Production, Investment, and the Current Account

1. Production, Investment, and the Current Account Roberto Chang Rutgers University April 2013

2. Announcements • Problem Set 3 available now in my web page • Due: Next week (April 11th)

3. Motivation • Recall that the current account is equal to savings minus investment. • Empirically, investment is much more volatile than savings. • Reference: chapter 6, section 3 of FT

4. Recall: The Savings Function • Recall that we had derived a national savings function from a basic model of consumer choice

5. The Savings Function Interest Rate S r* S S* Savings

6. Interest Rate An increase in savings. This may be due to higher Y(1). S S’ S S’ Savings

7. 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.

8. 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))

9. 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.

10. Output F(K) F(K) Capital K

11. 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

12. MPK = F’(K) Capital K

13. 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.

14. 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

15. 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).

16. MPK = F’(K) Capital K

17. MPK = F’(K) r(t) + δ Capital K(t)

18. MPK = F’(K) r(t) + δ K(t) Capital

19. Note that K(t) will fall if r(t) increases.

20. MPK = F’(K) r(t) + δ K(t) Capital

21. MPK = F’(K) A Fall in r: r’(t) < r(t) r(t) + δ r’(t) + δ K(t) K’(t) Capital

22. K(t) will increase if MPK(t) increases

23. MPK = F’(K) r(t) + δ K(t) Capital

24. MPK = F’(K) An increase in MPK r(t) + δ K(t) K’(t) Capital

25. Investment • The amount of capital in the economy at the beginning of period 2 is given by: K(2) = (1-δ)K(1) + I(1) • Hence investment in period one is I(1) = K(2) - (1-δ)K(1)

26. Now recall • K(1) is given as an initial condition • K(2) is a decreasing function of r(2) • Hence the equation I(1) = K(2) - (1-δ)K(1) implies that I(1) is a decreasing function of r(2)

27. The Investment Function • But in an open economy, r(t) must be equal to the world interest rate r*  Investment in period 1 is a decreasing function of the world interest rate r*

28. The Investment Function Interest Rate I r* I I* Investment

29. An increase in investment, May be due to an increase in the future MPK Interest Rate I’ I r* I’ I I* I** Investment