Testing the permanent-income hypothesis: new evidence from West-German states (Länder)

Abstract

This paper investigates whether time-series data from 11 West-German states (Länder) provide evidence in accord with the implication of the permanent-income hypothesis (PIH) for the stochastic relationship between consumption and income innovations. The empirical results do not support this hypothesis, in the sense that the response of consumption to income innovations is found to be much weaker than is predicted by the PIH. Moreover, for each individual state as well as for Germany as a whole, the response was found to be asymmetric, being stronger for negative than positive income innovations. This evidence of asymmetry is consistent with a model in which consumers are liquidity constrained.

Appendix

Time-varying interest rates

With deterministic time-varying interest rates, the difference equation for W is

$$W_{{i,t + 1}} = {\left( {1 + r_{t} } \right)}W_{{it}} + Y_{{it}} - C_{{it}} $$

The value of W at any time after t is

$$\begin{array}{*{20}c} {W_{{i,t + 1}} }{ = W_{{it}} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)} + {\sum\limits_{j = 0}^{i - 1} {E{\left[ {{\left( {Y_{{i,t + j}} - C_{{i,t + j}} } \right)}} \right]}{\prod\limits_{m = j}^{i - 1} {{\left( {1 + r_{{t + 1 + m}} } \right)}} }} }} }} \\ {}{ = {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}{\left\{ {W_{{it}} + {\sum\limits_{j = 0}^{i - 1} {{\left[ {E{\left[ {{\left( {Y_{{i,t + j}} - C_{{i,t + j}} } \right)}} \right]}{\prod\limits_{k = 0}^j {{\left( {1 + r_{{t + 1k}} } \right)}^{{ - 1}} } }} \right]}} }} \right\}}} }} \\ \end{array} $$

Discounting both sides gives

$$W_{{i,t + i}} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } } = W_{{it}} + {\sum\limits_{j = 0}^{i - 1} {{\left[ {E{\left[ {{\left( {Y_{{i,t + j}} - C_{{i,t + j}} } \right)}} \right]}{\prod\limits_{k = 0}^j {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right]}} }$$

The No-Ponzi condition is that the left side go to zero as i goes to infinity, which implies that

$${\sum\limits_{j = 0}^{i - 1} {{\left[ {E{\left( {C_{{i,t + j}} } \right)}{\prod\limits_{k = 0}^j {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right]} = W_{{i,t}} + {\sum\limits_{j = 0}^{i - 1} {{\left[ {E{\left( {Y_{{i,t + j}} } \right)}{\prod\limits_{k = 0}^j {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right]}} }} } = A_{t} $$

where A is total household assets. The general first-order condition is

$$\frac{{{\left( {1 + \rho } \right)}^{{ - 1}} E_{t} {\left[ {U\prime {\left( {C_{{t + j + 1}} } \right)}} \right]}}}{{E_{t} {\left[ {U\prime {\left( {C_{{t + j}} } \right)}} \right]}}} = {\left( {1 + r_{{t + j}} } \right)}^{{ - 1}} $$

For the quadratic utility function

$$U{\left( {C_{t} } \right)} = aC_{t} - bC^{2}_{t} $$

the first-order condition is

$$\frac{{{\left( {1 + \rho } \right)}^{{ - 1}} E_{t} {\left[ {a - 2bC_{{t + j + 1}} } \right]}}}{{E_{t} {\left[ {a - 2bC_{{t + j}} } \right]}}} = {\left( {1 + r_{{t + j}} } \right)}^{{ - 1}} $$

In period t, the value of C _t (i.e., for j=0) is known, so we can solve for E[C _t+1 ] in terms of C _t :

$$E{\left[ {C_{{t + 1}} } \right]} = \frac{a}{{2b}}\frac{{\rho - r_{t} }}{{1 + r_{t} }} + \frac{{1 + \rho }}{{1 + r_{t} }}C_{t} $$

We then can iterate forward to obtain the solution for any future C:

$$\begin{array}{*{20}c} {E{\left[ {C_{{t + i}} } \right]}}{ = \frac{a}{{2b}}{\sum\limits_{j = 0}^{i - 1} {{\left[ {{\left( {\rho - r_{{t + {\left( {i - 1} \right)} - j}} } \right)}{\left( {1 + \rho } \right)}^{j} {\prod\limits_{m = {\left( {i - 1} \right)} - j}^{i - 1} {{\left( {1 + r_{{t + m}} } \right)}^{{ - 1}} } }} \right]}} }} \\ {}{ + {\left( {1 + \rho } \right)}^{i} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} C_{t} } }} \\ \end{array} $$

