Pictures L.0 Dirty good X.0 tltl Private mc mc Labor Demand Income tax distorts labor market while externality distorts goods market
GPB Model 3 Goods Dirty X Clean Y Leisure H Dirty good externality PPF: T=X+Y+H Producer prices are all 1. T – H is labor Taxes t X for X t l for T-H, labor Govt revenue TR= t l (T-H) + t x X Given back to consumer lump sum. Is constant
Consumer Problem Consumer problem max U(X,Y,H) s.t. (1+t x )X + Y =(1-t l )(T-H) + TR good X costs more than good Y labor (T-H) is taxed at rate t l foc: U x =(1+t x ) ; U Y = ; U H =(1-t l ) is marginal utility of income Demands are X(t x,t l ), Y(), H(). Write X(t), Y(t), H(t) for short.
Consumer prices for Goods Approx 1/(1-t l ) as 1+t l Budget constraint is then (1+t l )(1+t x )X + (1+t l ) Y =(T-H) + TR (1+t l ) So is equivalent to a tax on both goods and a subsidy on TR (not appealing, but shows that Y really isnt untaxed.
More setup Gov Rev Constraint + Budget imply PPF just substitute for TR in budget Demands Equations satisfy Budget by construction So only one equation remains TR= t l (T-H(t)) + t x X(t) Taking the total derivative and rearranging give
Social Problem U() + V(Q(X)) utility plus negative contribution from dirty good. V doesnt enter into consumer choice because it is aggregate X, not individual X that impairs breathing
Change in utility D = 1/ V Q x Num of M is (1+t) – 1 times lost hours; partial equilib welfare loss Denom is partial equilib increase in tax rev from increase in labor tax
Intermediate Steps Now substitute: (1+t x ) for U x and so on. And D for VQ x (and note the sign reversal! My error, their error? And totally differentiate the ppf to get: dY/dt x = - dH/dtx- dX/dtx Putting this together with the definition of M gives the final expression on The next slide
Comments Empirical applications are via CGEs, which have lots of other things in them. When one raises a tax on labor it is equivalent to taxing both goods, to t x is the difference in the tax rate between the two goods with t l normalized to one. A standard doesnt have the revenue recycling effect, cause there is no revenue. The pigouvian tax is probably not the right tax, though one can argue for too low or too high, depending on parameters. Goulder says too high.
The Dual DM notation. PPF: py = 0 (sign of work is negative, of goods positive) Simple version has p fixed Budget: qx = 0 govt budget: R= pz = (q-p)x Treat z as fixed 3 equations
Down to one eq. R = (q-p)x Let x = x(q-p) = x(t), the demand equation. x(t) always satisfies x(t)q = 0. R = tx(t)
W = V(q) – Dx i (q) Indirect utility less damage is the marginal util of income dV/dt = dV/dq = - x by Roys identity What happens when only taxes i and j are perturbed. Like tax on dirt up and labor down.
Double dividend means first term is non- zero and original tax system is nonoptimal.
t = q-p p is constant, so can write q(t) = q(t+p) as the demand system q(t) satisfies budget constraint by construction
Direct Approach Form the indirect utility function IN(X(t),Y(t),H(t))= IN(t) Use Roys identity to get dIN/dt x = -aX –aH dt l /dt x Adding the Pigou term dU/dt x = -aX –aH dt l /dt x + VQ x dX/dt x Here the dwl in X market decreases by aX dX; in labor market by –aH dt l /dt x dX.