金声玉亮2.0模型评测报告

问题

You are supposed to provide a solution to a given problem.

Problem:
Carl the clothier owns a large garment factory on an isolated island. Carl's factory is the only source of employment for most of the islanders, and thus Carl acts as a monopsonist. The supply curve for garment workers is given by l = 80w, where l is the number of workers hired and w is their hourly wage. Assume also that Carl's labor demand (marginal revenue product) curve is given by l = 400 - 40MRP_l. How many workers will Carl hire to maximize his profits?
Solution:

答案

模型	模型答案	正确答案
金声玉亮2.0	为了确定Carl雇佣的工人数量以最大化利润，我们需要以下步骤： 1. 确定工资 \( w \) 和工人数量 \( l \) 之间的关系； 2. 确定边际成本（MC）和边际收益（MRP_l）之间的关系； 3. 通过等式 \( MRP_l = MC \) 确定平衡点，从而确定工人数量。 ### 1. 确定工资 \( w \) 和工人数量 \( l \) 之间的关系已知供应曲线 \( l = 80w \)，可以将其反转得到工资 \( w \) 与工人数量 \( l \) 之间的关系： \[ w = \frac{l}{80} \] ### 2. 确定边际成本（MC）对于一个垄断买方（单一雇主）来说，边际成本（MC）不仅包括支付给新增工人的工资，还包括提高所有现有工人的工资成本。总成本（TC）函数为： \[ TC = w \cdot l \] 将 \( w \) 替换为 \( \frac{l}{80} \)： \[ TC = \frac{l}{80} \cdot l = \frac{l^2}{80} \] 边际成本（MC）是总成本对工人数量 \( l \) 的导数： \[ MC = \frac{d(TC)}{d(l)} = \frac{d}{d(l)} \left(\frac{l^2}{80}\right) = \frac{l}{40} \] ### 3. 确定边际收益（MRP_l）已知需求曲线（边际收益产品曲线）为： \[ l = 400 - 40MRP_l \] 将其反转得到边际收益产品（MRP_l）与工人数量 \( l \) 之间的关系： \[ MRP_l = \frac{400 - l}{40} \] ### 4. 通过等式 \( MRP_l = MC \) 确定平衡点将边际收益产品（MRP_l）与边际成本（MC）相等： \[ \frac{400 - l}{40} = \frac{l}{40} \] 解这个方程： \[ 400 - l = l \] \[ 400 = 2l \] \[ l = 200 \] ### 5. 确定工资 \( w \) 将 \( l = 200 \) 代入供应曲线 \( l = 80w \)： \[ 200 = 80w \] \[ w = \frac{200}{80} = 2.5 \] ### 结论 Carl将雇佣200名工人，并支付每小时2.5元的工资以最大化他的利润。	200
deepseekr1	Carl is a monopsonist, meaning he is the only employer on the island and faces the entire market supply curve for labor. The supply curve for garment workers is given by \( l = 80w \), which can be rearranged to \( w = \frac{l}{80} \). The labor demand (marginal revenue product) curve is given by \( l = 400 - 40MRP_l \), which can be rearranged to \( MRP_l = 10 - \frac{l}{40} \). To find the profit-maximizing number of workers, we need to determine the marginal cost of labor (MCL) and set it equal to the marginal revenue product of labor (MRP_l). 1. Total Labor Cost (TLC) is given by: \[ TLC = w \cdot l = \left(\frac{l}{80}\right) \cdot l = \frac{l^2}{80} \] 2. Marginal Cost of Labor (MCL) is the derivative of TLC with respect to \( l \): \[ MCL = \frac{d(TLC)}{dl} = \frac{2l}{80} = \frac{l}{40} \] 3. Set MCL equal to MRP_l to find the profit-maximizing number of workers: \[ \frac{l}{40} = 10 - \frac{l}{40} \] 4. Solving for \( l \): \[ \frac{l}{40} + \frac{l}{40} = 10 \\ \frac{2l}{40} = 10 \\ \frac{l}{20} = 10 \\ l = 200 \] Thus, Carl will hire \boxed{200} workers to maximize his profits.	200
gpto1	To determine the number of workers Carl will hire to maximize his profits as a monopsonist, we'll follow these steps: 1. Understand the Labor Supply Curve: The supply curve for labor is given by: \[ l = 80w \] This can be rearranged to express the wage rate (\(w\)) as a function of labor (\(l\)): \[ w = \frac{l}{80} \] 2. Calculate the Total Expenditure (TE) on Labor: The total expenditure is the total wage bill paid by Carl: \[ \text{TE} = w \times l = \frac{l}{80} \times l = \frac{l^2}{80} \] 3. Find the Marginal Expenditure (ME) on Labor: Marginal Expenditure is the additional cost of hiring one more worker. It's the derivative of the total expenditure with respect to labor: \[ \text{ME} = \frac{d(\text{TE})}{dl} = \frac{d}{dl}\left(\frac{l^2}{80}\right) = \frac{2l}{80} = \frac{l}{40} \] Alternatively, using the definition of ME for a monopsonist: \[ \text{ME} = w + l \frac{dw}{dl} \] Since: \[ \frac{dw}{dl} = \frac{1}{80} \] Substitute back: \[ \text{ME} = \frac{l}{80} + l \times \frac{1}{80} = \frac{l}{80} + \frac{l}{80} = \frac{2l}{80} = \frac{l}{40} \] 4. Understand the Marginal Revenue Product (MRP) Curve: The marginal revenue product of labor is given by: \[ l = 400 - 40 \times \text{MRP}_l \] Rearranged, it becomes: \[ \text{MRP}_l = \frac{400 - l}{40} \] 5. Set ME Equal to MRP to Find the Profit-Maximizing Quantity of Labor: A monopsonist hires workers up to the point where ME equals MRP: \[ \text{ME} = \text{MRP}_l \] Substitute the expressions we found: \[ \frac{l}{40} = \frac{400 - l}{40} \] Multiply both sides by 40 to eliminate denominators: \[ l = 400 - l \] \[ 2l = 400 \] \[ l = 200 \] Conclusion: Carl will hire 200 workers to maximize his profits. Answer: 200	200
GPT-4o	To determine the number of workers Carl will hire to maximize his profits, we need to follow these steps: 1. Understand the supply and demand curves: - The supply curve for workers is given by: \[ l = 80w \] - The labor demand (marginal revenue product) curve is given by: \[ l = 400 - 40MRP_l \] 2. Express the wage in terms of the number of workers: - From the supply curve: \[ l = 80w \implies w = \frac{l}{80} \] 3. Determine the Marginal Factor Cost (MFC): - The MFC is the additional cost of hiring one more worker. Since Carl is a monopsonist, the MFC is not simply the wage rate but the derivative of the total cost of labor with respect to the number of workers. - The total cost (TC) of labor is: \[ TC = w \cdot l = \left(\frac{l}{80}\right) \cdot l = \frac{l^2}{80} \] - The MFC is the derivative of TC with respect to \( l \): \[ MFC = \frac{d(TC)}{dl} = \frac{d}{dl} \left( \frac{l^2}{80} \right) = \frac{2l}{80} = \frac{l}{40} \] 4. Determine the Marginal Revenue Product (MRP): - From the demand curve: \[ l = 400 - 40MRP_l \implies MRP_l = \frac{400 - l}{40} \] 5. Set MRP equal to MFC to maximize profits: - For profit maximization in a monopsony, the number of workers is determined where MRP equals MFC: \[ \frac{400 - l}{40} = \frac{l}{40} \] - Simplify and solve for \( l \): \[ 400 - l = l \implies 400 = 2l \implies l = 200 \] So, Carl will hire 200 workers to maximize his profits.	200
