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Sample Answers HW 3
1 a) , b) and c)
2) dU/dx = x-1/2y1/2
d2U/dx2 = -1/2 x-3/2y1/2 < 0 Diminishing marginal rates of utility w.r.t x
d2U/dy2 = -1/2 y-3/2x1/2 < 0 Diminishing marginal rates of utility w.r.t y
MRS = (dU/dx)/(dU/dy) = y/x
3) a) yes, because marginal utilities are positive for both goods b) No. d2U/dx2 = 4 > 0 and d2U/dy2 = 0.
Hence both goods do not exhibit diminishing MU's.
c) No. MRS = MUx/MUy = 4x/7 and d(MRS)/dx =
4/7 > 0. Also, d(MRS)/dy=0.
d)
In the above utility function, we see that it violates the law of diminishing marginal utility. Also, the
function doesn't display diminishing MRS. As X increases, the marginal rate of substitution between X
and Y also increases. Hence we do not get the convex, well behaved indifference curves.
5. For the 2nd function, the marginal utilities are negative. Utility is not an increasing function of the
commodities x and y. Violates "more is better" (monotonicity).
L = x1/2+ y1/2 + μ(x + 2y -200)
FOCs:
1/2x-1/2 + μ = 0
1/2y-1/2 + 2μ = 0
x + 2y = 200
Solving these simultaneously, we get y* = 33.33 and x* = 133.32
If we change the constraint to x + 2y = 300, only our third FOC will change and we will get y* = 50 and x*
= 200
For the first function, the goods are normal. dI/dx and dI/dy both are greater than 0 where I is the
income.
for the second utility function
L = x-2+ y-2 + μ(x + 2y -200)
FOCs:
-2x-3 + μ = 0
-2y-3 + 2μ = 0
x + 2y = 200
solving the FOCs we get y = 61.35 and x = 77.3.
If income is 300, then x = 115.95 and y = 92.0245. Hence x and y are normal goods with this utility
function as well.
4 b)
Since Ph = Pr, individuals who are risk-loving will accept the risky job while risk averse individuals will
accept the harmless job. Individuals who are concerned with safety will have an indifference curve
skewed in that direction (like Uh). A different worker, one who does not care as much about safety, and
is much more interested in wages, might have an indifference curve like Ur. Since we are assuming M to
be the same, all individuals achieve the same MRS but they maximize utility at different points on the
budget constraint according to their tastes and preferences.
a)
When Pr > Ph, every utility maximizing individual increases his/her utility by being on the budget
constraint of the harmless job. There is no incentive for anyone to be on the budget constraint of risky
job. By choosing the harmless job, everyone (whether risk loving or risk averse) has a higher payoff in
this case.
c)
The risk loving individuals will be worse off since they were "happier" with a higher wage and higher risk.
But now because of the minimum level of safety requirement, they have to settle for less wage and less
risk. If the law had not been implemented, risk loving individuals would have chosen more wage and less
safety.