<Tr> <Th> (show) Proof </Th> </Tr> <Tr> <Td> It is worth keeping in mind that an equilibrium for the model may not necessarily exist . If it exists and there are no taxes (I = 0, ∀ h), then demand equals supply, and the equilibrium is found by: <P> ∑ x _̄ h (q, I, z) − ∑ y _̄ f (p, z) = x _̄ (q, I, z) − ∑ y _̄ f (p, z) = 0 (\ displaystyle \ sum (\ bar (x)) ^ (h) (q, I, z) - \ sum (\ bar (y)) ^ (f) (p, z) = (\ bar (x)) (q, I, z) - \ sum (\ bar (y)) ^ (f) (p, z) = 0) </P> <P> Let's use ∂ E h ∂ q = E q h (\ displaystyle (\ frac (\ partial E ^ (h)) (\ partial q)) = E_ (q) ^ (h)) as a simplifying notation, where E h (q, z h, u h) (\ displaystyle E ^ (h) \ left (q, z ^ (h), u ^ (h) \ right)) is the expenditure function that allows the minimization of household expenditure for a certain level of utility . If there is a set of taxes, subsidies, and lump sum transfers that leaves household utilities unchanged and increase government revenues, then the above equilibrium is not Pareto optimal . On the other hand, if the above non taxed equilibrium is Pareto optimal, then the following maximization problem has a solution for t = 0: </P> <Dl> <Dd> maximize t, I R = t ⋅ x _̄ − ∑ I h s u b j e c t t o I h + ∑ a h f π f = E h (q, z h; u _̄ h) (\ displaystyle (\ begin (aligned) & (\ underset (t, I) (\ operatorname (maximize))) &&R = t \ cdot (\ bar (x)) - \ sum I ^ (h) \ \ & \ operatorname (subject \; to) &&I ^ (h) + \ sum a ^ (hf) \ pi ^ (f) = E ^ (h) (q, z ^ (h); (\ bar (u)) ^ (h)) \ \ \ end (aligned))) </Dd> </Dl> <P> This is a necessary condition for Pareto optimality . Taking the derivative of the constraint with respect to t yields: </P> <P> d I h d t + ∑ a h f (π z f d z f d t + π P f d p d t) = E q h d q d t + E z h d z h d t (\ displaystyle (\ frac (dI ^ (h)) (dt)) + \ sum a ^ (hf) \ left (\ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) + \ pi _ (P) ^ (f) (\ frac (dp) (dt)) \ right) = E_ (q) ^ (h) (\ frac (dq) (dt)) + E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt))) </P> <P> Where π z f = ∂ π ∗ f ∂ z f (\ displaystyle \ pi _ (z) ^ (f) = (\ frac (\ partial \ pi _ (*) ^ (f)) (\ partial z ^ (f)))) and π ∗ f (p, z f) (\ displaystyle \ pi _ (*) ^ (f) (p, z ^ (f))) is the firm's maximum profit function . But since q = t + p, we have that dq / dt = I + dp / dt . Therefore, substituting dq / dt in the equation above and rearranging terms gives: </P> <P> E q h + (E q h − ∑ a h f π P f) d p d t = d I h d t + (∑ a h f π z f d z f d t − E z h d z h d t) (\ displaystyle E_ (q) ^ (h) + \ left (E_ (q) ^ (h) - \ sum a ^ (hf) \ pi _ (P) ^ (f) \ right) (\ frac (dp) (dt)) = (\ frac (dI ^ (h)) (dt)) + \ left \ (\ sum a ^ (hf) \ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) - E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt)) \ right \)) </P> <P> Summing over all households and keeping in mind that ∑ a h f = 1 (\ displaystyle \ sum a ^ (hf) = 1) yields: </P> <P> ∑ E q h + (∑ E q h − ∑ π P f) d p d t = ∑ d I h d t + (∑ π z f d z f d t − ∑ E z h d z h d t) (\ displaystyle \ sum E_ (q) ^ (h) + \ left (\ sum E_ (q) ^ (h) - \ sum \ pi _ (P) ^ (f) \ right) (\ frac (dp) (dt)) = \ sum (\ frac (dI ^ (h)) (dt)) + \ left \ (\ sum \ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) - \ sum E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt)) \ right \)) </P> <P> By the envelope theorem we have: </P> <P> x ^ k h (q; z h, u h) = ∂ E h ∂ q z h, u h (\ displaystyle (\ widehat (x)) _ (k) ^ (h) (q; z ^ (h), u ^ (h)) = \ left. (\ frac (\ partial E ^ (h)) (\ partial q)) \ right _ (z ^ (h), u ^ (h))) </P> <P> ∂ π ∗ f ∂ p k 1 z f = y k f (\ displaystyle \ left. (\ frac (\ partial \ pi _ (*) ^ (f)) (\ partial p_ (k_ (1)))) \ right _ (z ^ (f)) = y_ (k) ^ (f)); ∀ k </P> <P> This allows the constraint to be rewritten as: </P> <P> x _̄ + (x _̄ − y _̄) d p d t = ∑ d I h d t + (∑ π z f d z f d t − ∑ E z h d z h d t) (\ displaystyle (\ bar (x)) + \ left ((\ bar (x)) - (\ bar (y)) \ right) (\ frac (dp) (dt)) = \ sum (\ frac (dI ^ (h)) (dt)) + \ left (\ sum \ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) - \ sum E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt)) \ right)) </P> <P> Since x _̄ = y _̄ (\ displaystyle (\ bar (x)) = (\ bar (y))): </P> <P> ∑ d I h d t = x _̄ − (∑ π z f d z f d t − ∑ E z h d z h d t) (\ displaystyle \ sum (\ frac (dI ^ (h)) (dt)) = (\ bar (x)) - \ left (\ sum \ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) - \ sum E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt)) \ right)) </P> <P> Differentiating the objective function of the maximization problem gives: </P> <P> d R d t = x _̄ + d x _̄ d t ⋅ t − ∑ d I h d t (\ displaystyle (\ frac (dR) (dt)) = (\ bar (x)) + (\ frac (d (\ bar (x))) (dt)) \ cdot t - \ sum (\ frac (dI ^ (h)) (dt))) </P> <P> Substituting ∑ d I h d t (\ displaystyle \ sum (\ frac (dI ^ (h)) (dt))) from the former equation in to latter equation results in: </P> <P> d R d t = d x _̄ d t ⋅ t + (∑ π z f d z f d t − ∑ E z h d z h d t) = d x _̄ d t ⋅ t + (Π t − B t) (\ displaystyle (\ frac (dR) (dt)) = (\ frac (d (\ bar (x))) (dt)) \ cdot t+ (\ sum \ pi _ (z) ^ (f) (\ frac (dz ^ (f)) (dt)) - \ sum E_ (z) ^ (h) (\ frac (dz ^ (h)) (dt))) = (\ frac (d (\ bar (x))) (dt)) \ cdot t+ (\ Pi ^ (t) - B ^ (t))) </P> <P> Recall that for the maximization problem to have a solution at = 0: </P> <P> d R d t = (Π t − B t) = 0 (\ displaystyle (\ frac (dR) (dt)) = \ left (\ Pi ^ (t) - B ^ (t) \ right) = 0) </P> <P> In conclusion, for the equilibrium to be Pareto optimal dR / dt must be zero . Except for the special case where ∏ and B are equal, in general the equilibrium will not be Pareto optimal, therefore inefficient . </P> </Td> </Tr> <P> ∑ x _̄ h (q, I, z) − ∑ y _̄ f (p, z) = x _̄ (q, I, z) − ∑ y _̄ f (p, z) = 0 (\ displaystyle \ sum (\ bar (x)) ^ (h) (q, I, z) - \ sum (\ bar (y)) ^ (f) (p, z) = (\ bar (x)) (q, I, z) - \ sum (\ bar (y)) ^ (f) (p, z) = 0) </P> <P> Let's use ∂ E h ∂ q = E q h (\ displaystyle (\ frac (\ partial E ^ (h)) (\ partial q)) = E_ (q) ^ (h)) as a simplifying notation, where E h (q, z h, u h) (\ displaystyle E ^ (h) \ left (q, z ^ (h), u ^ (h) \ right)) is the expenditure function that allows the minimization of household expenditure for a certain level of utility . If there is a set of taxes, subsidies, and lump sum transfers that leaves household utilities unchanged and increase government revenues, then the above equilibrium is not Pareto optimal . On the other hand, if the above non taxed equilibrium is Pareto optimal, then the following maximization problem has a solution for t = 0: </P>

An individual who wants others to pay for public goods