We next substitute this solution into the budget constraint:

$$A_{t} = {\sum\limits_{I = 0}^m {{\left\{ {{\left( {\frac{a}{{2b}}{\sum\limits_{j = 0}^{i - 1} {{\left[ {{\left( {\rho - r_{{t + {\left( {i - 1} \right)} - j}} } \right)}{\left( {1 + \rho } \right)}^{j} {\prod\limits_{m = {\left( {i - 1} \right)} - j}^{i - 1} {{\left( {1 + r_{{t + m}} } \right)}^{{ - 1}} } }} \right]} + {\left( {1 + \rho } \right)}^{i} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} C_{t} } }} }} \right)}{\prod\limits_{k = 0}^i {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right\}}} }$$

which can be expressed as

$$A_{t} = \alpha _{t} + \beta _{t} C_{t} $$

where

$$\begin{array}{*{20}c} {\alpha _{t} }{ = {\sum\limits_{i = 0}^m {\left\{ {{\left[ {\frac{a}{{2b}}{\sum\limits_{j = 0}^{i - 1} {{\left( {\rho - r_{{t + {\left( {i - 1} \right)} - j}} } \right)}{\left( {1 + \rho } \right)}^{j} {\prod\limits_{m = {\left( {i - 1} \right)} - j}^{i - 1} {{\left( {1 + r_{{t + m}} } \right)}^{{ - 1}} } }} }} \right]}} \right.} }} \\ {}{\left. {{\prod\limits_{k = 0}^i {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right\}} \\ \end{array} $$

$$\beta _{t} = {\sum\limits_{i = 0}^m {{\left\{ {{\left[ {{\left( {1 + \rho } \right)}^{i} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right]}{\prod\limits_{k = 0}^i {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right\}}} }$$

Finally, we solve for the optimal value of C _t

$$C_{t} = \beta ^{{ - 1}}_{t} A_{t} - \beta ^{{ - 1}}_{t} \alpha _{t} $$

From this expression, we can obtain the response of C _t to a change in any future income:

$$\Delta C_{t} = {\left[ {\beta ^{{ - 1}}_{t} {\prod\limits_{k = 0}^{i - 1} {{\left( {1 + r_{{t + k}} } \right)}^{{ - 1}} } }} \right]}\Delta Y_{{t + i}} = \gamma _{{t,i}} \Delta Y_{{t + i}} $$

Notice that, when r _t+k =ρ for all k, then γ _t,i reduces to the usual (1+r)^−(t+1). Because ρ is an attraction for the interest rate, we expect to see r fluctuate closely around ρ in any time series sample of reasonable length with an average value of approximately ρ. Thus in any such sample, we expect γ _t,i to be close to (1+r)^−(t+1). If Y follows an ARIMA process, then the response of C _t to an innovation in Y involves a weighted sum of the γ _t,i , where the weights depend on the AR and MA coefficients. Because the γ _t,i are time-varying, the weighted sum also will be time-varying, but because the γ _t,i are close to (1+r)^−(t+1), we expect the weighted sum of the γ _t,i to be close to the expression given in Eq. (15) in the main text. For this reason, assuming that the interest rate equals ρ for all time periods is a reasonable simplification that does not alter the results in this paper in any important way.

Absence of certainty equivalence

The general household choice problem without certainty equivalence (i.e., quadratic utility) and uncertainty in labor income is typically not solvable analytically. We can obtain analytical solutions in special cases. For example, suppose that there is uncertainty only in interest rates. Labor income either has no uncertainty or any uncertainty in it can be diversified away through life insurance, unemployment insurance, and other such schemes. We then can obtain analytic solutions for C _t for many forms of the utility function, such as members of the HARA class. For example, logarithmic utility yields the result (see Blanchard and Fischer, 1989)

$$C_{t} = \frac{\rho }{{1 + \rho }}W_{t} $$

The problem here is how to define permanent income. Permanent income is the annuity value of lifetime wealth, but it is unclear how to compute that value with a random interest rate. One solution is to use the risk-free interest rate, R. That rate may vary over time (deterministically) but should equal the time preference rate Δ on average. We can use the average value of R to define permanent income as

$$Y^{P}_{t} = \frac{{\overline{R} }}{{1 + \overline{R} }}W_{t} = \frac{\rho }{{1 + \rho }}W_{t} $$

We then immediately have

$$C_{t} = W^{P}_{t} $$

implying that

$$\Delta C_{t} = \Delta Y^{P}_{t} $$

which is the same result as that obtained in the main text with quadratic utility and constant interest rates. Admittedly, this case requires several strong restrictions, but it does show that our assumptions in the main text are at least defensible.