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**Announcements**• Presidents’ Day: No Class (Feb. 19th) • Next Monday: Prof. Occhino will lecture • Homework: Due Next Thursday (Feb. 15)**Production, Investment, and the Current Account**Roberto Chang Rutgers University February 2007**Motivation**• Recall that the current account is equal to savings minus investment. • Empirically, investment is much more volatile than savings. • Reference here: chapter 3 of Schmitt Grohe - Uribe**The Setup**• Again, we assume two dates t = 1,2 • Small open economy populated by households and firms. • One final good in each period. • The final good can be consumed or used to increase the stock of capital. • Households own all capital.**Firms and Production**• Firms produce output with capital that they borrow from households. • The amount of output produced at t is given by a production function: Q(t) = F(K(t))**Production Function**• The production function Q(t) = F(K(t)) is increasing and strictly concave, with F(0) = 0. We also assume that F is differentiable. • Key example: F(K) = A Kα, with 0 < α < 1.**Output F(K)**F(K) Capital K**The marginal product of capital (MPK) is given by the**derivative of the production function F. • Since F is strictly concave, the MPK is a decreasing function of K (i.e. F’(K) falls with K) • In our example, if F(K) = A Kα, the MPK is MPK = F’(K) = αA Kα-1**MPK = F’(K)**Capital K**Profit Maximization**• In each period t = 1, 2, the firm must rent (borrow) capital from households to produce. • Let r(t) denote the rental cost in period t. • In addition, we assume a fraction δ of capital is lost in the production process. • Hence the total cost of capital (per unit) is r(t) + δ.**In period t, a firm that operates with capital K(t) makes**profits equal to: Π(t) = F(K(t)) – [r(t)+ δ] K(t) • Profit maximization requires: F’(K(t)) = r(t) + δ**F’(K(t)) = r(t) + δ**• This says that the firm will employ more capital until the marginal product of capital equals the marginal cost. • Note that, because marginal cost is decreasing in capital, K(t) will fall with the rental cost r(t).**MPK = F’(K)**Capital K**MPK = F’(K)**r(t) + δ Capital K(t)**MPK = F’(K)**r(t) + δ K(t) Capital**MPK = F’(K)**r(t) + δ K(t) Capital**MPK = F’(K)**A Fall in r: r’(t) < r(t) r(t) + δ r’(t) + δ K(t) K’(t) Capital**Households**• The typical household owns K(1) units of capital at the beginning of period 1. • The amount of capital it owns at the beginning of period 2 is given by: K(2) = (1-δ)K(1) + I(1)**At the end of period 2, the household will choose not to**hold any capital (since t =2 is the last period), and hence I(2) = -(1-δ) K(2)**In addition, households own firms, and hence receive the**firms’ profits.**Closed Economy case**• Suppose that the economy is closed. Then the household’s budget constraints are: C(1) + I(1) = Π(1) + K(1)(r(1) + δ) C(2) + I(2) = Π(2) + K(2)(r(2) + δ) And, recall, K(2) = (1-δ)K(1) + I(1) I(2) = -(1-δ) K(2)**But all of these constraints are equivalent to the single**constraint: C(1) + C(2)/(1+r(2)) = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2))**Proof**• From: C(1) + I(1) = Π(1) + K(1)(r(1) + δ) and K(2) - (1-δ)K(1) = I(1) We obtain C(1) + K(2) = Π(1) + K(1)[1+ r(1) ]**Likewise,**C(2) + I(2) = Π(2) + K(2)(r(2) + δ) and I(2) = -(1-δ) K(2) yield C(2) = Π(2) + K(2)(1+ r(2))**Now,**C(1) + K(2) = Π(1) + K(1)[1+ r(1) ] C(2) = Π(2) + K(2)(1+ r(2)) can be combined to get the intertemporal budget constraint: C(1) + C(2)/(1+r(2)) = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2))**The household’s budget constraint**C(1) + C(2)/(1+r(2)) = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2)) = Z is similar to the ones we have seen before, with Z = the present value of income. • The household will choose consumption so that the marginal rate of substitution between C(1) and C(2) equals (1+r(2)).**C(2)**Household’s Optimum Z (1+r(2)) C* C*(2) C*(1) O Z C(1)**C(2)**Household’s Optimum Here, Z = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2)) is the present value of income. Z (1+r(2)) C* C*(2) C*(1) O Z C(1)**C(2)**Household’s Optimum Z (1+r(2)) In the closed economy, the slope is –(1+r(2)) C* C*(2) C*(1) O Z C(1)**Productive Possibilities**• The resource constraints in the closed economy are: C(1) + I(1) = F(K(1)) C(2) + I(2) = F(K(2)) But K(2) = (1-δ)K(1) + I(1) I(2) = -(1-δ) K(2)**The first and third equations give**Y(1) = F(K(1))+(1-δ)K(1) = C(1) + K(2) while the second and fourth give F(K(2)) + (1-δ)K(2) = C(2)**Production Possibilities**• Since K(2) = Y(1) – C(1), C(2) = F(K(2)) + (1-δ)K(2) = F(Y(1) – C(1)) + (1-δ)(Y(1) – C(1)) This gives the combinations (C(1),C(2)) that the economy can produce (the production possibility frontier)**A special case is when δ = 1 (complete depreciation of**capital), so the PPF is simply: C(2) = F(K(2)) = F(Y(1) – C(1)) And its slope is ∂C(2)/ ∂C(1) = -F’(Y(1)-C(1))**C(2)**C(2) = F(Y(1) – C(1)) F(Y(1)) O Y(1) C(1)**Production Equilibrium**• Recall that the slope of the PPF is F’(Y(1)-C(1)) = F’(K(2)). But also, profit maximization requires: (1+r(2)) = F’(K(2)) In equilibrium, production must be given by the PPF point at which the slope of the PPF equals 1+r(2) I I I**C(2)**F(Y(1)) O Y(1) C(1)**C(2)**If r(2) is the rental rate, production equilibrium is at P: The slope of the PPF at P is -(1+r(2)) P C*(2) C*(1) O C(1)**Finally: General Equilibrium in the Closed Economy**• In equilibrium in the closed economy, production must be equal to consumption. • But we saw that both production and consumption depend on 1+r(2). • Hence r(2) must adjust to ensure equality of supply and demand.**C(2)**Household’s Optimum Z(1+r(2)) Slope = - (1+r(2)) C* C*(2) C*(1) O Z C(1)**C(2)**Production Equilibrium Slope = -(1+r(2)) P C*(2) C*(1) O C(1)**C(2)**Equilibrium in the Closed Economy: r(2) adjusts to ensure the equality of production and consumption in equilibrium. Slope = -(1+r(2)) P = C C*(2) C*(1) O C(1)**Note that the rental rate r(2) must adjust to ensure**equilibrium.**C(2)**P = C C*(2) C*(1) O C(1)**C(2)**If r(2) were higher, production would be at P’ and consumption at C’, So markets would not clear. C’ P = C C*(2) P’ C*(1) O C(1)**Adjustment to an Income Shockin the Closed Economy**• Suppose that Y(1) falls by Δ (because, for example, there is less capital in period 1)**C(2)**P = C O Y(1) C(1)**C(2)**P Δ Δ O Y(1) Y(1) - Δ C(1)**C(2)**P and P’ must have the same slope and their horizontal distance is Δ. P P’ O Y(1) Y(1) - Δ C(1)**Why is the horizontal distance between P and P’ equal to**Δ? • P and P’ correspond to the same value of C(2), and hence the same value of K(2). But K(2) = Y(1) – C(1), so if Y(1) is lower at P’ than at P by Δ, C(1) must be lower by Δ too.