# Class Notes, Econ 8801 Lump Sum Taxes are Awesome

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1 Class Notes, Econ 8801 Lump Sum Taxes are Awesome Larry E. Jones 1 Exchange Economies with Taxes and Spending 1.1 Basics 1) Assume that there are n goods which can be consumed in any non-negative amounts; 2) I consumers indexed by i; 3) e i 2 R+ n is i 0 s endowment; 4) x i is the consumption of i; 5) g 2 R+ n is government provided goods; 6) U i = U i (x i ; g) : R+ 2n! R is i 0 s utility function. Allocation: (x 1 ; :::; x I ; g) 2 R In+n + Feasibility: g + i x i = i e i Conditional areto Optimality (given g, x 0 is are optimal): A feasible allocation is CO if there is not another feasible allocation, (x 0 1; :::; x 0 I ; g) 2 RIn+n + (note that g has not been changed) with U i (x 0 i; g) U i (x i ; g) for all i and strict for at least one i. areto Optimality (both the choice of g and the distribution of the x 0 is are optimal): A feasible allocation is O if there is not another feasible allocation (x 0 1; :::; x 0 I ; g0 ) 2 R+ In+n (note that g has been changed to g 0 ) with U i (x 0 i; g 0 ) U i (x i ; g) for all i and strict for at least one i. Lemma 1 If (x 1 ; :::; x I ; g) is.o., then (x 1 ; :::; x I ) is C..O. given g. proof: Obvious. 1

2 1.2 Implementation How can CO and O allocations be implemented? I.e., how can they be reached via decentralized means? One obvious way is to just con scate and redistribute resources: Lemma 2 If the government can con scate and redistribute resources in any way, any CO can be implemented in this way, and there will be no incentives for the private agents to trade after that. Implementation with transfers only: Implementation with Linear Taxes General Implementation: Given 0 ns, and a g a TDCE implements its equilibrium allocation You can tax all consumption goods? Consumer s roblem becomes: Max U i () s.t. n p n(1 + n )c n n p ne in So, the TDCE is p 2 R n +, an allocation, (x 1 ; :::; x I ; g) 2 R In+n + so that: 1) x i solves C 2) Feas: i x i + g = i e i 3) Gov t BB: n p ng n = n i p n n x in 2

3 1.2.2 Implementation with Transfers Only An ADT equilibrium (Arrow-Debreu equilibrium with Transfers) is given by prices, p 2 R+, n an allocation, (x 1 ; :::; x I ; g) 2 R+ In+n, and a set of transfers, (T 1 ; :::; T I ), such that: (A) For all i x i solves Max U i () s.t. pc pe i + T i... non-negativity of all variables. (B) <Accounting> (Supply = Demand) g + i x i = i e i <Government Budget Balance> i T i + pg = Results: Lump Sum Taxes are Awesome! HH1) U i is continuous, strictly increasing and strictly concave in x holding g xed. HH2) Endowments are positive: i e i >> 0. roposition 3 Under HH1 and HH2, given any TDCE allocation, there is a areto Superior allocation implemented by transfers only. roof: If it s CO already use Theorem below. If not, nd a S one that is CO and use Theorem below. Theorem 4 If the feasible allocation, (x 1 ; :::; x I ; g), is CO, and HH1 and HH2 hold, then, there is a choice of transfers, (T 1 ; :::; T I ) and prices, p, that implement (x 1 ; :::; x I ; g) as an ADT equilibrium allocation. 3

4 roof: Step 1. Notice that the restricted utility U i holding g xed satis es the assumptions of the 2nd Welfare Theorem where the relevant aggregate endowment is given by i e i g. Step 2. Depends on exactly what you think the 2nd Welfare Theorem says. For some versions, they automatically give the result given Step 1. If you think that the 2nd Welfare Theorem says: "If (x 1 ; :::; x I ) is O, then there is a price system p, such that (x 1 ; :::; x I ) with p is a CE equilibrium with no trade." Then you still need to show that you can implement this with transfers and a di erent budget constraint. In either case, it s step 1 that is the critical step. Do you know enough about people to do this? I.e., indiv speci c transfers So, this shows that the optimal way of nancing government spending is through the use of lump sum taxes. I.e., it s not just the best way to do redistribution, but also paying for spending. roblem: Show that this holds with rms/production too. roblem: Show that this holds in an in nite horizon too. Intuition "Lump sum taxes are non-distortionary they don t distort any decisions at the margin." Anything beyond this??? You should use lump sum taxes to nance g even if g is not chosen correctly. I.e., you can t do any better than this. 1.4 Are They Ever Equivalent? How about if you can only tax consumption goods? Having all of the taxes equal also implements some (?) of the lump sum allocations: Consumer s roblem becomes: Max U i () s.t. n p n(1 + n )c n n p ne in OR, if n = for all n, 4

