Introduction to Modern Economic Growth utility given initial capital stock K (0) and the time path of prices [w (t) , R (t)]∞ t=0 , and all markets clear Notice that in equilibrium we need to determine the entire time path of real quantities and the associated prices This is an important point to bear in mind In dynamic models whenever we talk of “equilibrium”, this refers to the entire path of quantities and prices In some models, we will focus on the steady-state equilibrium, but equilibrium always refers to the entire path Since everything can be equivalently defined in terms of per capita variables, we can state an alternative and more convenient definition of equilibrium: Definition 8.2 A competitive equilibrium of the Ramsey economy consists of paths of per capita consumption, capital-labor ratio, wage rates and rental rates of capital, [c (t) , k (t) , w (t) , R (t)]∞ t=0 , such that the representative household maximizes (8.3) subject to (8.7) and (8.10) given initial capital-labor ratio k (0) and factor prices [w (t) , R (t)]∞ t=0 with the rate of return on assets r (t) given by (8.8), and factor prices [w (t) , R (t)]∞ t=0 are given by (8.5) and (8.6) 8.2.2 Household Maximization Let us start with the problem of the representative household From the definition of equilibrium we know that this is to maximize (8.3) subject to (8.7) and (8.11) Let us first ignore (8.11) and set up the current-value Hamiltonian: ˆ (a, c, µ) = u (c (t)) + µ (t) [w (t) + (r (t) − n) a (t) − c (t)] , H with state variable a, control variable c and current-value costate variable µ This problem is closely related to the intertemporal utility maximization examples studied in the previous two chapters, with the main difference being that the rate of return on assets is also time varying It can be verified that this problem satisfies all the assumptions of Theorem 7.14, including weak monotonicity Thus applying Theorem 7.14, we obtain the following necessary conditions: ˆ c (a, c, µ) = u0 (c (t)) − µ (t) = H ˆ a (a, c, µ) = µ (t) (r (t) − n) = −µ˙ (t) + (ρ − n) µ (t) H lim [exp (− (ρ − n) t) µ (t) a (t)] = t→∞ 379