Question

For the production function $Y = 2000 + 7X - 0.02X^2$, find the level of resource use which would give maximum output:

• A. 125
• B. 175
• C. 2000
• D. 140

Hint:

For a quadratic function of the form $Y = a + bX - cX^2$, the output is always maximized at $X = \frac{b}{2c}$. Substituting the values: $X = \frac{7}{2 \times 0.02} = \frac{7}{0.04} = 175$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
In microeconomic production theory, output ($Y$) reaches its maximum level when the Marginal Physical Product (MPP) of the input resource ($X$) drops to zero.

Step 2: Key Formula or Approach:
To find the input level that maximizes output, we find the first derivative of the function and set it to zero:
\[ \frac{dY}{dX} = 0 \]

Step 3: Detailed Explanation:
Given the production function:
\[ Y = 2000 + 7X - 0.02X^2 \]
Let us take the first derivative of $Y$ with respect to $X$:
\[ \frac{dY}{dX} = 7 - 0.04X \]
Set the derivative (MPP) equal to zero to find the maximizing level of $X$:
\[ 7 - 0.04X = 0 \]
\[ 0.04X = 7 \]
\[ X = \frac{7}{0.04} \]
\[ X = \frac{700}{4} = 175 \]
To verify that this point is indeed a maximum, we check the second derivative:
\[ \frac{d^2Y}{dX^2} = -0.04 < 0 \]
Since the second derivative is negative, $X = 175$ represents a maximum.
Final Answer:
The level of resource use that maximizes output is 175, matching Option (B).