Sorry, the last Question didn't post

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? Be sure to show all your work.

Ive aready drawn my graph and i know that under my budget constraint i could buy 100 of X only and 50 of y only. i also know that my slop is -2. i just can not figure out how to solve to find how much of good X and Y he will buy.
thanks

Your slope of -2 of the budget constraint is because Py/Px = 2.

To maximize, Fred will want (MUy/MUx) = Py/Px. So, first find marginal utilities. For MUx take the first derivative with respect to X.
MUx = (1/2)*X^(-1/2)*Y^(1/2)
MUy = (1/2)*X^(1/2)*Y^(-1/2)

So, MUy/MYx collapses to X/Y. And this ratio is constant for all levels of consumption. So, Fred maximizes when X/Y = 2. For every 2 Xs purchase, Fred will buy 1 Y. Fred, spending $100, will consume 50 X and 25 Y.

To determine how much of good X and good Y Fred will buy, we need to find the point where his utility is maximized under the given income and prices.

First, let's set up the problem. We have the utility function U = X^0.5 * Y^0.5, an income of $100, and the prices of good X and good Y, which are $1.00 and $2.00, respectively.

To maximize utility, we need to find the combination of X and Y that maximizes the utility function U, while also staying within Fred's budget constraint.

Let's start by expressing the budget constraint in terms of Y. Since the price of good Y is $2.00 and Fred's income is $100, we have:

2Y + X = 100

Now we can rearrange this equation to express Y in terms of X:

Y = (100 - X) / 2

Next, substitute this expression for Y into the utility function:

U = X^0.5 * ((100 - X) / 2)^0.5

Simplify this further by multiplying the terms inside the brackets:

U = X^0.5 * ((100 - X) ^ 0.5) / 2^0.5

Simplify the square roots:

U = X^0.5 * ((100 - X)^0.5) / √2

To maximize utility, we differentiate the utility function with respect to X and set it equal to zero:

dU/dX = 0.5 * X^(-0.5) * (100 - X)^0.5 / √2 - 0.5 * (100 - X)^(-0.5) * X^0.5 / √2 = 0

Simplifying this equation, we get:

X / (2 * (100 - X)) = (100 - X) / (2 * X)

Cross-multiplying and simplifying, we have:

X^2 = (100 - X)^2

Expanding and rearranging, we get a quadratic equation:

X^2 = 100^2 - 200X + X^2

Simplifying, we have:

200X = 100^2

X = 100^2 / 200

X = 50

Now that we have the value of X, we can substitute it back into the budget constraint to find the value of Y:

2Y + 50 = 100

2Y = 50

Y = 25

Therefore, Fred will buy 50 units of good X and 25 units of good Y in order to maximize his utility under the given circumstances.

Please note that this solution assumes that Fred's utility function is strictly concave, which is generally the case for most individuals.