llama3.1-70B-Instruct	To maximize profits, Carl needs to equate the marginal revenue product (MRP) of labor with the wage (w) he pays to the workers. This is because the MRP represents the additional revenue generated by hiring one more worker, and Carl will want to hire workers until the additional revenue generated equals the additional cost (the wage). Given the labor demand (MRP) curve: l = 400 - 40MRP_l And the supply curve: l = 80w We can set MRP_l = w, since Carl will hire workers until the MRP equals the wage. Substituting l from the supply curve into the labor demand curve, we get: 80w = 400 - 40w Simplifying the equation: 120w = 400 w = 400/120 w = 3.33 Now that we have the wage, we can find the number of workers Carl will hire by substituting w into the supply curve: l = 80w l = 80(3.33) l = 266.4 Since we can't have a fraction of a worker, we'll round down to the nearest whole number. Carl will hire approximately 266 workers to maximize his profits.	200
llama3.1-405B-Instruct	To find the number of workers Carl will hire to maximize his profits, we need to find the point where the marginal revenue product (MRP) of labor equals the marginal cost (MC) of labor. First, let's find the marginal cost of labor. The supply curve for garment workers is given by l = 80w, where l is the number of workers hired and w is their hourly wage. To find the marginal cost of labor, we need to find the derivative of the supply curve with respect to the number of workers hired (l). MC = d(l)/d(w) = 80 So, the marginal cost of labor is $80 per worker. Next, let's find the marginal revenue product of labor. The labor demand (marginal revenue product) curve is given by l = 400 - 40MRP_l. To find the marginal revenue product of labor, we need to find the derivative of the labor demand curve with respect to the number of workers hired (l). MRP_l = d(l)/d(MRP_l) = -40 So, the marginal revenue product of labor is -$40 per worker. Now, we need to find the point where the marginal revenue product of labor equals the marginal cost of labor. MRP_l = MC -40 = 80 Solving for l, we get: l = 400 - 40(80) l = 400 - 3200 l = -2800 Since the number of workers cannot be negative, this means that Carl will not hire any workers. This is because the marginal cost of labor ($80) is greater than the marginal revenue product of labor (-$40), which means that hiring workers would result in a loss for Carl. Therefore, Carl will not hire any workers to maximize his profits.	200
DeepSeek-Coder-V2	To solve the problem of how many workers Carl the clothier will hire to maximize his profits, we need to follow these steps: 1. Understand the given equations: - Supply curve for garment workers: \( l = 80w \) - Labor demand (marginal revenue product) curve: \( l = 400 - 40MRP_l \) 2. Express the wage rate \( w \) in terms of the number of workers \( l \) from the supply curve: \[ l = 80w \implies w = \frac{l}{80} \] 3. Express the marginal revenue product \( MRP_l \) in terms of the number of workers \( l \) from the demand curve: \[ l = 400 - 40MRP_l \implies MRP_l = \frac{400 - l}{40} \] 4. Set the marginal revenue product equal to the wage rate to find the equilibrium point where Carl maximizes his profits: \[ MRP_l = w \] Substitute the expressions for \( MRP_l \) and \( w \): \[ \frac{400 - l}{40} = \frac{l}{80} \] 5. Solve the equation for \( l \): \[ \frac{400 - l}{40} = \frac{l}{80} \] Multiply both sides by 80 to clear the denominator: \[ 2(400 - l) = l \] Simplify and solve for \( l \): \[ 800 - 2l = l \] \[ 800 = 3l \] \[ l = \frac{800}{3} \approx 266.67 \] 6. Since the number of workers must be an integer, round to the nearest whole number: \[ l \approx 267 \] Therefore, Carl the clothier will hire approximately 267 workers to maximize his profits.	200
Qwen2-72B-Instruct	To find the number of workers Carl will hire to maximize his profits, we need to equate the marginal revenue product (MRP) of labor to the wage rate (w). Given: - Labor supply curve: l = 80w - Labor demand (MRP) curve: l = 400 - 40MRP_l First, we need to express the wage rate (w) in terms of labor (l) using the labor supply curve: l = 80w w = l/80 Next, we substitute this expression for w into the labor demand equation to get the MRP in terms of l: l = 400 - 40MRP_l MRP_l = (400 - l) / 40 To maximize profits, the MRP of labor must equal the wage rate (w): MRP_l = w (400 - l) / 40 = l/80 Solving for l: 400 - l = 2l 3l = 400 l = 400 / 3 ≈ 133.33 Since the number of workers must be a whole number, Carl will hire approximately 133 workers to maximize his profits. However, in a real-world scenario, the number of workers would likely be rounded to the nearest whole number, either 133 or 134, depending on the specifics of the situation. For the sake of this problem, we can say Carl will hire 133 workers.	200
baichuan4	To maximize his profits, Carl the clothier will hire workers up until the point where the marginal revenue product (MRP) of labor equals the wage rate (w). This is because, as a monopsonist, Carl wants to minimize his labor costs while still producing enough output to meet his revenue goals. The supply curve for garment workers is given by: l = 80w And the labor demand (marginal revenue product) curve is given by: l = 400 - 40MRP_l To find the number of workers Carl will hire to maximize profits, we need to set the wage rate (w) equal to the marginal revenue product of labor (MRP_l) and solve for l. First, let's express w in terms of l using the supply curve equation: w = l / 80 Next, substitute this expression for w into the labor demand equation: l = 400 - 40(l / 80) Now, we can solve for l: l = 400 - (40l / 80) l = 400 - 0.5l 1.5l = 400 l = 400 / 1.5 l = 266.67 Therefore, Carl will hire approximately 266.67 workers to maximize his profits.	200
doubao	To maximize profits, Carl will hire workers up to the point where the marginal revenue product of labor (MRP_l) is equal to the wage (w). Set the supply curve equal to the demand curve: 80w = 400 - 40MRP_l. But MRP_l = w, so substitute that in: 80w = 400 - 40w. 120w = 400. w = 400 / 120 = 10 / 3. Now substitute this wage back into the supply curve to find the number of workers: l = 80 * (10 / 3) = 800 / 3 ≈ 267 workers. So Carl will hire approximately 267 workers to maximize his profits.	200
