We modify the household’s budget constraint to take into account the taxes they pay. It remains exactly the same as before, except for the fact that less income is available for consumption:

$$k_{t+1} + c_{t} = w_{t} + r_{t}k_{t} + (1-\delta) k_{t} - \color{red}{T_{t}}. \tag{1, Budget Const.} \label{eq:budget}$$

Except for this modification, households still present the same utility formulation and maximise:

$$\max \sum_{t=0}^{\infty} \beta^{t} u(c_{t})$$ $$\mathrm{s.t.} \quad k_{t+1} + c_{t} = w_{t} + r_{t} + (1-\delta) k_{t} - \color{red}{T_{t}}.$$

Building up the Lagrangian (as in the basic case) leads to the following equation:

$$\mathcal{L} = \sum_{t=0}^{\infty} \beta^t u(c_{t}) + \sum_{t=0}^{\infty} \lambda_{t}(w_{t} + r_{t}k_{t} + (1-\delta)k_{t} - c_{t} - k_{t+1} - \color{red}{T_{t}}).$$

Despite the introduction of taxes $T_{t}$, the optimality conditions are exactly the same as before. Indeed, we introduced lump-sum taxes, and these do not affect optimality conditions, only allocations.1

Deriving and solving we obtain again the Euler equation:

$$\frac{\partial \mathcal{L}}{\partial c_{t}} = \beta^{t} u^{\prime}(c_{t}) - \lambda_{t}$$ $$\frac{\partial \mathcal{L}}{\partial k_{t+1}} = - \lambda_{t} + \lambda_{t+1}(r_{t+1}+(1-\delta))$$

Setting both equal to zero to obtain the maximum, and combining the equations to get:

$$u^{\prime}(c_{t}) = \beta u^{\prime}(c_{t+1})(r_{t+1}+1-\delta).$$

As before, we shall also impose the Transversality condition, which remains unaltered:

$$\lim_{t \rightarrow \infty} \beta^{t}u^{\prime}(c_{t})k_{t+1} = 0.$$

It may seem that introducing government expenditures is innocuous: the Euler equation remains, market clearing will ensure that $r_{t} = f^{\prime}(k_{t})$ and $w_{t} = f(k_{t}) - f^{\prime}(k_{t}) k_{t}.$ However, it changes the steady state equilibrium: households will consume less.

  1. From a mathematical perspective, we introduce a constant which has a nil derivative. Of course, the optimal allocation will be affected because less resources are available, but it will not impact the optimal ratios. ↩︎