Government in the Diamond model

Let’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.

Household

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.

Household optimization

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

Solution

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?

Implication

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

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