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the following is a monetary theory overlapping generations model question and I wonder whether I have the correct answer. Can someone please let me know? Thanks!

Question: Suppose there is a fixed population $N$ and a fixed amount of fiat money $M$. Each young person is endowed with $y_t$ goods when they are born. The person's endowment grows overtime such that $y_t = \alpha y_{(t-1)}, \alpha > 1$. People desire to hold money balances of half their endowments so that $v_tm_t = y_t/2$.

a) Find the lifetime budget constraint.

b) Write down the condition that represents the clearing of the money market. Find the real rate of return of fiat money.

My proposed answer:

a)

$$ Young: C_{1,t} + v_tm_t \leq y $$ $$ Old: C_{2,t+1} \leq v_{t+1}m_t $$ $$ Young: C_{1,t} + y_t/2 \leq y $$ $$ Old: C_{2,t+1} \leq y_{t+1}/2 $$ $$ 1/2 y_{t+1} = 1/2 \alpha y_t $$ $$ Old: C_{2,t+1} \leq 1/2 \alpha y_t $$ $$ Lifetime: C_{1,t} + C_{2,t+1}/ \alpha \leq y $$

b) $\alpha$ is the real rate of return.

I think that these answers make sense, but they are not what I saw at the following chegg link, so I would appreciate if anyone could help out. Thanks!

https://www.chegg.com/homework-help/questions-and-answers/chapter-modeled-growth-economy-growing-population-could-also-achieve-growing-economy-endow-q14826361

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