minimax	To solve this problem, we need to find the equilibrium point where Carl's labor demand equals the supply of labor. This is where the marginal revenue product (MRP) of labor equals the wage rate (w). First, let's express the labor demand curve in terms of wage rate (w). We have the labor demand curve as: l = 400 - 40MRP_l We need to find the MRP_l in terms of w. However, we don't have the direct relationship between MRP_l and w given in the problem. Normally, the MRP curve would be derived from the marginal product of labor (MPL) and the marginal revenue (MR) that the labor helps to produce. Since we don't have the MPL or MR, we can't directly calculate MRP_l. But we can use the given labor supply curve to find the wage rate that equates labor demand and labor supply. The labor supply curve is given as: l = 80w Now, we set the labor demand equal to the labor supply to find the equilibrium number of workers (l) and the corresponding wage rate (w): 400 - 40MRP_l = 80w Since we don't have the MRP_l, we can't solve for w directly. However, we can use the fact that Carl is a monopsonist to understand that he will hire workers up to the point where the wage rate equals the marginal cost of labor, which is the derivative of the total labor cost with respect to the number of workers. The total labor cost (TLC) for Carl is: TLC = w * l Since l = 80w, we can substitute l in the TLC equation: TLC = w * (80w) TLC = 80w^2 The marginal cost of labor (MCL) is the derivative of the total labor cost with respect to l: MCL = d(TLC)/dl But since w is a function of l (l = 80w), we need to use the chain rule: MCL = d(80w^2)/dl MCL = 160w * (dw/dl) Since dw/dl is the inverse of dl/dw, and dl/dw is the derivative of the supply curve, which is 80, we have: dw/dl = 1/80 Substituting this into the MCL equation: MCL = 160w * (1/80) MCL = 2w Carl will hire workers up to the point where the MCL equals the wage rate (w). Therefore, we have: MCL = w 2w = w w = 0 This result doesn't make sense in the context of the problem because it implies that Carl would not pay any wage, which is not realistic. It seems we have a missing piece of information or a misunderstanding in the problem as presented. In a typical monopsony problem, we would have the MRP curve and could equate it to the MCL to find the wage rate and the number of workers hired. However, without the MRP curve or additional information on how MRP relates to w, we cannot solve this problem as it stands. If you have additional information or if there was a mistake in the problem statement, please provide it so we can attempt to solve the problem again.	200
kimi	To solve this problem, we need to find the number of workers Carl will hire to maximize his profits. As a monopsonist, Carl will hire workers up to the point where the value of the marginal product (VMPL) equals the wage rate. First, let's find the labor demand curve for Carl's factory. The labor demand curve is given by: \[ l = 400 - 40MRP_l \] We need to find the value of \( MRP_l \) (marginal revenue product of labor). However, we don't have the full information to calculate \( MRP_l \) directly. Instead, we can use the fact that the marginal revenue product is the additional revenue generated by hiring one more worker. Since we don't have the revenue function, we can't calculate \( MRP_l \) directly. But we do have the supply curve for garment workers, which is given by: \[ l = 80w \] This equation tells us that the number of workers willing to work (the supply of labor) is a function of the wage rate \( w \). To find the number of workers Carl will hire, we need to equate the labor demand to the labor supply: \[ 400 - 40MRP_l = 80w \] Since Carl is a monopsonist, he will hire workers until the wage rate equals the marginal revenue product. However, without the actual value of \( MRP_l \), we can't solve for \( w \) directly. But we can infer that Carl will set the wage rate such that the number of workers he hires is equal to the number of workers willing to work at that wage. To maximize profits, Carl will set the wage rate \( w \) such that the marginal revenue product of labor equals the marginal cost of labor. Since the supply curve is \( l = 80w \), the marginal cost of labor is the wage rate \( w \). Therefore, Carl will hire workers until: \[ MRP_l = w \] Given that we don't have the actual value of \( MRP_l \), we can't find the exact number of workers Carl will hire. However, we can say that Carl will hire the number of workers where the labor demand curve intersects the labor supply curve. If we assume that Carl sets the wage rate such that the marginal revenue product equals the marginal cost (which is the wage rate), then we can solve for the number of workers by equating the labor demand and labor supply equations: \[ 400 - 40MRP_l = 80w \] \[ MRP_l = w \] Substituting \( MRP_l \) for \( w \) in the labor demand equation: \[ 400 - 40w = 80w \] \[ 400 = 120w \] \[ w = \frac{400}{120} \] \[ w = \frac{10}{3} \text{ or approximately } 3.33 \] Now, we can find the number of workers by plugging \( w \) back into the supply equation: \[ l = 80w \] \[ l = 80 \times \frac{10}{3} \] \[ l = \frac{800}{3} \] \[ l \approx 266.67 \] Since Carl can't hire a fraction of a worker, he will round down to the nearest whole number, which means he will hire approximately 266 workers to maximize his profits.	200
wenxinyiyan	To determine the number of workers Carl will hire to maximize his profits, we need to set the supply curve equal to the labor demand curve and solve for the wage (w) and then use that to find the number of workers (l). Given: Supply curve: l = 80w Labor demand curve: l = 400 - 40MRP_l Since the supply curve represents the workers' willingness to work at a given wage and the labor demand curve represents Carl's willingness to hire workers at a given marginal revenue product (MRP_l), we can equate the two to find the equilibrium point where both parties are satisfied. 80w = 400 - 40MRP_l However, we are given that MRP_l is equal to w (the hourly wage), so we can substitute w for MRP_l: 80w = 400 - 40w Now, we solve for w: 120w = 400 w = 400 / 120 w = 10/3 Now that we have the equilibrium wage, we can find the number of workers Carl will hire by plugging this wage back into either the supply curve or the labor demand curve (since they are equal at equilibrium): l = 80 * (10/3) l = 800/3 l ≈ 267 So, Carl will hire approximately 267 workers to maximize his profits.	200
