If utility is U(x,y) = xy + y, what is the indirect utility function and expenditure function?
The answer to this takes several steps (and beyond what I am willing to do here).
first, calculate MUx and MUy, noting that for maximization MUx/MUy = Px/Py
Second, solve for Px*X or Py*Y. Plug this into the budget constraint equation I=Px*X + Py*Y. Rearrange terms to get X (or Y) by itself. This will be X* or X when utility is maximized. Ditty for Y*. Plug this X* and Y* into your utility equation and you are done.
Repost, showing your work and I, or others will critique.
Solve Microeconomics
Answer
To find the indirect utility function and the expenditure function, we need to first understand what these concepts represent.
Indirect utility function measures the maximum utility a person can achieve given a certain level of prices and income. It tells us the maximum utility attainable for a given level of income and prices.
Expenditure function, on the other hand, represents the minimum amount of money a person needs to spend in order to achieve a given level of utility. It is essentially the cost or monetary value associated with reaching a particular level of utility.
To find the indirect utility function, we need to maximize the utility function U(x, y) = xy + y with respect to x and y, subject to the constraint that the individual's income is I.
The first step is to optimize the utility function by taking the partial derivatives with respect to x and y.
∂U/∂x = y
∂U/∂y = x + 1
Now, set these partial derivatives equal to 0 and solve the resulting equations.
y = 0 (equation 1)
x + 1 = 0 (equation 2)
From equation 1, we can see that y = 0. Substituting this into equation 2, we find x = -1.
So, the optimal values for x and y are x = -1 and y = 0.
To find the indirect utility function, we substitute these optimal values back into the utility function:
U(-1, 0) = (-1)*(0) + 0 = 0
Therefore, the indirect utility function is U(I, p) = 0, where I represents the income and p represents the prices.
Moving on to the expenditure function, we need to find the minimum expenditure required to reach a given level of utility. Since the utility function does not depend on the quantities of x and y, there is a wide range of quantities that will yield the same utility level.
Thus, the expenditure function for this utility function U(x, y) = xy + y is not defined.