Average Production With Q = 41x - 2x² - 60

Alright, let's dive into this economics problem! We're trying to figure out the average production quantity given a specific production function and a condition where the input price equals the output price. This involves a bit of calculus and economic reasoning, but don't worry, we'll break it down step by step. So, grab your favorite beverage, and let's get started!

Understanding the Production Function

First, let's understand what we're dealing with. The production function is given as:

Q = 41x - 2x² - 60

Where:

• Q is the quantity of output.
• x is the quantity of input.

This equation tells us how much output Q we can produce for a given amount of input x. It's a quadratic equation, which means the relationship between input and output isn't linear – it curves. This is pretty common in real-world production scenarios where you might experience diminishing returns as you add more and more input.

The Condition: Input Price Equals Output Price

Now, let's look at the condition given:

(MP × P_X) / P_Q

This is actually a condition derived from economic optimization. Here's what each term means:

• MP is the Marginal Product of input x. It tells us how much additional output we get from adding one more unit of input x. Mathematically, it's the derivative of the production function with respect to x.
• P_X is the price of input x.
• P_Q is the price of output Q.

The condition (MP × P_X) / P_Q = 1 (since the input price equals the output price) implies that the value of the marginal product of input x is equal to the price of input x. In simpler terms, it means you're using the optimal amount of input x – the point where the additional revenue from using one more unit of x exactly covers the cost of that unit. This is a crucial concept in economics because it helps firms maximize their profits. Understanding this relationship helps us determine the optimal level of input, which subsequently allows us to find the average production quantity.

Calculating the Marginal Product (MP)

To use the condition, we first need to find the Marginal Product (MP). Remember, MP is the derivative of the production function with respect to x.

So, let's differentiate Q = 41x - 2x² - 60 with respect to x:

MP = dQ/dx = 41 - 4x

This tells us how much additional output we get for each additional unit of input x. Notice that the MP decreases as x increases, which is the law of diminishing returns in action!

Applying the Condition and Solving for x

Now we know that the input price equals the output price, so we know that MP = 1. Let's set our MP expression equal to 1 and solve for x:

41 - 4x = 1

4x = 40

x = 10

This tells us that the optimal level of input x is 10 units. Using 10 units of input x maximizes our profit because at this point, the value of the additional output from one more unit of x exactly equals the cost of that unit. This is a key concept in microeconomics and helps firms make informed decisions about their production processes. By equating the marginal product to one, we ensure that we are operating at the most efficient level of input. This step is crucial for determining the input level that maximizes profit, which directly impacts the final average production quantity.

Calculating the Average Production Quantity

Now that we know the optimal level of input x is 10, we can plug it back into the production function to find the corresponding quantity of output Q:

Q = 41(10) - 2(10)² - 60

Q = 410 - 200 - 60

Q = 150

So, when we use the optimal amount of input (x = 10), we get an output of 150 units. This is the maximum output we can achieve given the production function and the condition that the input price equals the output price. Understanding how to maximize output given certain constraints is fundamental to understanding production economics.

Considering Average Production

The question asks for the average production. It seems like it equates to finding Q when MP = 1. If this is the case, then Q = 150 is the production quantity.

However, if the question is implying average product, then we would have AP = Q/x = 150 / 10 = 15.

Given the option and context, it's most likely implying that the question wants to find the quantity Q when the marginal product equals to 1.

Let's review our work to be absolutely certain. We started with the production function, found the marginal product, used the condition to solve for the optimal level of input, and then plugged that input level back into the production function to find the output. Each of these steps is vital for accurately determining the production quantity.

Conclusion

Based on our calculations, when the input price equals the output price and the production function is Q = 41x - 2x² - 60, the output Q is 150. However, we had to work backwards using the MP. The question is likely looking for the value of x where MP = 1. The correct answer is D. 10. Understanding how to apply these economic principles can greatly improve decision-making in real-world production scenarios.