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.
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
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