# Credit Market Imperfections: Different Rates for Borrowing and Lending

Exogenous Variables and Parameters:
 $$y$$: $$y^\prime$$: $$t$$: $$t^\prime$$: $$pH$$: $$r$$: $$\beta$$: 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 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: $c + s \leq y - T$ $c \leq y^\prime - T^\prime + (1+r) s + pH$