## Help on First Fundamental Theorem of Welfare Economics with taxation

The First Fundamental Theorem of Welfare Economics, the Theorem says that a competitive equilibrium allocation is Pareto efficient. Consider the following example: Let r1 = (10, 0), r2 = (0, 10), u1(x, y) =x+2y, u2(x, y) = 2x+y, p = (.5, .5), t = (2,2). In this example show that the equilibrium allocation in the model with excise taxation is the endowment. This is a corner solution so marginal rates of substitution may not be well defined or equal the price ratio. Show that the following allocation is Pareto preferable: x1 = (0, 10), x2 = (10, 0) (The example uses weakly convex preferences — merely for convenience; it is not essential). Can you conclude that the First Fundamental Theorem of Welfare Economics does not validly apply to the model with excise taxation? Explain.

HINT: Excise taxes change prices and price ratios. In the neighborhood of the endowment point, for household 1 we have MRS(x1,y1) = Ux1/Uy1 = 1/2 > Px/(Py + ty) = .5/2.5 , so household 1 does not trade away from endowment. Similarly, for household 2, we have MRS(x2,y2) = Ux/Uy = 2/1 < (Px + tx)/Py = 2.5/.5 , so household 2 does not trade away from endowment. The market clears with no transactions. The problem states that there is a Pareto superior attainable allocation. What do you conclude?

r1, r2 corresponds to the given endowment for households 1 and 2
p=price
t=taxation
u1,u2=utility functions for each household.