Economics and Ethics

First order condition
dU/dx₂ =-v'(I-x₂) + g'(x₂)

rearranging and solving for the optimal value of x₂

v' (I-x₂)=g' (x₂)  --> m*₂=x₂*

where U' = 0

Substituting into the dictator's utility function

U= v( I-x₂-t) + g(x₂)

Applying the first order condition
dU/dx₂ = -v' (I-x₂-t)+g'(x₂) = 0

rearranging: v'(I-x₂-t) = g'(x₂)  --> x₂^t

at the optimum, marginal material utility equals marginal warm glow

prove crowding out by contribution (trying to find no crowding out)
standard: v'(I-x₂) = g'(x*₂)
tax:            v'(I-x₂-t) = g'(x^t₂)

if there is NO crowding out, x₂* must be equal x₂^t,which is impossible, so there is crowiding out
 

if crowding out is complete (one for one €)
x₂^t = x₂^*-t

but substitution can't be made, so: crowding out is incomplete under warm glow

pure altriusm: Donors care about total benefit to recipeints
they respond to to tax transfer by reducing their donations by an equal amount
Crowding out is complete: c^p= -1

warm glow: Donors care only about the contribution they make voluntary
taxes reduces their net income, so they reduce giving
since they derive no utility from taxtransfer, thex reduce givung by less than the tax.
Crowding our is incomplete -1< c^w<0

pure altruism + warm glow
U=u(m₁, m₂, x₂)

Additively seperable utility function
U= v(m₁) +f(m₂) _+g(x2₎

Crowding out is intermediate
-1=c^p<cⁱ<c^w<0