Announcements Presidents’ Day: No Class (Feb. 19 th ) 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 K α, with 0 < α < 1.

Capital K Output F(K) F(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 K α, the MPK is MPK = F’(K) = αA K α-1

Capital K MPK = F’(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

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).

Capital K MPK = F’(K)

Capital K(t) MPK = F’(K) r(t) + δ

Capital MPK = F’(K) r(t) + δ K(t)

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

Capital MPK = F’(K) r(t) + δ K(t)

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

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(1) C(2) OZ Z (1+r(2)) C*(1) C*(2) C* Household’s Optimum

C(1) C(2) OZ Z (1+r(2)) C*(1) C*(2) C* Household’s Optimum Here, Z = Π(1) + K(1)[1+ r(1) ] + Π(2)/(1+r(2)) is the present value of income.

C(1) C(2) OZ Z (1+r(2)) C*(1) C*(2) C* Household’s Optimum In the closed economy, the slope is –(1+r(2))

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(1) C(2) O C(2) = F(Y(1) – C(1)) Y(1) F(Y(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(1) C(2) OY(1) F(Y(1))

C(1) C(2) O C*(1) C*(2) P If r(2) is the rental rate, production equilibrium is at P: The slope of the PPF at P is -(1+r(2))

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(1) C(2) OZ Z(1+r(2)) C*(1) C*(2) C* Household’s Optimum Slope = - (1+r(2))

C(1) C(2) O C*(1) C*(2) P Production Equilibrium Slope = -(1+r(2))

C(1) C(2) O C*(1) C*(2) P = C Equilibrium in the Closed Economy: r(2) adjusts to ensure the equality of production and consumption in equilibrium. Slope = -(1+r(2))

Note that the rental rate r(2) must adjust to ensure equilibrium.

C(1) C(2) O C*(1) C*(2) P = C

C(1) C(2) O C*(1) C*(2) P = C P’ C’ If r(2) were higher, production would be at P’ and consumption at C’, So markets would not clear.

Adjustment to an Income Shock in the Closed Economy Suppose that Y(1) falls by Δ (because, for example, there is less capital in period 1)

C(1) C(2) O P = C Y(1)

C(1) C(2) O P Y(1) Y(1) - Δ Δ Δ

C(1) C(2) O P Y(1) Y(1) - Δ P’ P and P’ must have the same slope and their horizontal distance is Δ.

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.

To see that P and P’ have the same slope, recall that the PPF must satisfy: C(2) = F(Y(1) – C(1)) So, since K(2) is the same at both P and P’, Y(1) – C(1) must also be the same. And, since, the slope of the PPF is ∂C(2)/ ∂C(1) = -F’(Y(1)-C(1)) it is also the same at P and P’.

C(1) C(2) O P Y(1) Y(1) - Δ P’ P and P’ must have the same slope

C(1) C(2) O P Y(1) Y(1) - Δ P’ Because C(1) and C(2) are normal, The new consumption point would be a point such as C’, if r(2) stayed the same. But then markets would not clear. C’

C(1) C(2) O P Y(1) Y(1) - Δ P’ C’’=P’’ Equilibrium is given by C’’ = P’’, where an indifference curve is tangent to the PPF. The slope of the PPF gives the new value of r(2), which must be higher than before. C(1) falls by less than Δ.

Hence: if Y(1) falls, The rental rate r(2) (the return on savings) increases. Consumption falls in both periods. Savings and Investment fall.

Open Economy Suppose that households can borrow and lend internationally at the interest rate r*. Let W(t) denote the wealth of the typical household at the end of period t. Then, if B*(t) denotes foreign assets at the end of t, W(t) = K(t+1) + B*(t)

In addition, since the household can save either by holding capital or holding foreign bonds, the return on both kinds of assets must be the same, that is, r(t) = r*  The world interest rate pins down the rental rate of capital.

Hence, since the marginal product of capital is a function only of capital, K(2) is determined solely by the world interest rate. And, since K(2) = (1-δ)K(1) + I(1), and K(1) is exogenously given, investment in period 1 (I(1)) is also determined by the world interest rate.

In particular, from F’(K(2)) = r(2) + δ It follows that F’(K(2)) = r* + δ That is, K(2) = K*, where F’(K*) = r* + δ And I(1) = K* - (1-δ)K(1).

Capital MPK = F’(K) r* + δ K(2) = K*

Note that K(2) and I(1) then depend inversely on r*. The previous graph can then be seen as an investment function.

