This post goes over a question regarding the economics of utility functions and budget constraints:

Matt has the utility function U = √XY (where Y represents pears and X
represents hamburgers), income of $20, and is deciding how to allocate that
income between pears and hamburgers. Both hamburgers and pears cost $1.00 each.

(i) Write the equation for Matt’s budget line in slope, intercept form (y =
mx +
c).

(ii) Matt’s utility function implies
that the marginal utility of pears is 0.5√X/Y and the marginal utility of hotdogs is 0.5√Y/X . How many pears will Matt buy? How many hamburgers will he buy?

(iii) This year, the price of hamburgers
rise to $3 each while Matt’s income is
unchanged. Matt’s father decides to help him by giving him a gift of $20. Consider an indifference curve-budget line diagram with hamburgers on the
x-axis and pears on the y-axis. After Matt gets the $20 gift, will his new budget line lie above, lie below, or pass through his initial optimum? Justify your
choice. Will Matt be better or worse off than he was last year?

Ok, let's start with the information given to us:

Utility function is (XY)^0.5

The budget constraint is equal to M = PxX + PyY

Since the budget is equal to 20, and the price of each good
is $1, we want the budget line to intercept each axis at a value of 20 (meaning
20 pears and 20 hamburgers). To get
this, our intercept has to be 20, and 20 times our coefficient value has to
result in 0. This gives us:

Y = 20 – 1*X (which gives us our budget line)

The marginal utility of hamburgers and pears is given to us,
but we could also figure it out by taking the appropriate derivative (if you
know calculus). In this case we would
want to take the derivative of the utility function with respect to either X or
Y, and this would give us the marginal utility associated with that good.

One of the rules of utility maximization is that the
marginal utility per dollar spent has to be equal (the other is that the entire
budget is exhausted). Since the price of
both goods is one dollar, we don’t have to worry about the per dollar part and
can simply equate the two marginal utilities (but I will include it in the
first step anyways to show what I mean).

So, marginal utility per dollar spent on good X is (NOTE:
the square root of X is equal to X^0.5, they mean the same thing):

MUx/Px = 0.5(Y/X)^0.5/Px = 0.5(Y/X)^0.5

And marginal utility per dollar spent on good Y is:

MUy/Py = 0.5(X/Y)^0.5/Py = 0.5(X/Y)^0.5

Now if we set these two equations equal to each other we
get:

0.5(Y/X)^0.5 = 0.5(X/Y)^.05

Now we can multiply both sides by 2, and then square both
sides to get:

Y/X = X/Y and we can multiply both sides by Y, and divide
both sides by X then take the square root of both sides to get:

X = Y

This means that we should consume the same amount of X and
Y, since we have $20 and the price of each good is $1, we should consume ten of
each.

Now the price of hamburgers rises to $3, but his budget
stays the same. This will definitely
make the consumer worse off. But what
happens if we increase the consumer’s budget by $20? Then we will see a shift out of the budget
line, and potentially the consumer could be better or worse off. This is pretty easy to see if we construct
both budget lines and make note of where the original consumption point existed.

The mathematical way to do this is to again set marginal
utility per dollar spent equal, except now the price of one good has increased
to three so we have to make a slight modification. Now:

The marginal utility per dollar spent of X (hamburgers) is:

Mux/Px = 0.5(Y/X)^0.5/3

We set this equal to the marginal utility per dollar spent
of Y to get:

0.5(Y/X)^0.5/3 = 0.5(X/Y)^.05/1 and we now multiply both
sides by 6, then square both sides to get:

Y/X = 9X/Y again multiply both sides by X and Y then take
the square root of both sides to get:

Y = 3X

This simply means that for every hotdog we consume, we
should consume 3 pears. This makes sense
because the price of hamburgers is 3 times the price of pears. If our budget constraint is 40, and the price
of pears is 1, and the price of hamburgers is 3, then we should still devote
$20 each to hot dog and apple consumption except now $20 buys 20 pears, and $20
buys 6.667 hamburgers. This is the
utility maximizing choice.

The trick now is to plug these values into the utility
function to see whether or not your utility is higher under the first or second
scenario. By consuming 10 of each good
our utility is equal to (10*10)^0.5 which is equal to 10, while in the second
scenario our utility is equal to (20*6)^0.5 which is about equal to 10.95 which
is higher. So we are willing to let the
price of hamburgers triple if we can also double our budget, in fact we prefer
it. You can confirm this by looking at the graph to the right.