# Monetary Theory Money Model Help

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.

$$Young: C_{1,t} + v_tm_t \leq y$$ $$Old: C_{2,t+1} \leq v_{t+1}m_t$$ $$Young: C_{1,t} + y_t/2 \leq y$$ $$Old: C_{2,t+1} \leq y_{t+1}/2$$ $$1/2 y_{t+1} = 1/2 \alpha y_t$$ $$Old: C_{2,t+1} \leq 1/2 \alpha y_t$$ $$Lifetime: C_{1,t} + C_{2,t+1}/ \alpha \leq y$$
b) $$\alpha$$ is the real rate of return.