Credit Market Imperfections: Different Rates for Borrowing and Lending
Exogenous Variables and Parameters:
\(y\):
\(y^\prime\):
\(t\):
\(t^\prime\):
\(r_b\):
\(r_s\):
\(\beta \):
Endogenous Variables and Parameters:
\(c\):
\(c'\):
\(s\):
\(U\):
Algebraic Details
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:
\[c + s \leq y - T\]
\[c \leq y^\prime - T^\prime + (1+r_s) s \quad\text{ if } s \geq 0\]
\[c \leq y^\prime - T^\prime + (1+r_b) s \quad\text{ if } s \leq 0\]