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and 
. Abigail’s preference relation can be represented by the utility function 
while Brian’s preference relation can be represented by the utility function 
.(a) Derive the uncompensated demand functions and indirect utility functions for both Abigail and Brian;
(b) Given that aggregate wealth in this economy is divided evenly between Abigail and Brian, show that the representative indirect utility function
can “rationalize” aggregate demand; (c) Recall that the equivalent variation of a change in prices and income from
is defined as 
. If EV>0, what does this signify and why? If 
, and the change in prices are caused by the imposition of commodity taxes, then the deadweight loss (DWL) or excess burden of the taxes is given by 
, where 
;(d) Briefly explain why this measure may be viewed as a deadweight loss to (social) economic efficiency;
(e) Suppose that the initial aggregate budget constraint has
and 
. Using the representative indirect utility function given in (b) calculate the “aggregate” DWL of the imposition of a specific tax of 1 on good
, that leads to the price of good
rising to 2 (with the price of good
remaining unchanged);(f) Using the individuals’ indirect utility function derived in (a) calculate the two individual deadweight losses,
. Explain why 
does or does not equal DWL. 
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