Competitive consumption and the mansion $($adapted from Goodstein Application $11\mathrm{.}1)$

Consider the following social welfare function for a small community of five people.

SW $=$ Ua$($Na$,$ Ca$,$ C~a$,$ Pa$)+$ Ub$($Nb$,$ Cb$,$ C~b$,$ Pb$)+$ Uc$($Nc$,$ Cc$,$ C~c$,$ Pc$)+$ Ud$($Nd$,$ Cd$,$ C~d$,$ Pd$)+$ Ue$($Ne$,$ Ce$,$ C~e$,$

Pe$)$

where:

N $=$ noncompetitive goods

C $=$ competitive goods

P $=$ pollution

a$,$b$,$c$,$d$,$e $=$ people

~a $=$ all people besides a

$1.$ Place positive and negative signs above all the terms in the function, illustrating whether

they increase or decrease each person$$s utility.

$2.$ Assume that producing all types of goods $($both N and C$)$ requires some level of pollution.

Will an increase in the consumption of non$-$competitive goods always increase social

welfare?

Now, let Cb be a new mansion on Cape Cod. Let P be the environmental costs of building the

mansion. Finally, let C~a$,$ C~c$,$ C~d$,$ and C~e be the status impact on folks who are not Ms$.$ b arising from

the new mansion. Assume that the positive increment to individual utility for Ms$.$ b of a one$-$unit

increase in Cb equals $$10,000$ in consumer surplus and that the negative status increment to other

people is $-$$$2,000.$ Assume also that a one$-$unit increase in Cb produces one unit of P$,$ which

decreases everyone$$s consumer surplus by $$500.$

$3.$ What are the private benefits to Ms$.$ b of consuming one more unit of Cb$?$

$4.$ What are the environmental costs to Ms$.$ b consuming one more unit of Cb$?$

$5.$ What are the status costs to others to Ms$.$ b consuming one more unit of Cb$?$

$6.$ Will social welfare increase if Ms$.$ b builds her mansion?