Describe analytically and graphically how much of the additional expenditure is captured by the tax revenue R = txpxhx (px; py; U).
Compensating Variation. In class we discussed commodity taxation versus lump sum taxation (using the notion of compensating variation). Suppose the individual has income I and to obtain the utility level U and prices px; py the expenditures necessary are given by E (px; py; U). We Öx the price py of commodity y and vary the price of commodity x by commodity taxation. The tax rate is tx > 0 and hence the new price is Px = (1 + tx) px. (a) Commodity Taxation. Using the notion of compensating variation to describe analytically and graphically how much more the consumer needs to expend to reach the utility level U at the new prices (Px; py). (Use the graph with px on the xaxis and hx (px; py; U) on the yaxis). Describe analytically and graphically how much of the additional expenditure is captured by the tax revenue R = txpxhx (px; py; U). (b) Lump Sum Taxation. If the price for commodity x where instead reduced from Px to px, by how much would the necessary expenditure of the consumer be reduced, use again the notion of compensating variation to describe analytically and graphically your result. Call the di§erence T = E (Px; py; U) E (px; py; U), and let this be the lump sum tax that the government could raise from the consumer. (c) Comparison. Now argue that the consumer has as much total expenditure in the lump sum tax system than in the commodity tax system, and that he reaches exactly the same utility, but that T R > 0, i.e. the lump sum taxes are a more e¢ cient way to raise tax revenue. Use again the notion of compensating variation to describe analytically and graphically your result and in particular T R.
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