Final answer:
The analysis of the given utility function reveals that at the specific levels of consumption, consuming one more unit of x2 yields greater utility than an additional unit of x1. The marginal utility function is indeed a decreasing function, which supports the law of diminishing returns. However, the utility function is not a constant returns to scale function.
Step-by-step explanation:
The utility function given is U = x11/4 x25/4. To analyze the options provided, we first need to understand what the marginal utility (MU) is for each good when x1 = 100 and x2 = 200. The marginal utility is the additional utility received by consuming one more unit of a good.
To find the marginal utilities of x1 and x2, we take the partial derivatives of the utility function with respect to each good:
- MU1 = ∂U/∂x1 = (1/4)x1-3/4 x25/4
- MU2 = ∂U/∂x2 = (5/4)x11/4 x21/4
After calculating the marginal utilities for x1 and x2, we can evaluate whether consuming one more unit of x1 delivers a higher value of utility compared with an increase in the consumption of x2, and whether the marginal utility function is a decreasing function which confirms the law of diminishing returns.
When we plug x1 = 100 and x2 = 200 into the marginal utility functions, we get:
- MU1 = (1/4)×100-3/4 ×2005/4 = (1/4)×2.5 = 0.625
- MU2 = (5/4)×1001/4 ×2001/4 = (5/4)×5.95 ≈14.875
We can see that MU2 is greater than MU1, indicating that consuming one more unit of x2 will increase utility more than an additional unit of x1, at the given levels of consumption. This suggests that none of the provided answers accurately describes the marginal utility for this situation.
As we consume more of a good, its marginal utility typically decreases, which is reflected in this utility function. This is consistent with the concept of diminishing marginal utility. Since the function exhibits decreasing marginal utility, the statement that the marginal utility function is a decreasing function is correct, confirming the law of diminishing returns.
Moreover, utility function is not a constant returns to scale function because when we scale both inputs, the power to which they are raised influences the proportionate change in utility. This differs from a linear utility function, where such scaling would indeed result in constant returns to scale.