Decentralizing the Ramsey/Cass-Koopmans Model

Consider first the problem of an individual infinitely lived consumer indexed by i who has some predetermined set of expectations for how the aggregate net interest rate rt and wage rate Wt will evolve. At date t household i owns some capital kt,i, and can in principle also borrow; designate the net debt of household i in period t as dt,i; since we will be examining a perfect foresight solution with perfect capital markets, the interest rate on debt must match the rate of return on assets. This means that all that really matters is the household’s total net asset position, xt,i = kt,i − dt,i. (1) Each household is endowed with one unit of labor, which it supplies exogenously, earning a wage rate Wt,i. Each household solves: