Consider the standard OLG model with money. Individuals are endowed with y units of a perishable consumption good when young and nothing when old. There are N individuals in every generation. Each generation has identical preferences where u(c 1,l
​
,c 2,ℓ11
​
)=c 1,l
1/2
​
+c 2,ℓ11
1/2
​
. There exists one asset in the economy - money. The money supply grows at a constant rate z, where M l
​
=zM l−1
​
and z>1. The new money created is used to finance government purchases of g goods per young individual in every period. The initial old are endowed with M 0
​
units of money. In the following, we focus on stationary allocations. (a) Find an individual's budget constraints when young and when old. Combine them to form the individual's lifetime budget constraint. (2 marks) (b) Solve for the optimal consumption allocation (c 1
∗
​
,c 2
∗
​
) chosen by the individual in a stationary monetary equilibrium. How do (c 1
∗
​
,c 2
∗
​
) depend on z ? (2 marks) (c) Find the government budget constraint. Express government purchases g as a function of z and other parameters in the model. (2 marks) Now instead of being endowed with y units of the consumption good, individuals can supply labour ℓ 1,t
​
only when young. That is, the young supply labour ℓ 1,t
​
and consume c 1,t
​
, but the old can only consume c 2,ℓ11
​
. One unit of labour supply produces one unit of the consumption good. Each generation has identical preferences where u(c 1,ℓ
​
,c 2,l11
​
,ℓ 1,ℓ
​
)=c 1,t
1/2
​
+c 2,ℓ∤1
1/2
​
−ℓ 1,ℓ
​
. (d) Find an individual's budget constraints when young and when old. Combine them to form the individual's lifetime budget constraint. (1 mark) (c) Solve for the optimal consumption and labour supply (c 1
∗
​
,c 2
∗
​
,ℓ 1
∗
​
) chosen by the individual in a stationary monetary equilibrium. ( 2 marks) (f) How do (c 1
∗
​
,c 2
∗
​
) depend on z ? How does ℓ 1
∗
​
depend on z ? Briefly explain the intuition for your answer. ( 1 mark)