 + 0.002(1,050 - 50P). - Distribute the 0.002: P = $10 + 2.1 - 0.1P. - Combine like terms: P + 0.1P = $10 + 2.1. - Simplify further: 1.1P = $12.1. - Solve for P: P = $12.1 / 1.1 = $11. Therefore, the equilibrium price of copied tapes in the long-run equilibrium is $\boxed{11}$. |
11 |
| Qwen2-72B-Instruct | To find the equilibrium price of copied tapes in a long-run equilibrium, we need to find the point where the quantity demanded equals the quantity supplied. First, let's find the supply function. The total cost of copying a tape is the sum of the average cost of copying and the royalty fee per tape. The average cost of copying a tape is $10, and the royalty fee per tape is r = 0.002Q. Therefore, the total cost per tape is: Total cost per tape = $10 + 0.002Q In the long-run equilibrium, the price of copied tapes will be equal to the total cost per tape. Therefore, we can set the price equal to the total cost per tape: P = $10 + 0.002Q Now we need to find the quantity supplied at this price. To do this, we need to solve for Q in terms of P: P = $10 + 0.002Q P - $10 = 0.002Q Q = (P - $10) / 0.002 This is the supply function. Next, we need to find the demand function. We are given that demand is given by Q = D(P) = 1,050 - 50P. To find the equilibrium price, we need to set the quantity demanded equal to the quantity supplied: D(P) = S(P) 1,050 - 50P = (P - $10) / 0.002 Solving for P, we get: 1,050 - 50P = 500P - 500 550P = 1,550 P = $2.82 Therefore, the equilibrium price of copied tapes in the long-run equilibrium is $2.82. Note: This solution assumes that the firms in the industry are price takers, meaning they have no market power and take the market price as given. It also assumes that there are no barriers to entry or exit in the industry, so that firms can enter or exit the industry freely until economic profits are zero. |
11 |
| baichuan4 | To find the equilibrium price of copied tapes in a perfectly competitive market, we need to equate the marginal cost (MC) of production to the market price (P). In this case, the average cost (AC) is given as $10 per tape, and since it's a constant, the marginal cost is also $10 per tape. The demand function is given as: \[ Q = D(P) = 1,050 - 50P \] The royalty cost per tape is: \[ r = 0.002Q \] Since the industry is perfectly competitive, firms are price takers. Therefore, the price (P) is equal to the average total cost (ATC), which includes both the average copying cost and the average royalty cost. The ATC can be represented as: \[ ATC = AC + r \] \[ ATC = 10 + 0.002Q \] Substituting the demand function for Q gives us: \[ ATC = 10 + 0.002(1,050 - 50P) \] \[ ATC = 10 + 2.1 - 0.1P \] \[ ATC = 12.1 - 0.1P \] Setting P equal to ATC to find the equilibrium price: \[ P = 12.1 - 0.1P \] Solving for P: \[ 1.1P = 12.1 \] \[ P = \frac{12.1}{1.1} \] \[ P = 11 \] Therefore, the equilibrium price of copied tapes in the long-run equilibrium of a perfectly competitive market would be $11. |
11 |
| doubao | In a perfectly competitive long-run equilibrium, price equals average cost. The average cost of copying a tape is $10. The total cost for each firm also includes the royalty. The royalty per tape is 0.002Q. Since each firm copies 5 tapes per day, the total output of the industry (Q) is 5 times the number of firms. Let's say there are n firms, then Q = 5n. So the royalty per tape is 0.002 * 5n = 0.01n. The total cost per tape is then $10 + 0.01n. In long-run equilibrium, price = average cost, so P = $10 + 0.01n. We also know that demand is Q = 1050 - 50P. Substituting P = $10 + 0.01n into the demand equation gives: Q = 1050 - 50($10 + 0.01n) Q = 1050 - 500 - 0.5n Q = 550 - 0.5n But Q = 5n, so 5n = 550 - 0.5n 5.5n = 550 n = 100. So the equilibrium price P = $10 + 0.01 * 100 = $11. |
11 |
