Question

For the production function $Y = X + 4.2 X^2 - 0.2 X^3$, what will be the level of resource use corresponding to point of inflexion on TPP curve?

• A. 70.00
• B. 10.00
• C. 0.14
• D. 7.00

Hint:

To locate the point of inflexion of a cubic production function $Y = aX + bX^2 - cX^3$, you can use the shortcut formula: $X = \frac{b}{3c}$. Here, $b=4.2$ and $c=0.2$, so $X = \frac{4.2}{3 \times 0.2} = \frac{4.2}{0.6} = 7$.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The point of inflexion on the Total Physical Product (TPP) curve is the point where the rate of change of output shifts from increasing to decreasing.
At this point, the Marginal Physical Product (MPP) reaches its maximum level, which corresponds mathematically to where the second derivative of the production function equals zero.

Step 2: Key Formula or Approach:
For a production function $Y = f(X)$, we must find $X$ where:
\[ \frac{d^2Y}{dX^2} = 0 \]

Step 3: Detailed Explanation:
Given the production function:
\[ Y = X + 4.2 X^2 - 0.2 X^3 \]
First, let us find the first derivative (which is the MPP):
\[ \text{MPP} = \frac{dY}{dX} = 1 + 8.4 X - 0.6 X^2 \]
Next, let us compute the second derivative:
\[ \frac{d^2Y}{dX^2} = \frac{d(\text{MPP})}{dX} = 8.4 - 1.2 X \]
To find the point of inflexion, we set this second derivative equal to zero:
\[ 8.4 - 1.2 X = 0 \]
\[ 1.2 X = 8.4 \]
\[ X = \frac{8.4}{1.2} = 7 \]
Thus, the level of input resource use ($X$) corresponding to the point of inflexion is 7.00.
Final Answer:
The required level of input is 7.00, matching Option (D).