Asked by Danique Mcfarlane on Jun 16, 2024
Verified
Al's production function for deer is f(x1, x2) (2x1 x2) 1/2, where x1 is the amount of plastic and x2 is the amount of wood used.If the cost of plastic is $4 per unit and the cost of wood is $4 per unit, then the cost of producing 8 deer is
A) $16.
B) $96.
C) $256.
D) $128.
E) $32.
Production Function
An equation or model that describes the relationship between the inputs used in production and the output of goods or services.
Plastic
A wide range of synthetic or semi-synthetic materials that use polymers as a main ingredient and are malleable, making them suitable for various applications.
Wood
A natural, fibrous material derived from trees, widely used for construction, manufacturing, and fuel.
- Execute the application of production function principles to real-life circumstances, involving a range of inputs and financial implications.
- Acknowledge the effect of input costs on the manufacturing cost of products.
Verified Answer
LS
Layla SobhanpanahJun 18, 2024
Final Answer :
D
Explanation :
To produce 8 deer, we need to find the optimal combination of plastic and wood that minimizes the cost of production.
Al's production function is:
f(x1, x2) (2x1 x2)1/2
Taking the partial derivative with respect to x1 and setting it equal to zero, we get:
∂f/∂x1 = 1/2(4x1 + 2x2 ) = 0
Simplifying this, we get:
2x1 + x2 = 0
2x1 = -x2 /
Taking the partial derivative with respect to x2 and setting it equal to zero, we get:
∂f/∂x2 = 1/2(2x1 + x2) = 0
Simplifying this, we get:
x1 + 0.5x2 = 0
x2 = -2x1 /
Substituting this into the first equation, we get:
2x1 = 2x1
This means that we can use any combination of plastic and wood that satisfies the above equation. For example, we can use x1 = 4 and x2 = -8, which satisfies the above equation, and produces 8 deer.
The cost of using 4 units of plastic and 8 units of wood is:
Cost = 4(4) + 4(8) = $32
Therefore, the answer is D) $128.
Al's production function is:
f(x1, x2) (2x1 x2)1/2
Taking the partial derivative with respect to x1 and setting it equal to zero, we get:
∂f/∂x1 = 1/2(4x1 + 2x2 ) = 0
Simplifying this, we get:
2x1 + x2 = 0
2x1 = -x2 /
Taking the partial derivative with respect to x2 and setting it equal to zero, we get:
∂f/∂x2 = 1/2(2x1 + x2) = 0
Simplifying this, we get:
x1 + 0.5x2 = 0
x2 = -2x1 /
Substituting this into the first equation, we get:
2x1 = 2x1
This means that we can use any combination of plastic and wood that satisfies the above equation. For example, we can use x1 = 4 and x2 = -8, which satisfies the above equation, and produces 8 deer.
The cost of using 4 units of plastic and 8 units of wood is:
Cost = 4(4) + 4(8) = $32
Therefore, the answer is D) $128.
Learning Objectives
- Execute the application of production function principles to real-life circumstances, involving a range of inputs and financial implications.
- Acknowledge the effect of input costs on the manufacturing cost of products.