(40 points) Consider the two-period economy with endogenous production For the sake of simplicity, assume that the utility is time separable and that the per-period utility has the usual properties, namely it is increasing in both consumption and leisure (which are normal goods), and the indifference curves are strictly convex. Assume also that the technology to produce the consumption goods is well-behaved and has constant returns to scale. Moreover, labour is the only input to production and the current and future total factor productivities z and z' are equal to 1. The government finances the current and future expenditure G and G' with proportional taxes on labour income in the current and future periods. That is, if the labour incomes earned by the representative consumer in the current and future periods are wN and w'N' and 7 and 7' are the current- and future-period labour income tax rates, wN and T'w'N' are the taxes paid by the representative consumer in the current and future periods. There are no lump-sum taxes. (a) (10 points) Write down the current- and future-period budget constraints of the representative consumer. Interpret them. Derive the lifetime budget constraint. (b) (8 points) Write down the Lagragian of the consumer's problem, derive the cor- responding optimality conditions and interpret them. (c) (12 points) Using the optimality conditions found on point (b), derive and inter- pret the intertemporal optimality condition that compares the marginal cost of renouncing to one unit of leisure in the current period with the marginal benefit that this unit can generate if used to increase leisure in the future period. (e) (10 points) Is the competitive equilibrium Pareto optimal? Explain your answer and provide an economic interpretation.