Final answer:
The production function f(x₁,x₂)=9 ³√x₁x₂ exhibits diminishing returns for both inputs x₁ and x₂. It also demonstrates constant returns to scale. The slope of the isoquant at the point (8,5) represents the MRTS, which indicates the substitution rate between the inputs while maintaining the same output.
Step-by-step explanation:
The output of a factory can be described using a production function. The given production function is f(x₁,x₂)=9 ³√x₁x₂. In terms of the Law of Diminishing Returns, the function appears to imply that as either x₁ or x₂ is increased while holding the other constant, production may increase. However, because both inputs are under the cube root, the marginal product (additional output produced by additional unit of input) of each factor will eventually decrease, exhibiting diminishing returns.
The production function can showcase different returns to scale, which refers to what happens to output if we proportionately increase all inputs. Given that the inputs x₁ and x₂ are under a cube root in our function, this implies constant returns to scale, meaning that if we double the inputs, the output also doubles.
The slope of the isoquant through the input combination (8,5) can be found by taking the partial derivative of f with respect to x₁ and x₂ then evaluating at the point (8,5). The slope of the isoquant, in economic terms, represents the rate at which one input can be substituted for another while keeping the output level constant, known as the marginal rate of technical substitution (MRTS).