Download Solutions to Question 1 from midterm

Solutions to Question 1 from midterm
1a.) L shaped indifference curves where the vertices are along the line yx/5. By far the
most common mistake was graphing them along the line y5x. Also, I wanted to see some
amount of minimal labelling. By minimal, I mean at least the axes had to be labelled.
Otherwise, people who graphed y5x could have claimed that they had the graph right, but had
just put x on the vertical axis.
b.) Yes. They do represent the same preferences. Some people said that if they were
monotonic transformations of each other, then they would be the same preferences and then
said that these were not MTs of each other, suggesting that perhaps not everyone knows what a
MT is. Also suggesting this was that some people plugged in bundle into both, got different
numbers, and concluded that they were not the same preferences. So let’s take a second to
review what an MT is. An MT is any transformation that doesn’t change the ORDERING of
preferences. That is, for any bundles A and B, if I originally prefer A to B, then after a
monotonic transformation, I still prefer A to B. Therefore, let’s think operations that are MTs:
1. adding (or subtracting) a constant. If originally, U(A)x and U(B)y where xy, if I
add a constant (call it k) to the utility function, I get: U(A)kxk and U(B)kyk. Since
xy, we know that even after adding the constant, bundle A will still give more utility than B,
therefore the ordering is still the same.
2. multiplying by a constant. again, start with U(A)x and U(B)y where xy. Now
multiply by a constant: kU(A)kx and kU(B)ky. Again, since xy, we know A is still
preferred to B, so the ordering is still the same after the transformation.
3. taking logs and exponentiating. Since both e x and log are strictly increasing, we know
that e x  e y and ln(x)ln(y) iff xy. Therefore, if we take logs of or exponentiate a utility
function, if a bundle originally gave more utility, it will still give more utility after the
transformation.
So, using these rules, we can easily go from 3xy2 to lnxlny9:
3xy2
add a constant (-2): 3xy
take logs: ln(3)ln(x)ln(y)
add a constant (9-ln3) and voila: ln(x)ln(y)9
Since all of these transformations were monotonic, the 2 functions must be MTs of each
other.
Another acceptable methods was to check to see if the 2 had the same MRS. Since setting
MRS equal to prices is what determines demand functions, if two utility functions give the
same MRS, they generate the same demand functions and therefore represent the same
preferences.
c.) In all but one case, Hicksian demand curves for well behaved utility functions slope
downwards. Hicksian demand curves show only the substitution effect. The substitution
effect is always negative in the sense that quantity consumed and price move in opposite
directions because we are keeping you on the same indifference curve. To show why this is
true, imagine the price of x goes up. the slope of the BC is now steeper. Hicksian
compensation dictates we find the tangency point where the slope of the BCMRS. Along a
well behaved indifference curve, the MRS gets steeper as we reduce x along an indifference
curve. Therefore, when price of x goes up, we buy less X. The reverse story if price of x goes
down. A graphical answer was also acceptable. Just saying that the substitution effect is
always negative with no argument as to why did not earn full credit.
The exception is the case of perfect complements. Since you never substitute between
goods with these indifference curves, there is no substitution effect. Therefore, after being
compensated, you end up with the exact same bundle as where you started. So no matter how
the price changes for x, after being compensated, you end up with the exact same amount of
X. This means that the Hicksian compensated demand curve for x when x is part of a perfect
complements utility function is a vertical line which is neither upward nor downward sloping.
d.) False. The Hicksian and Marshallian demand curves coincide in this case, so they are
equally steep. These are quasi-linear preferences and the trick was to remember that in this
case, demand for x does not depend on income at all. Therefore, there is zero income effect on
x when prices change, there is only substitution effect. Therefore, both the Marshallian and
the Hicksian demands for x are only showing the substitution effect. Therefore, they are the
same curve in this case and are equally steep. X is neither normal nor inferior in this case.
e.) For Marshallian demand to slope up, the good must be a Giffen good. To be a Giffen
good, the good must be inferior. Since homothetic preferences generate straight line upward
slopping income offer curves (or Engel’s curves...see notes from TA session for why this is
true), we know that when income rises, we must buy more of good X. Therefore, x must be
normal. Normal goods obviously can’t be a Giffen good. Therefore, if preferences are
homothetic, etc, all goods must be normal, ruling out Giffen goods. Therefore demand must
slope down. Graphical responses exploiting the definition of homothetic accepted.