5 Max s.t. U i () n p nc n 1 1+ where T i = n p ne ni 1 1+ n p ne ni = n p ne ni n p ne ni T i Notice that you can t get all combinations of T i out of this. As an example, if all agents are identical, it follows that all transfers will be equal. Also note that p n depends on. So, in some sense, Ramsey problems always have the solution tax everything the same and it will be equivalent to lump-sum. At least unless there is some other motive than making things O e.g., redistribution trying to pick out one particular O allocation. roblem: Show this result with production in it to. roblem: What does it say about taxing labor income? Show that you should tax leisure at the same rate as all other consumption goods in order to replicate lump sum taxes. Beyond that, everything comes from restricting the set of instruments. uzzle: What about consumption taxes that are di erent? I.e., suppose we have n di erent. And suppose we have an equilibrium with these taxes for some g i.e., 1. all the max problems are satis ed at x i and 2. i x i + g = i e i (This can t hold for all p 0 s, for some private demand is bigger than endowment, so g n would have to be negative!) Then all the MRS s are equalized (at p ), so the allocation must be CO. So, it must be true that you can also implement it with lump sum taxes what are the prices? robably p n = p n(1 + n ). And T i is de ned accordingly. 5

6 1.5 articular O allocations and interpretations ex post vs. ex ante Looking at speci c O allocations the one that maxes 1 i U I i notice that this adds some extra stu redistribution or social insurance. Even if g = 0. Why is this important? 1. One interpretation of this: one particular choice of redistribution 2. Second interpretation of this: ex ante insurance ex ante each person is equally likely to end up being one of the I types (U i ; e i ). This is a representative consumer economy (ex ante) and hence there is an obvious choice of the O allocation, the ex ante symmetric one. Alternatives would be to put more weight on names NOT types. Speci c example, U i = u for all i, n = 1, only one good. I = 2, H and L, e H > e L If you can t trade insurance contracts, c L = e L and c H = e H. max L u(c L ) + H u(c H ) s.to H c H + L c L = H e H + L e L = e Solution is c L = c H = H e H + L e L = e Is this redistribution? (i.e., Ex ost) Or is it Social Insurance? (i.e., Ex Ante) Ex Ante, all agents would prefer to solution to this problem over the ex post allocation as long as u is strictly concave. What if people aren t born yet, so that they can t trade these kinds of insurance contracts? I.e., born, nd out your type and then trade. This gives the ex post outcome as above. So, to implement the insurance outcome requires some sort of government intervention? 6

7 1.5.1 More on the Example So, to implement the insurance outcome requires some sort of government intervention ex post if these contracts can t be signed privately ex ante? What would the Lump Sum Transfers be: T H = e e H < 0 T L = e e L > 0 I.e., this requires TYE SECIFIC transfers. What if the lanner or government can t observe the types/endowments? Set up a game or mechanism. REVELATION RINCILE: Any equilibrium outcome can be implemented as a Bayesian Nash Equilibrium in a Direct Revelation Game in which telling the truth is an equilibrium strategy. Each person reports their type, and transfers are made based on reported type. T () where = H; L. Truth telling is an equilibrium implies that the following two Incentive Constraints must hold: (IC H ) (IC L ) U H ( = H) = u(e H + T (H)) U H ( = L) = u(e H + T (L)) U L ( = L) = u(e L + T (L)) U L ( = H) = u(e L + T (H)) These say, respectively: T (H) T (L) T (L) T (H) Thus, T (H) = T (L). Hence, feasibility ( H T (H) + L (T (L)) = 0) implies that T (H) = T (L) = 0. Thus, there is no social insurance possible in this example. 7

8 1.5.2 Other Thoughts/Issues What about maxing i (i)g(u i)? Maybe more redistribution? So, ex ante O is sort of a combination of classic S (ex post) and redistribution due to an ASSUMTION of ex ante identical chances of being each type combined with no ability to trade contracts before that type is realized. What if we are not all ex ante identical? 8

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