# Lump sum taxation in the overlapping generations model

I am working through my textbook's problems, using the overlapping generation's model. In this economy, people are endowed with $y$ goods when young, and nothing when old. They will choose to exchange some of their endowment for money, and hence real money demand is given by $$N(y-c_{1, t})$$ for the population $N$ of young people, and the value $v^t$ of money.

In this question, the money supply is shrinking such that $M_{t+1} = zM_{t}$ with $0<z<1$. The old are taxed $\tau$ goods by the government, payable in terms of fiat money. It then destroys this revenue.

I have already found that the rate of return of money is given by $$\frac{v_{t+1}}{v_t} = \frac{1}{z} > 1$$

The question asks for proof that this situation will not maximise individuals' welfare. I first find the lifetime budget constraint:

$$c_{1, t} + zc_{2, t+1} \leq y - z\tau$$

and then find the feasible set for consumption for each individual:

$$c_1 + c_2 + \tau \leq y$$

When I now compare the first and second period consumption intersections, I appear to find that consumption for both periods will be more than is feasible. For instance my feasible set intersections are

$$c_1 = y - \tau, c_2 = y - \tau$$

whereas for the equilibrium, I get

$$c_1 = y - z\tau, c_2 = \frac{y}{z} - \tau$$

both of which are more than their feasible equivalents, since $z < 1$. What have I done wrong here?

• By the "value of money" $v_t$ do you mean the reciprocal of the price level $v_t = 1/p_t$? – Alecos Papadopoulos Mar 25 '15 at 21:50
• Also: Are both $z$ and $\tau$ decided by the government? Or one of them? Which? Also: the government decision variable(s) are treated simply as exogenous, or the government solves some optimization problem? – Alecos Papadopoulos Mar 25 '15 at 23:54

If I understand correctly the set up, the government does not destroy real goods, it just collects fiat money and destroys it to reduce the available quantity of money (since money is in the hands of the individuals). The fact that the government calculates the amount of fiat money to collect by using a real-goods base, $\tau$, does not alter this fact. But then, all endowment is consumed and the government's actions just affect the price level.

Some assumptions:
a) population is constant at $N$ per generation
b) all individuals are identical, so we can carry the analysis by setting $N=1$
c) There are no altruistic motives, so no inheritance
d) Individuals do not carry over periods real goods: they consume some of their endowment, and what they do not consume, they sell in exchange for money
e) the subscript $1$ denotes consumption when Young, and the subscript $2$ consumption when Old.

Assume that we are in economy's period $t+1$. The Young come to the Goods market as suppliers, bringing with them $y-c_{1,t+1}$ goods. The Old come to the Goods market as buyers, bringing with them fiat money.

Market clearing implies in real terms that all real goods will change hands, and then that the Old that bought them will consume them. So we obtain

$$c_{2,t+1} = y- c_{1,t+1} \implies c_{2,t+1} + c_{1,t+1} = y \tag{1}$$ In nominal terms equilibrium requires that Total Revenue of Suppliers equals Total Expenditure of Buyers. Total Revenue is

$$TR_{t+1} = p_{t+1}(y-c_{1,t+1}) \tag{2}$$

The Old will want to spend all their available money (since money is not consumed per se, and they don't leave any inheritance). The money they got in the previous period (prior to being taxed) was Total Revenue of Suppliers in period $t$. The tax is paid on $t+1$, has a real base $\tau$ and so its nominal value is $T_{t+1} = p_{t+1}\tau$. So Total Expenditure is

$$TE_{t+1} = p_t(y-c_{1,t}) - p_{t+1}\tau \tag {3}$$

Equating $(2)$ and $(3)$ we get

$$p_{t+1}(y-c_{1,t+1}) = p_t(y-c_{1,t}) - p_{t+1}\tau \implies p_{t+1}(y-c_{1,t+1} +\tau) = p_t(y-c_{1,t})$$

and using $(1)$ we obtain an expression that involves the components of life-time consumption of the same individual

