Introduction
In economics, a production function is a mathematical relationship that shows how inputs like labour and capital are used to produce output. The given production function helps us understand how a factory that manufactures toys can utilize its resources efficiently to maximize output. This question includes both numerical computation and a discussion on the concept of returns to scale.
Understanding the Production Function
The given production function is:
Q = 5√(L × K)
Where:
Q = Output (number of toys produced)
L = Labour hours used per day
K = Machine hours used per day
Maximum Number of Toys Produced
We are told that 9 labour hours and 9 machine hours are used per day. Substituting these into the production function:
Q = 5√(9 × 9) = 5√81 = 5 × 9 = 45
Maximum toys produced in a day: 45 toys
Marginal Product of Labour
The marginal product of labour (MPL) is the additional output produced when one more unit of labour is added, keeping capital constant.
Given:
Q = 5√(L × K)
We can rewrite √(L × K) as (L × K)^0.5.
Then, MPL = dQ/dL
dQ/dL = 5 × d/dL (L × K)^0.5 = 5 × 0.5 × (L × K)^(-0.5) × K
MPL = (5 × 0.5 × K) / √(L × K) = (2.5K) / √(L × K)
Substitute L = 9, K = 9:
MPL = (2.5 × 9) / √(9 × 9) = 22.5 / 9 = 2.5
Marginal Product of Labour = 2.5
Doubling Inputs: Labour and Capital
If the firm doubles both labour and machine hours:
L = 2 × 9 = 18
K = 2 × 9 = 18
Now, Q = 5√(18 × 18) = 5√324 = 5 × 18 = 90
New output = 90 toys
Previous output = 45 toys
Increase in output = 90 – 45 = 45 toys
Returns to Scale
Returns to scale describe how output changes when all inputs are increased in the same proportion.
- If output more than doubles, it’s Increasing Returns to Scale
- If output exactly doubles, it’s Constant Returns to Scale
- If output less than doubles, it’s Decreasing Returns to Scale
Here, doubling L and K led to exactly double the output (from 45 to 90). So, this production function exhibits:
Constant Returns to Scale
Conclusion
The production function Q = 5√(L × K) allows us to calculate how much output a firm can generate using labour and capital. With 9 hours each of labour and capital, the firm produces 45 toys per day. The marginal product of labour at this level is 2.5. Doubling both inputs results in double the output, indicating constant returns to scale. This understanding helps businesses plan their resource allocation more efficiently and assess how output changes in response to changes in input.