zhipu	To determine the number of workers Carl will hire to maximize his profits, we need to find the point where his marginal cost of labor (MCL) equals the marginal revenue product of labor (MRP_L). In a monopsonistic market, the firm (Carl, in this case) faces an upward-sloping supply curve, which means that to hire more workers, Carl must pay a higher wage, not just to the new workers but also to the existing ones. Let's start by analyzing the given supply and demand curves: 1. The supply curve for garment workers is given by: \( l = 80w \) where \( l \) is the number of workers and \( w \) is the hourly wage. Rearranging this equation to solve for \( w \) gives us the marginal cost of labor (MCL), since each additional worker costs more than the previous one due to the higher wage that must be paid to all workers: \( w = \frac{l}{80} \) 2. The labor demand curve (marginal revenue product of labor, MRP_L) is given by: \( l = 400 - 40MRP_L \) Rearranging this equation to solve for MRP_L gives us: \( MRP_L = \frac{400 - l}{40} \) To maximize profits, Carl will hire workers up to the point where MRP_L equals MCL. Setting the two equations equal to each other gives us: \( \frac{400 - l}{40} = \frac{l}{80} \) Now, let's solve for \( l \): \( 80(400 - l) = 40l \) \( 32000 - 80l = 40l \) \( 32000 = 120l \) \( l = \frac{32000}{120} \) \( l = \frac{800}{3} \) \( l ≈ 266.67 \) Since Carl cannot hire a fraction of a worker, he will hire either 266 or 267 workers. However, we need to check which of these options actually maximizes his profits by comparing the MRP_L and MCL at these two points. Let's check for 266 workers: \( MRP_L = \frac{400 - 266}{40} = \frac{134}{40} = 3.35 \) \( MCL = \frac{266}{80} = 3.325 \) Since MRP_L > MCL, hiring 266 workers does not maximize profit because the value of the marginal product of the 266th worker is greater than the cost, indicating that hiring one more worker would increase profit. Now let's check for 267 workers: \( MRP_L = \frac{400 - 267}{40} = \frac{133}{40} = 3.325 \) \( MCL = \frac{267}{80} = 3.3375 \) Here, MRP_L < MCL, which means that the cost of hiring the 267th worker is greater than the value of their marginal product, so hiring this worker would decrease profit. Therefore, Carl will maximize his profits by hiring 266 workers, as this is the point just before the cost of an additional worker exceeds the revenue generated by that worker. So the solution is: Carl will hire 266 workers to maximize his profits.	200
gpto1	To determine the number of workers Carl will hire to maximize his profits as a monopsonist, we'll follow these steps: 1. Understand the Labor Supply Curve: The supply curve for labor is given by: \[ l = 80w \] This can be rearranged to express the wage rate (\(w\)) as a function of labor (\(l\)): \[ w = \frac{l}{80} \] 2. Calculate the Total Expenditure (TE) on Labor: The total expenditure is the total wage bill paid by Carl: \[ \text{TE} = w \times l = \frac{l}{80} \times l = \frac{l^2}{80} \] 3. Find the Marginal Expenditure (ME) on Labor: Marginal Expenditure is the additional cost of hiring one more worker. It's the derivative of the total expenditure with respect to labor: \[ \text{ME} = \frac{d(\text{TE})}{dl} = \frac{d}{dl}\left(\frac{l^2}{80}\right) = \frac{2l}{80} = \frac{l}{40} \] Alternatively, using the definition of ME for a monopsonist: \[ \text{ME} = w + l \frac{dw}{dl} \] Since: \[ \frac{dw}{dl} = \frac{1}{80} \] Substitute back: \[ \text{ME} = \frac{l}{80} + l \times \frac{1}{80} = \frac{l}{80} + \frac{l}{80} = \frac{2l}{80} = \frac{l}{40} \] 4. Understand the Marginal Revenue Product (MRP) Curve: The marginal revenue product of labor is given by: \[ l = 400 - 40 \times \text{MRP}_l \] Rearranged, it becomes: \[ \text{MRP}_l = \frac{400 - l}{40} \] 5. Set ME Equal to MRP to Find the Profit-Maximizing Quantity of Labor: A monopsonist hires workers up to the point where ME equals MRP: \[ \text{ME} = \text{MRP}_l \] Substitute the expressions we found: \[ \frac{l}{40} = \frac{400 - l}{40} \] Multiply both sides by 40 to eliminate denominators: \[ l = 400 - l \] \[ 2l = 400 \] \[ l = 200 \] Conclusion: Carl will hire 200 workers to maximize his profits. Answer: 200	200
deepseekv3	To determine the number of workers Carl will hire to maximize his profits, we need to analyze the monopsony labor market. A monopsonist is a single buyer of labor (in this case, Carl), and they have the power to influence wages. ### Step 1: Understand the Supply and Demand Curves - Supply Curve: The supply of labor is given by \( l = 80w \), where \( l \) is the number of workers and \( w \) is the hourly wage. We can rewrite this as \( w = \frac{l}{80} \). - Demand Curve: The demand for labor is given by \( l = 400 - 40 \text{MRP}_l \), where \( \text{MRP}_l \) is the marginal revenue product of labor. We can rewrite this as \( \text{MRP}_l = \frac{400 - l}{40} \). ### Step 2: Marginal Cost of Labor (MCL) In a monopsony, the marginal cost of labor (MCL) is higher than the wage rate because hiring an additional worker increases the wage for all workers. The MCL is the derivative of the total labor cost with respect to the number of workers. The total labor cost (TLC) is given by: \[ \text{TLC} = w \times l = \left(\frac{l}{80}\right) \times l = \frac{l^2}{80} \] The marginal cost of labor (MCL) is the derivative of TLC with respect to \( l \): \[ \text{MCL} = \frac{d(\text{TLC})}{dl} = \frac{2l}{80} = \frac{l}{40} \] ### Step 3: Profit Maximization Condition A monopsonist maximizes profit by equating the marginal cost of labor (MCL) to the marginal revenue product of labor (MRP_l): \[ \text{MCL} = \text{MRP}_l \] \[ \frac{l}{40} = \frac{400 - l}{40} \] ### Step 4: Solve for \( l \) Multiply both sides by 40 to eliminate the denominator: \[ l = 400 - l \] Add \( l \) to both sides: \[ 2l = 400 \] Divide both sides by 2: \[ l = 200 \] ### Step 5: Conclusion Carl will hire 200 workers to maximize his profits. ### Step 6: Determine the Wage Rate To find the wage rate \( w \) corresponding to \( l = 200 \), use the supply curve: \[ w = \frac{l}{80} = \frac{200}{80} = 2.5 \] So, Carl will pay a wage of $2.50 per hour to each worker. ### Final Answer Carl will hire 200 workers to maximize his profits.	200