Investment r*+δ I(1) = K*

The National Budget Line In the open economy case, the budget constraint is given by: C(1) + I(1) + B(1) = Y(1) + (1+r*)B(0) C(2) + I(2) = Y(2) + (1+r*)B(1)

Assume again δ = 1, for simplicity. Then K(2) = I(1). But we saw that K(2) = K*. Also, I(2) = 0. Assuming that B(0) = 0, the two constraints above reduce to: C(1) + K* + B(1) = Y(1) C(2) = F(K*) + (1+r*)B(1) Which imply: C(1) + K* + C(2)/(1+r*) = Y(1) + F(K*)/(1+r*)

In other words, the economy’s consumption possibilities in the open economy are given by a conventional budget line: C(1) + C(2)/(1+r*) = Y(1) – K* + F(K*)/(1+r*) = Z

C(1) C(2) O

C(1) C(2) OZ Z = Y(1) – K* + F(K*)/(1+r*) (Recall that K* is uniquely defined by r*)

C(1) C(2) OZ This is the national budget line Slope= -(1+r*)

C(1) C(2) OZ By construction, B must be on the budget Line. F(K*) Y(1) – K* B

C(1) C(2) OZ F(K*) Y(1) – K* B Importantly, the PPF must go through B (since B is feasible in the closed economy) and have slope -(1+r*)

What determines consumption? Because (1+r*) is the return on savings, optimal consumption will require that the marginal rate of substitution between C(1) and C(2) equal (1+r*).

C(1) C(2) OZ F(K*) Y(1) – K* B A C*(1) C*(2) Equilibrium consumption is at Point A.

Note that the ability to borrow and lend internationally causes changes in consumption and production.

C(1) C(2) OI P In a closed economy, consumption and production are at P and the return on savings is the slope of the green line.

C(1) C(2) O F(K*) Y(1) – K* B P If the economy can borrow and lend at rate r* (cheaper than in the closed economy), there is more investment and production moves to B.

C(1) C(2) O F(K*) Y(1) – K* B A C*(1) C*(2) P International capital markets also allow an optimal allocation of income between current and future consumption, as in A.

The Current Account Balance Budget constraints in each period are: C(t) + I(t) + B(t) = (1+r*) B(t-1) + Y(t) Recalling that the current account is : CA(t) = B(t) – B(t-1) = r*B(t-1) + Y(t) – C(t) – I(t) = savings - investment

The trade balance is given by net exports: TB(t) = Y(t) – C(t) – I(t) Note that CA(t) = TB(t) + r*B(t-1)

In our example, in period 1 (recall B(0) = 0 and I(1) = K(2) = K*), CA(1) = TB(1) = Y(1) – K* - C(1)

C(1) C(2) O F(K*) Y(1) – K* B A C*(1) C*(2)

C(1) C(2) O F(K*) Y(1) – K* B A C*(1) C*(2) Current Account Deficit

Adjustment to an Income Shock in the Open Economy Same Experiment as Before: Suppose that Y(1) falls by Δ (because, for example, there is less capital in period 1)

C(1) C(2) O P = C Y(1) Now we assume that the world interest rate is such that, before the shock, trade is balanced. Slope= -(1+r*)

C(1) C(2) O P Y(1) Y(1) - Δ Δ Δ Exactly as in the closed economy case, the PPF shifts to the left.

C(1) C(2) O P Y(1) Y(1) - Δ P’ After the shock, the world interest rate is still given by r*. This means that the new production point is P’.

C(1) C(2) O P Y(1) Y(1) - Δ P’ The national budget line is given by the blue line.

C(1) C(2) O P Y(1) Y(1) - Δ P’ Because C(1) and C(2) are normal, consumption moves to a point such as C’. C’

C(1) C(2) O P Y(1) Y(1) - Δ P’ Because C(1) and C(2) are normal, consumption moves to a point such as C’. Note that C(1) falls by less than Δ. C’

Summarizing, the fall in Y(1): Leaves I(1) and K(2) unchanged (at K*) C(2) must fall. C(1) falls, but by less than Y(1) If B(0) = 0, this means that the trade balance and current account go into deficit in period 1

Note, in particular, that a fall in Y(1): Does not affect I(1) Reduces savings in period 1 (S(1) = Y(1) – C(1)) Causes a trade deficit and a current account deficit (CA(1) = TB(1) = S(1) – Y(1))

Changes in World Interest Rate Now consider a change in the world interest rate: an increase in r*.

C(1) C(2) O P = C Y(1) Again, assume that the world interest rate is such that, before the shock, trade is balanced. Slope= -(1+r*)

C(1) C(2) O P Y(1) Suppose that the world interest rate increases. Then the national budget line would be the red line, if production equilibrium remained at P.

C(1) C(2) O P Y(1) Production, however, will change to P’, where the national budget line is tangent to the PPF. I(1), in particular, must fall. P’

C(1) C(2) OY(1) The new consumption point is C’. Here, this means that savings in period 1 increase. Since investment falls, the trade balance goes into surplus. P’ C’

C(1) C(2) OY(1) The adjustment can be regarded as the sum of a substitution effect (C to C’’) and an income effect (C’’ to C’) P’ C’ C=P C’’

An increase the interest rate produces: A substitution effect: future consumption becomes relatively cheaper  induces more savings An income effect: production reallocation which increases the value of GNP  induces less savings, if both goods are normal

Finally, there is a wealth effect, ignored so far. If the country is initially a debtor, the cost of the debt increases, which reduces the net present value of income, and goes against the income effect. If the country is initially a creditor, the effect is the opposite, and the wealth effect reinforces the income effect.

So, the impact of an increase in r* on national savings is ambiguous. Our “normal” assumption will be that savings increase with the interest rate. The savings function (or schedule) relates savings to the interest rate, other things equal.

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

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