Assume Bree has the utility function: u=min{X, Y}.
a) To graph Bree’s market demand for X you must first solve for X*(.) i.e., the demand function for X. With this utility function, you can’t set MRS equal to ERS, so you look at the indifference curve/budget line graph and notice that at an optimal bundle it must be true that X=Y. Now use the budget line to get: PxX+PyY=I and by substitution, PxX+PyX=I so that X*=I/(Px+Py).
Now you would just graph the inverse of this function, where you rearrange to have Px on the left hand side of the equation to get: Px=(I/X)-Py. See graph.
b) Note, with this utility there is NO substitution effect so “d hicks” is vertical.
c) Suppose there is an increase in the price of X from Po=$8 to P1=$12 (arbitrarily chosen).
i) Dupuit change in Bree’s consumer’s surplus = area PoP1CA.
ii) Bree’s compensating variation for this price increase = area PoP1BA
- From last time we know that: X*=Py2/4Px2 and Y*=I/Py – (Py/4Px). Now you just take the appropriate partials:
a) (∂X/∂Px)M = (-2/4)Py2Px-3
b) (∂X/∂Py)M = (2/4)PyPx-2
c) (∂Y/∂Px)M = (1/4)PyPx-2
d) (∂Y/∂Py)M = -IPy-2 –(1/4)Px-1
- Ralph has the following utility function: u(X,Y)=XY.
a) This is standard Cobb-Douglas with α=1/2, so X*=I/2Px and Y*=I/2Py.
b) If I=100 and Px=Py=1, then X*=50 and Y*=50.
c) If I=100 and Px=2 and Py=1, then X*=25 and Y*=50.
d) Revenue = $25, so now if there is a lump sum tax, Ralph has I=75 and with Px=Py=1 you get X*=37.5 and Y*=37.5
e) So, with the unit tax, Ralph has u=25*50=1250 utils. If he makes the bribe and gets the income tax passed his income is $72 and Px=Py=1 and his bundle is X*=36 and Y*=36 so that utility = 1296 utils. Since this utility is great than 1250, he should cut the deal with the politician.