模型

模型答案

正确答案

金声玉亮2.0

为了确定Carl雇佣的工人数量以最大化利润，我们需要以下步骤：

1. 确定工资 $ w $ 和工人数量 $ l $ 之间的关系；
2. 确定边际成本（MC）和边际收益（MRP_l）之间的关系；
3. 通过等式 $ MRP_l = MC $ 确定平衡点，从而确定工人数量。

### 1. 确定工资 $ w $ 和工人数量 $ l $ 之间的关系

已知供应曲线 $ l = 80w $，可以将其反转得到工资 $ w $ 与工人数量 $ l $ 之间的关系：

\[ w = \frac{l}{80} \]

### 2. 确定边际成本（MC）

对于一个垄断买方（单一雇主）来说，边际成本（MC）不仅包括支付给新增工人的工资，还包括提高所有现有工人的工资成本。总成本（TC）函数为：

\[ TC = w \cdot l \]

将 $ w $ 替换为 $ \frac{l}{80} $：

\[ TC = \frac{l}{80} \cdot l = \frac{l^2}{80} \]

边际成本（MC）是总成本对工人数量 $ l $ 的导数：

\[ MC = \frac{d(TC)}{d(l)} = \frac{d}{d(l)} \left(\frac{l^2}{80}\right) = \frac{l}{40} \]

### 3. 确定边际收益（MRP_l）

已知需求曲线（边际收益产品曲线）为：

\[ l = 400 - 40MRP_l \]

将其反转得到边际收益产品（MRP_l）与工人数量 $ l $ 之间的关系：

\[ MRP_l = \frac{400 - l}{40} \]

### 4. 通过等式 $ MRP_l = MC $ 确定平衡点

将边际收益产品（MRP_l）与边际成本（MC）相等：

\[ \frac{400 - l}{40} = \frac{l}{40} \]

解这个方程：

\[ 400 - l = l \]
\[ 400 = 2l \]
\[ l = 200 \]

### 5. 确定工资 $ w $

将 $ l = 200 $ 代入供应曲线 $ l = 80w $：

\[ 200 = 80w \]
\[ w = \frac{200}{80} = 2.5 \]

### 结论

Carl将雇佣200名工人，并支付每小时2.5元的工资以最大化他的利润。

200

deepseekr1

Carl is a monopsonist, meaning he is the only employer on the island and faces the entire market supply curve for labor. The supply curve for garment workers is given by $ l = 80w $, which can be rearranged to $ w = \frac{l}{80} $. The labor demand (marginal revenue product) curve is given by $ l = 400 - 40MRP_l $, which can be rearranged to $ MRP_l = 10 - \frac{l}{40} $.

To find the profit-maximizing number of workers, we need to determine the marginal cost of labor (MCL) and set it equal to the marginal revenue product of labor (MRP_l).

1. **Total Labor Cost (TLC)** is given by:
\[
TLC = w \cdot l = \left(\frac{l}{80}\right) \cdot l = \frac{l^2}{80}
\]

2. **Marginal Cost of Labor (MCL)** is the derivative of TLC with respect to $ l $:
\[
MCL = \frac{d(TLC)}{dl} = \frac{2l}{80} = \frac{l}{40}
\]

3. **Set MCL equal to MRP_l** to find the profit-maximizing number of workers:
\[
\frac{l}{40} = 10 - \frac{l}{40}
\]

4. **Solving for $ l $**:
\[
\frac{l}{40} + \frac{l}{40} = 10 \\
\frac{2l}{40} = 10 \\
\frac{l}{20} = 10 \\
l = 200
\]

Thus, Carl will hire \boxed{200} workers to maximize his profits.

200

gpto1

To determine the number of workers Carl will hire to maximize his profits as a monopsonist, we'll follow these steps:

**1. Understand the Labor Supply Curve:**
The supply curve for labor is given by:
\[ l = 80w \]
This can be rearranged to express the wage rate ($w$) as a function of labor ($l$):
\[ w = \frac{l}{80} \]

**2. Calculate the Total Expenditure (TE) on Labor:**
The total expenditure is the total wage bill paid by Carl:
\[ \text{TE} = w \times l = \frac{l}{80} \times l = \frac{l^2}{80} \]

**3. Find the Marginal Expenditure (ME) on Labor:**
Marginal Expenditure is the additional cost of hiring one more worker. It's the derivative of the total expenditure with respect to labor:
\[ \text{ME} = \frac{d(\text{TE})}{dl} = \frac{d}{dl}\left(\frac{l^2}{80}\right) = \frac{2l}{80} = \frac{l}{40} \]

Alternatively, using the definition of ME for a monopsonist:
\[ \text{ME} = w + l \frac{dw}{dl} \]
Since:
\[ \frac{dw}{dl} = \frac{1}{80} \]
Substitute back:
\[ \text{ME} = \frac{l}{80} + l \times \frac{1}{80} = \frac{l}{80} + \frac{l}{80} = \frac{2l}{80} = \frac{l}{40} \]

**4. Understand the Marginal Revenue Product (MRP) Curve:**
The marginal revenue product of labor is given by:
\[ l = 400 - 40 \times \text{MRP}_l \]
Rearranged, it becomes:
\[ \text{MRP}_l = \frac{400 - l}{40} \]

**5. Set ME Equal to MRP to Find the Profit-Maximizing Quantity of Labor:**
A monopsonist hires workers up to the point where ME equals MRP:
\[ \text{ME} = \text{MRP}_l \]
Substitute the expressions we found:
\[ \frac{l}{40} = \frac{400 - l}{40} \]
Multiply both sides by 40 to eliminate denominators:
\[ l = 400 - l \]
\[ 2l = 400 \]
\[ l = 200 \]

