Investment and production

• Our objective, eventually, is to have a general equilibrium model with both preferences and production (and maybe frictions or limited market structure) specified. But it makes sense to study preferences and production separately. Walk before you try to run. (See the graph of preferences, technology, and separating prices.) Theory • Here, I'll write down the simplest standard preferences, and let's see how they fit in to asset pricing. Analogously, the consumption problem is, maximize utility in contingent claim markets with an endowment stream e t max E ∞ X t=0 β t u(c t) s.t. E 0 ∞ X t=0 Λ t Λ 0 (c t − e t) = 0. The familiar answer is, β u 0 (c t) u 0 (c 0) = Λ t Λ 0 So, the question is, what happens if we do exactly the same sort of thing from the producer's side? • Theory preview: y t+1 = θ t+1 f (k t) max {k t } E(m t+1 y t+1) − k t = max {k t } E(m t+1 θ t+1 f (k t)) − k t ∂/∂k : 1 = E(m t+1 θ t+1 f 0 (k t)) 1. Interpretation 1: " price the investment return consistently with the other returns " 1 = E(m t+1 R I t+1 (k t)) 2. Interepretation 2: marginal q mc of one unit of capital = ∂V ∂k t = ∂ ∂k t E(m t+1 θ t+1 f (k t)) 3. With constant returns to scale, f (k t) = m k k t then we have two special additional results: 109