1.  (15 points) War financing in a two-period model

Suppose that the world only lasts for two periods.  Subscript denotes time.  Country Ur is involved in a war at the first period.  The spending needs by the government of Ur are g1  = 1 and g2  = 0.  Production in country Ur only needs labor and Y = 10L. Households in country Ur value consumption and leisure with a utility function u(c,h) =^c + 4.4^h and without time discounting, i.e. β = 1. Country Ur only has labor income tax; tax rates in two periods are τ1(l)  and τ2(l), respectively.

(a)  (6 points) Country Ur can finance the war by tax in the first period only. The

tax plan is τ1(l)   = 45.57% and τ2(l)   = 0%.  Let w1  and w2  denote wage rates and r the real interest rate.  Households’ budget constraints in two periods are the following:

c1 + s = (1 − τ1(l))w1 (1 − h1 )

c2  = (1 + r)s + (1 − τ2(l))w2 (1 − h2 )

Households’ intertemporal budget constraint is

c1 +  = (1 − τ1(l))w1 (1 − h1 ) + (1 − τ2(l) − h2 )

Since all household are alike, at the equilibrium we have additional market clearing condition, where s = 0.  Define and compute the competitive equilibrium under this tax regime.

(b)  (6 points) Country Ur can also finance half of war spending by tax in the first

period then issue a patriotic war bond with 0 interest rate to finance the other half; every family is required to buy the same amount of war bond, denoted by B . (Requiring government to pay the market interest rate will make this problem too hard to be solved by hand.)  The fiscal plan in this case is τ1(l)   =  14.41%, τ2(l)  = 20.20%, and B = 0.5. Households’ budget constraints in two periods become the following:

c1 + B + s = (1 − τ1(l))w1 (1 − h1 )

c2  = B + (1 + r)s + (1 − τ2(l))w2 (1 − h2 )

Intertemporal budget is the following:

c1 +  = (1 − τ1(l))w1 (1 − h1 ) + (1 − τ2(l) − h2 ) − B

Of course, it is still the case that s = 0 at equilibrium. Compute the competitive equilibrium in this case.

(c)  (3 points) Compute household welfare under the two tax regimes. Which one is better? Comment on your result from the optimal taxation perspectives.

2.  (15 points) No taxation on intermediate goods, please!

This is a one-period economy.  Household preference over consumption and leisure is u(c,h). There are two kinds of firms: type 1 firms produce final good and type 2 firms produce intermediate good used in the final good production. Both types of firms use labor as input, denoted by L1  and L2 , respectively. Let Y denote the final good and Z the intermediate good. The technology used by type 1 firms is Y = F(L1 ,M) and the technology used by type 2 firms is Z = G(L2 ). Both technology are constant returns to scale.

The representative household has a total time endowment of 1. Households are free to work in either type of firm. Wage rate is normalized to be 1. The price of final good is p and that of intermediate goods is q . One unit of labor can either buy 1/p units of

final good or 1/q units of intermediate good. Households consume final good only.     Tax can be levied on either final good or intermediate good with a tax rate τ c  and τ z respectively. Tax revenue is used to finance government spending g, which also comes from final good production.

(a)  (3 points) The representative household solves the following problem:

maxu(c,h)

c,h

subject to   p(1 + τ )cc  ≤ 1 − h

Use household marginal condition to eliminate price and tax in household budget and derive the implementability constraint.

(b)  (3 points) A final good producing firm solves the following.

max pY − (1 + τz )qM − L1

subject to   Y ≤ F(L1 ,M)

A intermediate good producing firm solves this problem

maxqZ − L2

subject to   Z ≤ G(L2 )

Find the marginal conditions for each type of firm.