**Conclusion:**
Carl will hire **200 workers** to maximize his profits.

**Answer:** 200

200

GPT-4o

To determine the number of workers Carl will hire to maximize his profits, we need to follow these steps:

1. **Understand the supply and demand curves:**
- The supply curve for workers is given by:
\[
l = 80w
\]
- The labor demand (marginal revenue product) curve is given by:
\[
l = 400 - 40MRP_l
\]

2. **Express the wage in terms of the number of workers:**
- From the supply curve:
\[
l = 80w \implies w = \frac{l}{80}
\]

3. **Determine the Marginal Factor Cost (MFC):**
- The MFC is the additional cost of hiring one more worker. Since Carl is a monopsonist, the MFC is not simply the wage rate but the derivative of the total cost of labor with respect to the number of workers.
- The total cost (TC) of labor is:
\[
TC = w \cdot l = \left(\frac{l}{80}\right) \cdot l = \frac{l^2}{80}
\]
- The MFC is the derivative of TC with respect to $ l $:
\[
MFC = \frac{d(TC)}{dl} = \frac{d}{dl} \left( \frac{l^2}{80} \right) = \frac{2l}{80} = \frac{l}{40}
\]

4. **Determine the Marginal Revenue Product (MRP):**
- From the demand curve:
\[
l = 400 - 40MRP_l \implies MRP_l = \frac{400 - l}{40}
\]

5. **Set MRP equal to MFC to maximize profits:**
- For profit maximization in a monopsony, the number of workers is determined where MRP equals MFC:
\[
\frac{400 - l}{40} = \frac{l}{40}
\]
- Simplify and solve for $ l $:
\[
400 - l = l \implies 400 = 2l \implies l = 200
\]

So, Carl will hire **200 workers** to maximize his profits.

200

llama3.1-70B-Instruct

To maximize profits, Carl needs to equate the marginal revenue product (MRP) of labor with the wage (w) he pays to the workers. This is because the MRP represents the additional revenue generated by hiring one more worker, and Carl will want to hire workers until the additional revenue generated equals the additional cost (the wage).

Given the labor demand (MRP) curve: l = 400 - 40MRP_l

And the supply curve: l = 80w

We can set MRP_l = w, since Carl will hire workers until the MRP equals the wage.

Substituting l from the supply curve into the labor demand curve, we get:

80w = 400 - 40w

Simplifying the equation:

120w = 400

w = 400/120
w = 3.33

Now that we have the wage, we can find the number of workers Carl will hire by substituting w into the supply curve:

l = 80w
l = 80(3.33)
l = 266.4

Since we can't have a fraction of a worker, we'll round down to the nearest whole number.

Carl will hire approximately 266 workers to maximize his profits.

200

llama3.1-405B-Instruct

To find the number of workers Carl will hire to maximize his profits, we need to find the point where the marginal revenue product (MRP) of labor equals the marginal cost (MC) of labor.

First, let's find the marginal cost of labor. The supply curve for garment workers is given by l = 80w, where l is the number of workers hired and w is their hourly wage. To find the marginal cost of labor, we need to find the derivative of the supply curve with respect to the number of workers hired (l).

MC = d(l)/d(w) = 80

So, the marginal cost of labor is $80 per worker.

Next, let's find the marginal revenue product of labor. The labor demand (marginal revenue product) curve is given by l = 400 - 40MRP_l. To find the marginal revenue product of labor, we need to find the derivative of the labor demand curve with respect to the number of workers hired (l).

MRP_l = d(l)/d(MRP_l) = -40

So, the marginal revenue product of labor is -$40 per worker.

Now, we need to find the point where the marginal revenue product of labor equals the marginal cost of labor.

MRP_l = MC
-40 = 80

Solving for l, we get:

l = 400 - 40(80)
l = 400 - 3200
l = -2800

Since the number of workers cannot be negative, this means that Carl will not hire any workers. This is because the marginal cost of labor ($80) is greater than the marginal revenue product of labor (-$40), which means that hiring workers would result in a loss for Carl.

Therefore, Carl will not hire any workers to maximize his profits.

200

DeepSeek-Coder-V2

To solve the problem of how many workers Carl the clothier will hire to maximize his profits, we need to follow these steps:

1. **Understand the given equations:**
- Supply curve for garment workers: $ l = 80w $
- Labor demand (marginal revenue product) curve: $ l = 400 - 40MRP_l $

2. **Express the wage rate $ w $ in terms of the number of workers $ l $ from the supply curve:**
\[
l = 80w \implies w = \frac{l}{80}
\]

3. **Express the marginal revenue product $ MRP_l $ in terms of the number of workers $ l $ from the demand curve:**
\[
l = 400 - 40MRP_l \implies MRP_l = \frac{400 - l}{40}
\]

4. **Set the marginal revenue product equal to the wage rate to find the equilibrium point where Carl maximizes his profits:**
\[
MRP_l = w
\]
Substitute the expressions for $ MRP_l $ and $ w $:
\[
\frac{400 - l}{40} = \frac{l}{80}
\]

5. **Solve the equation for $ l $:**
\[
\frac{400 - l}{40} = \frac{l}{80}
\]
Multiply both sides by 80 to clear the denominator:
\[
2(400 - l) = l
\]
Simplify and solve for $ l $:
\[
800 - 2l = l
\]
\[
800 = 3l
\]
\[
l = \frac{800}{3} \approx 266.67
\]

6. **Since the number of workers must be an integer, round to the nearest whole number:**
\[
l \approx 267
\]

Therefore, Carl the clothier will hire approximately 267 workers to maximize his profits.

200

Qwen2-72B-Instruct

To find the number of workers Carl will hire to maximize his profits, we need to equate the marginal revenue product (MRP) of labor to the wage rate (w).

