# 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:
 $$c$$: test $$c'$$: test $$s$$: test $$U$$: test

#### 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: $c + s \leq y - T$ $c \leq y^\prime - T^\prime + (1+r_s) s \quad\text{ if } s \geq 0$ $c \leq y^\prime - T^\prime + (1+r_b) s \quad\text{ if } s \leq 0$