| minimax | To solve this problem, we need to find the equilibrium price (P) in the perfectly competitive market where the demand equals the supply. Since the industry is in long-run equilibrium, firms are making zero economic profit, and price equals average cost. First, let's find the supply function. Each firm can copy five tapes per day at an average cost of $10 per tape. However, there is also a per-film royalty rate (r) that is an increasing function of total industry output (Q). The royalty rate is given by r = 0.002Q. The total cost per tape for each firm is the sum of the average cost and the royalty rate: Total cost per tape = Average cost + Royalty rate Total cost per tape = $10 + 0.002Q Since each firm can copy five tapes per day, the marginal cost (MC) for each additional tape is the same as the total cost per tape: MC = Total cost per tape = $10 + 0.002Q In a perfectly competitive market, the supply curve is the marginal cost curve above the average variable cost. Since the average cost is given as $10, and this is also the average variable cost (because the average cost is not specified to include fixed costs), the supply curve will be the marginal cost curve starting from $10. Now, let's find the demand function. The demand is given by Q = D(P) = 1,050 - 50P. In long-run equilibrium, the price equals the marginal cost, so we set the supply equal to the demand: 1,050 - 50P = $10 + 0.002Q Now we need to solve for P. However, we have two variables, P and Q, so we need to express one in terms of the other. We can use the demand function to express Q in terms of P: Q = 1,050 - 50P Now we substitute this expression for Q into the supply equation: 1,050 - 50P = $10 + 0.002(1,050 - 50P) Now we solve for P: 1,050 - 50P = $10 + 2.1 - 0.1P 1,050 - 50P = $12.1 - 0.1P 1,050 - 12.1 = 50P - 0.1P 1,037.9 = 49.9P P = 1,037.9 / 49.9 P ≈ $20.80 So, the equilibrium price of copied tapes in the perfectly competitive market is approximately $20.80. |
11 |
| kimi | To find the equilibrium price of copied tapes in a perfectly competitive industry, we need to consider both the demand and the cost conditions. In a perfectly competitive market, firms are price takers, meaning they cannot influence the market price. The equilibrium price will be where the market demand equals the market supply. First, let's find the market supply curve. Since each firm has an average cost of $10 per tape, this is the marginal cost (MC) for each firm. The royalty rate (r) is an increasing function of total industry output (Q), given by r = 0.002Q. The total cost per tape for a firm is the average cost plus the royalty cost per tape, which is r times the number of tapes copied per day. The total cost per tape for a firm is: \[ \text{Total Cost per Tape} = \text{Average Cost per Tape} + \text{Royalty Cost per Tape} \] \[ \text{Total Cost per Tape} = 10 + (0.002Q \times 5) \] \[ \text{Total Cost per Tape} = 10 + 0.01Q \] Since the industry is in long-run equilibrium, the price will equal the marginal cost, which is also equal to the average cost in a perfectly competitive market. So, the price (P) will be: \[ P = 10 + 0.01Q \] Now, let's find the market demand curve. The demand function is given by: \[ Q = 1,050 - 50P \] In equilibrium, the quantity supplied equals the quantity demanded, so we set the supply equation equal to the demand equation: \[ 1,050 - 50P = P \times (10 + 0.01Q) \] Since Q is the same on both sides, we can simplify the equation: \[ 1,050 - 50P = 10P + 0.01Q^2 \] Now, we need to solve for P. Rearrange the equation to isolate P: \[ 1,050 = 60P + 0.01Q^2 \] Since Q = D(P), we can substitute Q back in: \[ 1,050 = 60P + 0.01(1,050 - 50P)^2 \] This is a quadratic equation in terms of P. To solve for P, we can expand and simplify the equation: \[ 1,050 = 60P + 0.01(1,102,500 - 10,500P + 2,500P^2) \] \[ 1,050 = 60P + 11,025 - 105P + 25P^2 \] \[ 0 = 25P^2 - 45P + 105 \] Divide through by 25 to simplify: \[ 0 = P^2 - 1.8P + 4.2 \] Now, we can solve for P using the quadratic formula: \[ P = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] \[ P = \frac{1.8 \pm \sqrt{(1.8)^2 - 4 \times 1 \times 4.2}}{2 \times 1} \] \[ P = \frac{1.8 \pm \sqrt{3.24 - 16.8}}{2} \] Since the term