Given:
- Labor supply curve: l = 80w
- Labor demand (MRP) curve: l = 400 - 40MRP_l

First, we need to express the wage rate (w) in terms of labor (l) using the labor supply curve:
l = 80w
w = l/80

Next, we substitute this expression for w into the labor demand equation to get the MRP in terms of l:
l = 400 - 40MRP_l
MRP_l = (400 - l) / 40

To maximize profits, the MRP of labor must equal the wage rate (w):
MRP_l = w
(400 - l) / 40 = l/80

Solving for l:
400 - l = 2l
3l = 400
l = 400 / 3 ≈ 133.33

Since the number of workers must be a whole number, Carl will hire approximately 133 workers to maximize his profits. However, in a real-world scenario, the number of workers would likely be rounded to the nearest whole number, either 133 or 134, depending on the specifics of the situation. For the sake of this problem, we can say Carl will hire 133 workers.

200

baichuan4

To maximize his profits, Carl the clothier will hire workers up until the point where the marginal revenue product (MRP) of labor equals the wage rate (w). This is because, as a monopsonist, Carl wants to minimize his labor costs while still producing enough output to meet his revenue goals.

The supply curve for garment workers is given by:
l = 80w

And the labor demand (marginal revenue product) curve is given by:
l = 400 - 40MRP_l

To find the number of workers Carl will hire to maximize profits, we need to set the wage rate (w) equal to the marginal revenue product of labor (MRP_l) and solve for l.

First, let's express w in terms of l using the supply curve equation:
w = l / 80

Next, substitute this expression for w into the labor demand equation:
l = 400 - 40(l / 80)

Now, we can solve for l:
l = 400 - (40l / 80)
l = 400 - 0.5l
1.5l = 400
l = 400 / 1.5
l = 266.67

Therefore, Carl will hire approximately 266.67 workers to maximize his profits.

200

doubao

To maximize profits, Carl will hire workers up to the point where the marginal revenue product of labor (MRP_l) is equal to the wage (w).

Set the supply curve equal to the demand curve: 80w = 400 - 40MRP_l.

But MRP_l = w, so substitute that in: 80w = 400 - 40w.

120w = 400.

w = 400 / 120 = 10 / 3.

Now substitute this wage back into the supply curve to find the number of workers: l = 80 * (10 / 3) = 800 / 3 ≈ 267 workers.

So Carl will hire approximately 267 workers to maximize his profits.

200

minimax

To solve this problem, we need to find the equilibrium point where Carl's labor demand equals the supply of labor. This is where the marginal revenue product (MRP) of labor equals the wage rate (w).

First, let's express the labor demand curve in terms of wage rate (w). We have the labor demand curve as:

l = 400 - 40MRP_l

We need to find the MRP_l in terms of w. However, we don't have the direct relationship between MRP_l and w given in the problem. Normally, the MRP curve would be derived from the marginal product of labor (MPL) and the marginal revenue (MR) that the labor helps to produce. Since we don't have the MPL or MR, we can't directly calculate MRP_l.

But we can use the given labor supply curve to find the wage rate that equates labor demand and labor supply. The labor supply curve is given as:

l = 80w

Now, we set the labor demand equal to the labor supply to find the equilibrium number of workers (l) and the corresponding wage rate (w):

400 - 40MRP_l = 80w

Since we don't have the MRP_l, we can't solve for w directly. However, we can use the fact that Carl is a monopsonist to understand that he will hire workers up to the point where the wage rate equals the marginal cost of labor, which is the derivative of the total labor cost with respect to the number of workers.

The total labor cost (TLC) for Carl is:

TLC = w * l

Since l = 80w, we can substitute l in the TLC equation:

TLC = w * (80w)
TLC = 80w^2

The marginal cost of labor (MCL) is the derivative of the total labor cost with respect to l:

MCL = d(TLC)/dl

But since w is a function of l (l = 80w), we need to use the chain rule:

MCL = d(80w^2)/dl
MCL = 160w * (dw/dl)

Since dw/dl is the inverse of dl/dw, and dl/dw is the derivative of the supply curve, which is 80, we have:

dw/dl = 1/80

Substituting this into the MCL equation:

MCL = 160w * (1/80)
MCL = 2w

Carl will hire workers up to the point where the MCL equals the wage rate (w). Therefore, we have:

MCL = w
2w = w
w = 0

This result doesn't make sense in the context of the problem because it implies that Carl would not pay any wage, which is not realistic. It seems we have a missing piece of information or a misunderstanding in the problem as presented.

In a typical monopsony problem, we would have the MRP curve and could equate it to the MCL to find the wage rate and the number of workers hired. However, without the MRP curve or additional information on how MRP relates to w, we cannot solve this problem as it stands.

If you have additional information or if there was a mistake in the problem statement, please provide it so we can attempt to solve the problem again.

200

kimi

To solve this problem, we need to find the number of workers Carl will hire to maximize his profits. As a monopsonist, Carl will hire workers up to the point where the value of the marginal product (VMPL) equals the wage rate.

First, let's find the labor demand curve for Carl's factory. The labor demand curve is given by:

\[ l = 400 - 40MRP_l \]

We need to find the value of $ MRP_l $ (marginal revenue product of labor). However, we don't have the full information to calculate $ MRP_l $ directly. Instead, we can use the fact that the marginal revenue product is the additional revenue generated by hiring one more worker. Since we don't have the revenue function, we can't calculate $ MRP_l $ directly.

But we do have the supply curve for garment workers, which is given by:

\[ l = 80w \]

This equation tells us that the number of workers willing to work (the supply of labor) is a function of the wage rate $ w $. To find the number of workers Carl will hire, we need to equate the labor demand to the labor supply:

\[ 400 - 40MRP_l = 80w \]

Since Carl is a monopsonist, he will hire workers until the wage rate equals the marginal revenue product. However, without the actual value of $ MRP_l $, we can't solve for $ w $ directly. But we can infer that Carl will set the wage rate such that the number of workers he hires is equal to the number of workers willing to work at that wage.

To maximize profits, Carl will set the wage rate $ w $ such that the marginal revenue product of labor equals the marginal cost of labor. Since the supply curve is $ l = 80w $, the marginal cost of labor is the wage rate $ w $. Therefore, Carl will hire workers until:

\[ MRP_l = w \]

Given that we don't have the actual value of $ MRP_l $, we can't find the exact number of workers Carl will hire. However, we can say that Carl will hire the number of workers where the labor demand curve intersects the labor supply curve.

