Answer:To find the maximum amount of good x (denoted as x*) that Max can purchase, we can use the concept of utility maximization. Max wants to allocate his income in a way that maximizes his overall utility.
To start, we need to set up Max's budget constraint, which states that the total expenditure on goods x and y cannot exceed his income. In this case, Max's income is $20, and the prices of x and y are given as $4 and $2, respectively.
Budget constraint equation:
P_x * x + P_y * y = m
(4 * x) + (2 * y) = 20
Now, we can substitute Max's utility function U(x,y) = 4xy into the budget constraint equation to form a single equation:
4xy = 20 - 2y
Next, let's solve for x in terms of y by rearranging the equation:
4xy + 2y = 20
x(4y) = 20 - 2y
x = (20 - 2y) / (4y)
x = (10 - y) / (2y)
To find the maximum amount of x Max can purchase, we need to find the value of y that maximizes the equation for x. Since y cannot be negative (as we are dealing with quantities), we need to find the value of y that makes x* the highest possible positive value.
To find the maximum, we can take the derivative of x with respect to y and set it equal to zero:
d(x)/d(y) = [(10 - y) / (2y)]' = 0
To solve this equation, we can simplify it by multiplying through by 2y:
(10 - y)' = 0
-1 = 0
Since -1 does not equal zero, this means that there is no maximum value for x* within the given constraints. In other words, there is no maximum amount of good x that Max can purchase.
However, we can find the maximum amount of good y (denoted as y*) that Max can purchase by substituting x = 0 into the budget constraint equation:
(4 * 0) + (2 * y) = 20
2y = 20
y = 10
Therefore, the maximum amount of good y Max can purchase is 10 units.