under the square root is negative, we have a complex solution, which is not possible for a price. This suggests that there might be a mistake in the setup of the problem or the equations. Let's recheck the steps. The mistake is in the supply equation. The correct supply equation should be: \[ Q = P \times (10 + 0.01Q) \] Now, let's solve for P again: \[ 1,050 - 50P = P \times (10 + 0.01(1,050 - 50P)) \] \[ 1,050 - 50P = 10P + 0.01P^2 - 0.5P \] \[ 1,050 = 15.5P + 0.01P^2 \] Now, we have a quadratic equation in terms of P: \[ 0.01P^2 + 15.5P - 1,050 = 0 \] Using the quadratic formula: \[ P = \frac{-15.5 \pm \sqrt{15.5^2 + 4 \times 0.01 \times 1,050}}{2 \times 0.01} \ |
11 |
| wenxinyiyan | To find the equilibrium price of copied tapes in the long-run equilibrium of this perfectly competitive videotape-copying industry, we need to consider the industry's supply and demand. First, let's look at the cost structure for firms in this industry. Each firm can copy five tapes per day at an average cost of $10 per tape. However, they must also pay a royalty to film studios, which is a function of total industry output (Q). The per-film royalty rate (r) is given by r = 0.002Q. Now, let's consider the demand side. Demand is given by Q = D(P) = 1,050 - 50P, where P is the price of copied tapes. In a perfectly competitive market, firms will produce at the point where marginal cost equals marginal revenue. Since the market is in long-run equilibrium, firms will also be producing at the point where average cost equals the market price. To find the equilibrium price, we need to determine the point at which the supply and demand curves intersect. However, given the information provided, we don't have a direct supply curve. Instead, we know that firms will produce up to the point where the marginal cost of production (including royalties) equals the market price. Let's denote the equilibrium price as P_e and the equilibrium quantity as Q_e. At equilibrium, the average cost of production (including royalties) will equal P_e. Therefore, we can write: AC = P_e Where AC is the average cost of production, including royalties. We know that the average cost per tape without royalties is $10, so we can write the average cost including royalties as: AC = 10 + r Substituting the royalty rate function, we get: AC = 10 + 0.002Q_e Since AC = P_e at equilibrium, we can write: P_e = 10 + 0.002Q_e Now, we need to find Q_e using the demand function. Substituting P_e into the demand function, we get: Q_e = 1,050 - 50(10 + 0.002Q_e) Solving for Q_e, we find that: Q_e = 1,000 Substituting Q_e back into the equation for P_e, we get: P_e = 10 + 0.002(1,000) P_e = 12 Therefore, the equilibrium price of copied tapes in the long-run equilibrium of this perfectly competitive videotape-copying industry will be $12. |
11 |
| zhipu | To find the equilibrium price of copied tapes in a perfectly competitive industry in long-run equilibrium, we need to set the price equal to the long-run average cost (LRAC) for firms, because in the long run, firms in a perfectly competitive market make zero economic profit. Given data: - Each firm can copy five tapes per day at an average cost of $10 per tape. - The per-film royalty rate (r) is an increasing function of total industry output (Q): r = 0.002Q. - Demand is given by Q = D(P) = 1,050 - 50P. First, let's express the total cost for a firm: The average cost of copying a tape is $10, so for five tapes, the cost without the royalty would be 5 * $10 = $50. Now, we need to include the royalty cost. The royalty per tape for a firm will be the royalty rate times the number of tapes copied, which is 5r. Since r = 0.002Q, the royalty per firm becomes 5 * 0.002Q = 0.01Q. Thus, the total cost for a firm to copy five tapes, including the royalty, is: Total cost = $50 + 0.01Q. The average cost (AC) per tape, including the royalty, is then: AC = Total cost / Number of tapes AC = ($50 + 0.01Q) / 5 AC = $10 + 0.002Q. In long-run equilibrium, the price (P) will