$$p_{t+1}(c_{2,t+1}+ \tau) = p_t(y-c_{1,t})\tag{4}$$

We are moreover told that what fiat money the government collects as taxes, it destroys, targeting a fixed rate of reduction. So we have

$$m_{t+1} = m_t - T_{t+1} = m_t - p_{t+1}\tau = zm_t$$

$$\implies m_t = \frac {p_{t+1}\tau}{1-z} ,\;\; 0<z<1 \tag {5}$$

We know that $m_t = p_t(y-c_{1,t})$, so we obtain

$$p_t(y-c_{1,t}) = \frac {p_{t+1}\tau}{1-z} \tag {6}$$

Combining $(4)$ and $(6)$ we see that consumption of Old is determined exogenously

$$(4), (6) \implies p_{t+1}(c_{2,t+1}+ \tau) = \frac {p_{t+1}\tau}{1-z}$$

$$\implies c_{2,t+1} = \frac {z}{1-z}\tau \tag{7}$$

Since the endowment $y$ is also exogenous, consumption when Young is also determined exogenously (from $(1)$)

$$c_{1,t+1} = y-\frac {z}{1-z}\tau \tag{8}$$

But if $z$ and $\tau$ are constant over time, then the time subscripts become irrelevant. Each generation consumes exactly the same when Young and when Old as the previous and the next ones:

$$c_1 = y- \frac {z}{1-z}\tau ,\;\; c_2 = \frac {z}{1-z}\tau \tag{9}$$

Using $(9)$ together with $(4)$ we can determine the evolution of the price level

$$(4), (9) \implies p_{t+1}\left(\frac {z}{1-z}\tau+ \tau\right) = p_t\frac {z}{1-z}\tau$$

$$\implies \frac {p_{t+1}}{p_t} = z <1 \tag{10}$$

as should be expected. Constant amount of goods with shrinking amount of fiat money $\implies$ deflation.

As regards the issue of whether the equilibrium here is welfare maximizing.

Eq. $(4)$ combined with $(10)$ gives us the life-time budget constraint. Assuming logarithmic utility, the individual would solve

$$\max V = \ln(c_1)+ \frac 1{1+\rho}\ln(c_2)\\ s.t. \;\; y - zc_{2} - z\tau = c_{1} \tag {11}$$

The first order condition with respect to $c_2$ is

$$-z\frac 1 {c_1} + \frac 1{1+\rho}\frac 1 {c_2} = 0 \implies c_1 = (1+\rho)zc_2 \tag{12}$$

Combining with the life-time budget constraint as well as $(9)$ we end up with a quadratic in $z$ in order for the equilibrium to be utility-maximizing.

$$(11), (12) \implies (1+\rho)zc_2 = y - zc_{2} - z\tau \implies (2+\rho)zc_2^* = y - z\tau \tag{13}$$

$$(9), (13):\;\; c_2^{eq.} = c_2^* \implies \frac {(2+\rho)\tau z^2}{1-z} = y - z\tau$$

$$\implies (1+\rho)\tau z^2 +(y+\tau)z - y = 0$$

This has a single positive real solution

$$z^* = -\frac {y+\tau}{2(1+\rho)\tau} + \left[\left(\frac {y+\tau}{2(1+\rho)\tau}\right)^2+\frac {y}{(1+\rho)\tau}\right]^{1/2}$$

Now, for this value of $z$ to be feasible, not only do we want it smaller than unity, but we must also consider the non-negativity constraint on the level of consumption of the first period.

We want $$c_1 > 0 \implies y- \frac {z}{1-z}\tau > 0 \implies z < \frac {y}{y+\tau} <1$$

So one should check whether $$-\frac {y+\tau}{2(1+\rho)\tau} + \left[\left(\frac {y+\tau}{2(1+\rho)\tau}\right)^2+\frac {y}{(1+\rho)\tau}\right]^{1/2} < \;? \; \frac {y}{y+\tau}$$

which holds. So $z^*$ is feasible, and for this single value the imposed allocation of consumption will equal the utility-maximizing solution (at least for logarithmic utility).