If we assume that Carl sets the wage rate such that the marginal revenue product equals the marginal cost (which is the wage rate), then we can solve for the number of workers by equating the labor demand and labor supply equations:

\[ 400 - 40MRP_l = 80w \]
\[ MRP_l = w \]

Substituting $ MRP_l $ for $ w $ in the labor demand equation:

\[ 400 - 40w = 80w \]
\[ 400 = 120w \]
\[ w = \frac{400}{120} \]
\[ w = \frac{10}{3} \text{ or approximately } 3.33 \]

Now, we can find the number of workers by plugging $ w $ back into the supply equation:

\[ l = 80w \]
\[ l = 80 \times \frac{10}{3} \]
\[ l = \frac{800}{3} \]
\[ l \approx 266.67 \]

Since Carl can't hire a fraction of a worker, he will round down to the nearest whole number, which means he will hire approximately 266 workers to maximize his profits.

200

wenxinyiyan

To determine the number of workers Carl will hire to maximize his profits, we need to set the supply curve equal to the labor demand curve and solve for the wage (w) and then use that to find the number of workers (l).

Given:
Supply curve: l = 80w
Labor demand curve: l = 400 - 40MRP_l

Since the supply curve represents the workers' willingness to work at a given wage and the labor demand curve represents Carl's willingness to hire workers at a given marginal revenue product (MRP_l), we can equate the two to find the equilibrium point where both parties are satisfied.

80w = 400 - 40MRP_l

However, we are given that MRP_l is equal to w (the hourly wage), so we can substitute w for MRP_l:

80w = 400 - 40w

Now, we solve for w:

120w = 400
w = 400 / 120
w = 10/3

Now that we have the equilibrium wage, we can find the number of workers Carl will hire by plugging this wage back into either the supply curve or the labor demand curve (since they are equal at equilibrium):

l = 80 * (10/3)
l = 800/3
l ≈ 267

So, Carl will hire approximately 267 workers to maximize his profits.

200

zhipu

To determine the number of workers Carl will hire to maximize his profits, we need to find the point where his marginal cost of labor (MCL) equals the marginal revenue product of labor (MRP_L). In a monopsonistic market, the firm (Carl, in this case) faces an upward-sloping supply curve, which means that to hire more workers, Carl must pay a higher wage, not just to the new workers but also to the existing ones.

Let's start by analyzing the given supply and demand curves:

1. The supply curve for garment workers is given by:
$ l = 80w $
where $ l $ is the number of workers and $ w $ is the hourly wage.

Rearranging this equation to solve for $ w $ gives us the marginal cost of labor (MCL), since each additional worker costs more than the previous one due to the higher wage that must be paid to all workers:
$ w = \frac{l}{80} $

2. The labor demand curve (marginal revenue product of labor, MRP_L) is given by:
$ l = 400 - 40MRP_L $

Rearranging this equation to solve for MRP_L gives us:
$ MRP_L = \frac{400 - l}{40} $

To maximize profits, Carl will hire workers up to the point where MRP_L equals MCL. Setting the two equations equal to each other gives us:

$ \frac{400 - l}{40} = \frac{l}{80} $

Now, let's solve for $ l $:

$ 80(400 - l) = 40l $

$ 32000 - 80l = 40l $

$ 32000 = 120l $

$ l = \frac{32000}{120} $

$ l = \frac{800}{3} $

$ l ≈ 266.67 $

Since Carl cannot hire a fraction of a worker, he will hire either 266 or 267 workers. However, we need to check which of these options actually maximizes his profits by comparing the MRP_L and MCL at these two points.

Let's check for 266 workers:

$ MRP_L = \frac{400 - 266}{40} = \frac{134}{40} = 3.35 $

$ MCL = \frac{266}{80} = 3.325 $

Since MRP_L > MCL, hiring 266 workers does not maximize profit because the value of the marginal product of the 266th worker is greater than the cost, indicating that hiring one more worker would increase profit.

Now let's check for 267 workers:

$ MRP_L = \frac{400 - 267}{40} = \frac{133}{40} = 3.325 $

$ MCL = \frac{267}{80} = 3.3375 $

Here, MRP_L < MCL, which means that the cost of hiring the 267th worker is greater than the value of their marginal product, so hiring this worker would decrease profit.

Therefore, Carl will maximize his profits by hiring 266 workers, as this is the point just before the cost of an additional worker exceeds the revenue generated by that worker.

So the solution is:
Carl will hire 266 workers to maximize his profits.

200

gpto1

200

deepseekv3

To determine the number of workers Carl will hire to maximize his profits, we need to analyze the monopsony labor market. A monopsonist is a single buyer of labor (in this case, Carl), and they have the power to influence wages.

### Step 1: Understand the Supply and Demand Curves
- **Supply Curve**: The supply of labor is given by $ l = 80w $, where $ l $ is the number of workers and $ w $ is the hourly wage. We can rewrite this as $ w = \frac{l}{80} $.

- **Demand Curve**: The demand for labor is given by $ l = 400 - 40 \text{MRP}_l $, where $ \text{MRP}_l $ is the marginal revenue product of labor. We can rewrite this as $ \text{MRP}_l = \frac{400 - l}{40} $.

### Step 2: Marginal Cost of Labor (MCL)
In a monopsony, the marginal cost of labor (MCL) is higher than the wage rate because hiring an additional worker increases the wage for all workers. The MCL is the derivative of the total labor cost with respect to the number of workers.

The total labor cost (TLC) is given by:
\[ \text{TLC} = w \times l = \left(\frac{l}{80}\right) \times l = \frac{l^2}{80} \]

The marginal cost of labor (MCL) is the derivative of TLC with respect to $ l $:
\[ \text{MCL} = \frac{d(\text{TLC})}{dl} = \frac{2l}{80} = \frac{l}{40} \]

### Step 3: Profit Maximization Condition
A monopsonist maximizes profit by equating the marginal cost of labor (MCL) to the marginal revenue product of labor (MRP_l):
\[ \text{MCL} = \text{MRP}_l \]
\[ \frac{l}{40} = \frac{400 - l}{40} \]

### Step 4: Solve for $ l $
Multiply both sides by 40 to eliminate the denominator:
\[ l = 400 - l \]

Add $ l $ to both sides:
\[ 2l = 400 \]

Divide both sides by 2:
\[ l = 200 \]

### Step 5: Conclusion
Carl will hire **200 workers** to maximize his profits.

### Step 6: Determine the Wage Rate
To find the wage rate $ w $ corresponding to $ l = 200 $, use the supply curve:
\[ w = \frac{l}{80} = \frac{200}{80} = 2.5 \]

So, Carl will pay a wage of **$2.50 per hour** to each worker.

### Final Answer
Carl will hire **200 workers** to maximize his profits.

200