SUMMARY OF THE PERMANENT INCOME MODEL (Classes 5 and 6)

(Chapter 8 in Abel&Bernanke)

• Intertemporal problem of the household, for given real interest rate r and assuming there is no government
  • problem:
    
    $$\begin{eqnarray*}max_{C1,C2} U(C1,C2)=max_{C1,C2} \ln{C1}+ \beta \ln{C2} \\ \mbox{ s.t. } (1+r)(Y1+W-C1)=C2-Y2 \end{eqnarray*}$$
    
    
  • solution:
    
    $$C1= \frac{1}{1+ \beta}(W+ Y1+ \frac{Y2}{1+r})$$
    
    
    
    
    $$C2= \frac{(1+r) \beta}{1+ \beta}(W + Y1+ \frac{Y2}{1+r})$$
    
    

  • implications: the consumption decision does not depend on the time pattern of income, but only on the present discounted value of income $Y1+ \frac{Y2}{1+r}$, that is, the permanent income
  • in particular, if $(1+r) \beta=1$ we obtain C1=C2: regardless of the time pattern of income, the household consumes the same amount in both periods (consumption smoothing)

Aside: why "permanent income"?

$C1= \frac{1}{1+ \beta}(Y1+ \frac{Y2}{1+r})$ , $C2=\frac{(1+r)\beta}{1+ \beta}(Y1+ \frac{Y2}{1+r})$ assume $(1+r) \beta=1$, then $C1 = C2= \frac{1+r}{2+r}(Y1+ \frac{Y2}{1+r})$ Let's ask the question: what is the level of income $\bar{Y}$ that gives the same PDV (present discounted value)? $\bar{Y}+ \frac{\bar{Y}}{1+r} = Y1+ \frac{Y2}{1+r}$ or $\bar{Y} = \frac{1+r}{2+r}(Y1+ \frac{Y2}{1+r})$ according to the Permanent Income Theory the agent consumes exactly as if income were permanently constant at the level $\bar{Y}$, which is in fact called permanent income

• Equilibrium (Closed Economy)
  • C1=Y1+W, C2=Y2
  • the model delivers the equilibrium real interest rate:
    $(1+r)= \frac{Y2}{Y1 \beta}$
  • prediction: the real interest rate is counter-cyclical
• Intertemporal problem of the household, for given real interest rate r and with government
  • household's problem:
    

    $$max_{C1,C2} U(C1,C2) = max_{C1,C2} \ln{C1} +\beta \ln{C2}$$

    $$\mbox{ s.t. } (1+r)(Y1-T1+W-C1)+Y2-T2=C2$$

    
    

  • solution: households consumption depends on permanent disposable income
    
    

    $$C1= \frac{1}{1+ \beta}(W+ Y1+ \frac{Y2}{1+r}-T1-\frac{T2}{1+r})$$

    $$C2= \frac{(1+r) \beta}{1+ \beta}(W + Y1+ \frac{Y2}{1+r} -T1-\frac{T2}{1+r})$$

    
    

  • government's intertemporal budget constraint:
    $T1+\frac{T2}{1+r}=G1+\frac{G2}{1+r}$
  • solution:
    
    

    $$C1= \frac{1}{1+ \beta}(W+ Y1+ \frac{Y2}{1+r}-G1-\frac{G2}{1+r})$$

    $$C2= \frac{(1+r) \beta}{1+ \beta}(W + Y1+ \frac{Y2}{1+r} -G1-\frac{G2}{1+r})$$

    
    

  • implication (Ricardian equivalence): only the present value of government spending $G1+ \frac{G2}{1+r}$ affects consumption, and not the time pattern of taxes
    in other words, for given present value of government spending the time pattern of taxes is irrelevant