Ricardian Equivalence: Timing of taxes does not matter

• Household intertemporal budget constraint: c1  +             = y1  +             − T1  −

c2                              y2                               T2    

1 + R               1 + R               1 + R .

– Suppose financial wealth f1  = 0, and there are lump-sum taxes in both periods, T1  and T2 .

– Budget constraint in period 1: c1  + (f2  − f1 ) = y1  − T1 .

– Budget constraint in period 2: c2  = y2  + (1 + R)f2  − T2 .

• Government intertemporal budget constraint: G1  +   G2      = T1  +    T2    

– Suppose initial debt B1  = 0, and we put everything in real term.

– Budget constraint in period 1: B2  = (1 + R)B1  + G1  − T1 .

– Budget constraint in period 2: B3  = (1 + R)B2  + G2  − T2  = 0.

c2                              y2                               G2    

1 + R               1 + R                1 + R .

– If G1  and G2  remain unchanged, lifetime after-tax income does not change.

– A change in T1 , offset by an equal (in present-value terms) and opposite change  in T2 , holding G1  and G2  fixed, has no effect on consumption / the economy.

A Model of Flat Taxes

• Flat labor-income tax rate: τ .

• Household problem

l1+e  

– Utility function: u(c, l) = c − B

– Budget constraint: pc = (1 − τ )wl + pc0 .

• Firm problem

– Technology: y = Al .

– Profit: py − wl = pAl − wl = (pA − w)l . w

p

• Solution

– Objective function: maxl (1 − τ )    l + c0  − B

– Elasticity of labor supply:  d(w/p)/(w/p) = d log   = ϵ .

– Output: y = Al = A[(1 − τ )A )]1/e  = (1 − τ )1/eA e(1)+1

– Note that  = −  . The absolute value is the percentage increase in income from a tax cut, which increases as τ increases.

– Laffer Curve - total revenue from tax: T = τ y = τ (1 − τ )1/eA e(1)+1

Exchange Rates

• Exchange rate of currency i, ϵi , is the number of units of a currency that exchange for one dollar.

– The dollar value of 1 unit of currency i is  1

– Higher ϵi  means currency i is less valuable.

• Law of One Price and Purchasing-Power Parity

– Pi  is the price of a good in country i (in terms of currency i).

— ei  units of currency i, and ej   units of currency j both can change to one dollar,

ej

ei

• Relative PPP

— Define ∆ei  = et(i)+1  − et(i) .

∗ If ∆ei  > 0, currency i is less valuable over time - a depreciation.  ∗ If ∆ei  < 0, currency i is more valuable over time - an appreciation.

ej  + ∆ej           Pj  + ∆Pj

ei  + ∆ei           Pi  + ∆Pi .

∗ Assume ∆ · ∆ ≈ 0, and use PPP today: ei   = Pi     ⇐⇒ ej Pi  = ei Pj .

∗ ∆Pi ej  + Pi ∆ej  = ∆Pj ei  + Pj ∆ei .

∗ Divide LHS by ej Pi , and divide RHS by ei Pj   (they are equal!).

∗     +         =           +   ej

∆ei                      ∆ej           ∆ei ⇐⇒            − ei                           ej                ei ∆ei − ei      = πj  − πi .

=

∆Pj Pj

−

∆Pi Pi

= πj  − πi .

• Interest-Rate Parity

— Nominal interest rate in country i (in terms of currency i) is Ri .

∗ Note: In previous chapters, we use i for nominal interest rate and R for real interest rate, but we follow the notation in class here.

— Return of holding assets in different currencies:  =  . ∗ Note that  =  = 1 +  , we get  =  .

∗ Assume Ri ∆ej   ≈ 0, and Rj ∆ei   ≈ 0, we get Ri  + ∆ej   = Rj  + ∆ei

 

— IRP: Rj  − Ri  =         −

∗ If the PPP in relative form holds, we get Rj  − Ri  = πj  − πi    ⇐⇒ Rj  − πj  = Ri − πi , which shows that the real interest rates are the same in countries (in terms of currencies) i and j.