Kenji Sato

February 1, 2017

Letâ€™s introduce a government in the Diamond OLG model. We consider a government that finance its expense with

- lump-sum tax

We only modify the household behavior.

Young individuals, who are born without financial wealth, supply unit labor and earn wage. They divide their income between consumption, saving, **and tax**. When they become old, they simply consume what you can buy with the saving and the interest they will have earned. Note the difference in notations from the textbook.

- Let \(c_{t}^{Y}\) be the consumption of the representative consumer in generation \(t\) when they are young, (per capita consumption)
- \(G_t\) be the lump-sum tax levied on the young (per-capita basis,
*not*per-effective labor basis) - \(c_{t+1}^{O}\) be the consumption of the representative consumer in generation \(t\) when they are old.

\[ \begin{aligned} &\max_{c_{t}^{Y},c_{t+1}^{O},s_{t}} \frac{\left(c_{t}^{Y}\right)^{1-\theta}}{1-\theta} + \frac{1}{1+\rho}\frac{\left(c_{t+1}^{O}\right)^{1-\theta}}{1-\theta}\\ &\text{subject to }\\ &\qquad c_{t}^{Y}+s_{t}+G_t=w_{t},\\ &\qquad c_{t+1}^{O}=(1+r_{t+1})s_{t}. \end{aligned} \]

Assume

\[ \theta = 1, \quad \text{and} \quad F(K, AL) = K^\alpha (AL)^{1-\alpha} \]

According to the textbook, the equilibrium dynamics is given by

\[ \hat{k}_{t+1}=\frac{1}{(1+g)(1+n)(2+\rho)} \left[(1-\alpha) \hat{k}_{t}^{\alpha} - \hat G_t\right], \]

where \(\hat G_t = G_t / A_t\). How do you get this?

Unlike the Ramsey model, anticipation about future stream of the government purchases **does not** affect the individual behavior.

This is due to an individualâ€™s consumption in their old age is solely determined by the tax in the period when they are young.