2. Suppose Fred has a utility function of the form U = X ½ Y ½ . Fred has an income of $100, good X costs $1.00, and good Y costs $2.00. How much of good X and how much or good Y will Fred buy?
Ive already drawn my graph. i know that fred could buy 100 of x only and 50 of y only under his budget constraint. i know the slop is -2 but i cannot figure out how to calculate for good X and Y
thanks
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To determine how much of goods X and Y Fred will buy, we need to find the point on the budget line where his utility function is maximized. This can be done by solving an optimization problem.
First, let's set up the problem. We have Fred's utility function:
U = X^(1/2) * Y^(1/2)
We also know that Fred has an income of $100 and the prices of goods X and Y are $1.00 and $2.00, respectively.
Using the budget constraint equation:
$1.00X + $2.00Y = $100
We can solve this equation to find the relationship between X and Y on the budget constraint.
Next, we can substitute this relationship into the utility function:
U = X^(1/2) * (100 - $1.00X)^(1/2)
Now, we can maximize this utility function by taking the derivative with respect to X and setting it equal to zero:
dU/dX = 1/2 * (X^(-1/2)) * (100 - $1.00X)^(1/2) + 1/2 * X^(1/2) * (1/2) * (-$1.00) * (100 - $1.00X)^(-1/2) = 0
Simplifying this equation, we get:
(X / (100 - $1.00X))^(1/2) = $1.00/2
Squaring both sides yields:
X / (100 - $1.00X) = ($1.00/2)^2
Now we can solve for X:
X / (100 - $1.00X) = $0.50
X = ($0.50) * (100 - $1.00X)
X = 50 - $0.50X
1.50X = 50
X = 50 / 1.50
X = 33.33
Now we can substitute this value of X back into the budget constraint equation to find Y:
$1.00 * 33.33 + $2.00Y = $100
33.33 + $2.00Y = $100
$2.00Y = $66.67
Y = $66.67 / $2.00
Y = 33.33
Therefore, Fred will buy approximately 33.33 units of good X and 33.33 units of good Y to maximize his utility.