# Money Supply and the Budget Constraint in Lagos & Wright 2005 model

### Context

I'm studying the Lagos & Wright (2005) model as presented in the Chapter 3 of the book "Money, Payments & Liquidity" by Guillaume Rocheteau and Ed Nosal.

Simplifying the context (see page 45), the value function for an agent holding $$m\in\mathbb{R}_+$$ units of money, evaluated at the beginning of the market, satisfies: $$V_t(m) = \max_{m'\in\mathbb{R}_+,x,y}\{x-y+\beta V_{t+1}(m')\}$$ $$\text{s.t.}\quad\quad x + \phi_tm' = y + \phi_tm$$ Where $$x$$ is the amount of good consumed; $$y$$ is the amount of work done; $$\phi_t$$ is the price of money (in terms of goods) in $$t$$; $$m'$$ is the amount of money the agent holds for the next period.

In that context, since it is valid for all agents, it's easy to say that money supply is constant: $$M_t = M_{t+1}$$.

### But what if there's a transference based on the amount of money agents hold to the next period?

So I'm proposing an extra transference for the agents based on the amount of money they hold, but it is financed by an increase in the money supply.

So the budget constraint of the agent should be: $$\text{s.t.}\quad\quad x + \phi_t(1-\tau)m' = y + \phi_tm$$

And how to calculate the amount of money in the period $$t+1$$? Is the following correct? $$(1-\tau)M_{t+1} = M_t$$