General consumer optimization problem:
\(\mathcal{L} = \sum \limits_{s = t} ^T \frac{u(c)}{(1 + \delta)^{s - t}} - \sum \lambda _s [A_s - (1 + r)A_{s - 1} - y_s + c_s] + \mu A_T\)
Notice there are two constraints since current assets must be equal to the past period's assets (times interest) plus income and minus consumption AND final assets must be greater than zero.
The first order conditions imply:
\(\frac{u'(c_t)}{\frac{u'(c_s)}{(1 + \delta) ^{s - t}}} = (1 + r)^{s - t}\)
Modigliani's finding:
\(\frac{c_t}{y_t} = \alpha \frac{A_{t - 1}}{y_t} + \beta\)
This explains the empirical finding that consumption over income is constant in time series but declining in cross-sectional data. It turns out, with cross-sectional data, you are including people in different "cohorts" or stages in their life. Hence, you would not expect assets and income to move together necessarily.
Friedman's permanent income equations:
\(c = c^P + c^T \) and \(y = y^P + y^T\)
The econometric problem is that they will vary together and the coefficients will be downward-biased.