### Demographics and the Politics of Capital Taxation in a Life-Cycle Economy

Xavier Mateos-Planas
2010 The American Economic Review
This appendix considers a more general case for the model in Section 3 of the paper. Here I denote periods t = 1, 2. The population is divided into two types of agents, i, old and young, with sizes µ old and 1 − µ old respectively. Agents own productive inputs in the form of capital k i t and one unit of labour, which earn an interest rate R t and a wage rate w t respectively. The income from the two inputs is taxed at rates τ k t and τ l t respectively. The individual after-tax income can be
more » ... tax income can be divided between current consumption, c i t , and capital next period, k i t+1 . Firms produce output in each period with a Cobb-Douglass production function of labour and capital, with a capital share α ∈ (0, 1). The government uses the tax revenues to fund an amount of government spending equivalent to a share g of output. Assume 1 − g ≥ α. Markets for inputs and output are competitive. In the first period, the endowments of capital across types, k i 1 , as well as the tax rates, τ k 1 and τ l 1 , are given. Individuals choose investment and consumption to maximise their utility. In period 1, next-period tax rates will be chosen by an individual of a designated group i in order to maximise her own utility. An agent's utility depends only on her lifetime consumption, and is represented by log c t + β log c t+1 , with β > 0. Define the after-tax current income for an individual in group i at time t as Similarly, for the aggregate economy the after-tax income is y t ≡ (1−τ k t )R t k t +(1−τ l t )w t . In equilibrium, factor markets clear so k t = i µ i k i t , the government constraint holds so g = ατ k t + (1 − α)τ l t , and factor prices equal their marginal products so w t = (1 − α)k α t and R t = αk α−1 t . Then it follows that these measures of disposable income can be written as and Consider first the equilibrium determination of capital at t = 2 for an already given tax rate on capital τ k t , and initial incomes y t−1 and y i t−1 . The individual decision problem 1