equal the average cost (AC), so: P = $10 + 0.002Q. Now, we need to find the equilibrium quantity (Q) using the demand function: Q = D(P) = 1,050 - 50P. Substitute the expression for P from the AC equation into the demand equation: Q = 1,050 - 50($10 + 0.002Q) Q = 1,050 - 500 - 0.1Q Q + 0.1Q = 550 1.1Q = 550 Q = 550 / 1.1 Q = 500. Now that we have the equilibrium quantity (Q), we can find the equilibrium price (P) by plugging Q back into the equation where P equals the average cost: P = $10 + 0.002Q P = $10 + 0.002(500) P = $10 + $1 P = $11. Therefore, the equilibrium price of copied tapes in the long-run equilibrium will be $11. |
11 |
| gpto1 | To find the equilibrium price of copied tapes in this perfectly competitive market, we'll use the following steps: **1. Understand the Components of Total Cost per Tape:** Each firm has an average cost (excluding royalties) of **\$10** per tape. Additionally, there is a royalty cost per tape, which is a function of total industry output \( Q \): \[ \text{Royalty per tape} = r = 0.002Q \] Thus, the total cost per tape for a firm is: \[ \text{Total Cost per tape} = \text{Average Cost} + \text{Royalty per tape} = 10 + r \] **2. Long-Run Equilibrium Condition in Perfect Competition:** In the long-run equilibrium of a perfectly competitive market, firms make zero economic profit, which implies: \[ P = \text{Minimum Average Total Cost} \] Including the royalty, this becomes: \[ P = 10 + r \] **3. Relate Price to Total Quantity Demanded:** The market demand function is given by: \[ Q = D(P) = 1,050 - 50P \] **4. Express Price in Terms of Quantity:** From the equilibrium condition, we have: \[ P = 10 + 0.002Q \] This equation relates price to the total industry output \( Q \). **5. Substitute Price Back into the Demand Function:** Plugging the expression for \( P \) into the demand function: \[ Q = 1,050 - 50P = 1,050 - 50(10 + 0.002Q) \] **6. Solve for Q:** Simplify the equation: \[ Q = 1,050 - 500 - 0.1Q \\ Q + 0.1Q = 550 \\ 1.1Q = 550 \\ Q = \frac{550}{1.1} \\ Q = 500 \] **7. Solve for P:** Now use \( Q \) to find \( P \): \[ P = 10 + 0.002Q = 10 + 0.002 \times 500 = 10 + 1 = 11 \] **Answer:** The equilibrium price of copied tapes is **\$11** per tape. |
11 |
| deepseekv3 | To find the equilibrium price of copied tapes in the perfectly competitive videotape-copying industry, we need to consider the long-run equilibrium conditions. Here's a step-by-step solution: --- ### **Step 1: Understand the Costs** Each firm can copy **5 tapes per day** at an **average cost (AC)** of **$10 per tape**. This means the **total cost (TC)** for a firm producing \( q = 5 \) tapes is: \[ TC = AC \times q = 10 \times 5 = 50 \] In addition to the production cost, each firm must pay a **royalty** to film studios. The royalty rate \( r \) is a function of total industry output \( Q \): \[ r = 0.002Q \] Thus, the **total royalty cost** for a firm producing \( q \) tapes is: \[ \text{Royalty Cost} = r \times q = 0.002Q \times q \] --- ### **Step 2: Long-Run Equilibrium Condition** In long-run equilibrium, firms in a perfectly competitive market earn **zero economic profit**. This means the **price (P)** equals the **minimum average total cost (ATC)**. The **ATC** for a firm includes both the production cost and the royalty cost: \[ ATC = \text{Average Production Cost} + \text{Average Royalty Cost} \] \[ ATC = 10 + 0.002Q \] In equilibrium, the price \( P \) equals the ATC: \[ P = 10 + 0.002Q \] --- ### **Step 3: Demand Function** The demand function is given by: \[ Q = 1050 - 50P \] --- ### **Step 4: Solve for Equilibrium** Substitute the equilibrium condition \( P = 10 + 0.002Q \) into the demand function: \[ Q = 1050 - 50(10 + 0.002Q) \] \[ Q = 1050 - 500 - 0.1Q \] \[ Q + 0.1Q = 550 \] \[ 1.1Q = 550 \] \[ Q = 500 \] Now substitute \( Q = 500 \) into the equilibrium condition to find \( P \): \[ P = 10 + 0.002(500) \] \[ P = 10 + 1 \] \[ P = 11 \] --- ### **Final Answer** The equilibrium price of copied tapes is **$11**. |
11 |