\(y\): | \(y^\prime\): | \(t\): | \(t^\prime\): | \(pH\): | \(r\): | \(\beta \): |

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 also owns an illiquid asset which they can choose to sell in the second period. The value of this asset is \(pH\). (Think of it like the value of the housing which the consumers own.)
- 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 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 \((1+r)\) units of income in the second period. For each unit they borrow, they must pay back \((1+r)\) units in the second period. Borrowing is limited by the value of the consumer's asset, so \[-s \leq pH\]
- The consumer's budget constraints are given by: \[c + s \leq y - T\] \[c \leq y^\prime - T^\prime + (1+r) s + pH\]