\(y\): | \(y^\prime\): | \(t\): | \(t^\prime\): | \(r_b\): | \(r_s\): | \(\beta \): |

Endogenous Variables and Parameters: |

\(c\): | \(c'\): | \(s\): | \(U\): |

- In the first period. the consumer is endowed with income \(y\) and pays lump-sum tax \(T\). Likewise, in the second period, they are endowed with income \(y^\prime\) and pay lump sum tax \(T^\prime \).
- The consumer can choose to save or borrow in the first period. Savings are denoted by \(s\). For each unit they save, they will earn an additional \((1+r_s)\) units of income in the second period. For each unit they borrow, they must pay back \((1+r_b)\) units in the second period. We typically assume that \(r_b > r_s\).
- The consumer has preferences over consumption in the first period \(c\) and consumption in the second period \(c^\prime\) given by the utility function \[U(c,c^\prime) = \ln c + \beta \ln c^\prime\]
- The consumer's budget constraints are given by: \[c + s \leq y - T\] \[c \leq y^\prime - T^\prime + (1+r_s) s \quad\text{ if } s \geq 0\] \[c \leq y^\prime - T^\prime + (1+r_b) s \quad\text{ if } s \leq 0\]