Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected])
9 The two-period investment economy
Index:
9 The two-period investment economy ...... 1
9.1 Introduction ...... 2 9.2 National accounts and budget constraints ...... 2 9.2.1 Production and investment ...... 2 9.2.2 Savings and investment...... 3 9.2.3 The two-period case ...... 4 9.3 The Robinson Crusoe’ Problem ...... 6 9.3.1 Closed economy ...... 6 9.3.2 Open economy ...... 8 Box 1: infinite horizon ...... 14 Box 2: The Marginal q ...... 14 9.3.3 Savings and investment schedules ...... 16 9.4 Comparative statics ...... 18 9.4.1 Fall in the world interest rate ...... 18 9.4.2 Temporary output contraction...... 19 9.4.3 Anticipated productivity change ...... 22 Box 3. The Feldstein-Horioka puzzle ...... 26 Box 4: The marginal q and installation costs ...... 27 9.5 Households and firms and the debt-equity mix ...... 29 9.5.1 Households ...... 30 9.5.2 Firms ...... 31 9.5.3 Fisher separation theorem ...... 33 9.5.4 Dividend policy ...... 33 9.5.5 Value of the firm and capital structure ...... 35 9.5.6 Average q ...... 36 9.6 Taking stock ...... 37 Further reading ...... 39 Appendix 1: Tobin’s q, infinite horizon ...... 39 Review questions and exercises ...... 41 Review questions ...... 41 Exercises ...... 41
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9.1 Introduction
In this chapter, we extend the two-period model, allowing future production to be determined by today’s investment decisions. For simplicity, we assume that labour is supplied inelastically. In most of the chapter, we follow the simplest approach, assuming that the private economy is composed by agents that are simultaneously consumers and producers. In Section 2, we revisit the essential national accounts for the model with investment. In section 3, we compare the equilibrium allocations in the case of a closed economy, and in the case of an open economy. In section 4, we examine how savings and investment respond to a variety of shocks, in a closed and in an open economy. Finally, in Section 5 we split the private sector into households and firms to show that, in the absence of market frictions, separation of investment decisions from the ownership of capital does not alter the nature of the equilibrium. In Section 6 we summarise the main ideas.
9.2 National accounts and budget constraints
9.2.1 Production and investment
Instead of assuming that output is exogenous, we now assume that output is the result of a production process. Production each year is assumed to depend on the level of technology, z, and on the stock of capital1:
Qt zt FKt , with FK 0 and FKK 0 (1)
Capital takes time to build: that is, investment in period 1 only adds to the capital stock in period 2. By investment, we mean the purchase of goods (physical capital) not for
1 For simplicity, we assume that future productivity is known with certainty. We also abstract from decisions regarding employment of labour.
2 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) consumption, but rather to be employed in future production2. This means that the capital stock in each period is the result of past investment decisions.
In this economy, there is only one good. This good can be transformed into capital and back at no cost: in other words, one unit of consumption can be sacrificed to obtain exactly one unit of capital; conversely, one unit of installed capital can be transformed to obtain one unit of consumption good. The capital stock erodes over time due to depreciation. It is assumed that depreciation each year is a constant proportion of the existing capital stock. Hence:
Kt1 Kt 1 It (2)
where It denotes for “gross investment” in period t. Gross investment differs from net investment, Kt1 Kt due to depreciation. In the extreme case in which 0 , the capital stock each year is equal to the sum of all past investments. In alternative, when 1, the capital stock each year is equal to the previous year’ investment, only.
9.2.2 Savings and investment
Abstracting from valuation changes and international transfers, the change in a country’ net international investment position is given by:
* * * * CAt bt bt1 rt1bt1 TBt (3)
The equilibrium in the goods market takes the form:
Qt Ct It Gt TBt (4)
Substituting (4) in (3), one obtains
**** CAbbttt1 rb tt 1 1 QCG ttttt ISI 1 (3a)
2 Investment refers to acquisition of real assets and shall be distinguished from purchases of financial assets with the aim to obtain future returns. Confusion may arise, because the later is often labelled as financial investment.
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Where St refers to national savings:
** Sttt rb1 1 QCG t t t (5)
G The private sector holds financial assets in the form of government bonds ( dt ) and/or
* 3 foreign assets (bt ) :
* G bt b t d t (6)
The private sector net lending is equal to the difference between private savings and investment:
P bbtt1 rb tt 1 1 QCTIS tttttt I (7)
Where we used
P** G S1 rbd 0 0 0 QCT 1 1 1 (8)
The government sector net borrowing is equal to the government deficit, which is the symmetrical of government savings:
GGG* G d1 d 0 r 0 d 0 G 1 T 1 S 1 (9)
The sum of private savings and government savings is equal to national savings.
9.2.3 The two-period case
In the two-period model, current (period 1) capital stock is determined by past investment decisions. Future production (period 2) is endogenous, because it depends on current investment decisions. That is:
Q2 z2 FK2 (1a)
3 Note that this is less than the private sector Net Worth. The later includes the capital stock, which accumulates with investment.
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With:
K2 K11 I1 (2a)
Since there are only two periods, there is no point in accumulating capital to period 3. Hence, any remaining capital at the end of period 2 will be uninstalled:
I 2 K2 1 (2b)
In a two-period economy, the materializations of equation (7) are:
b11 rbQTCI 0 0 1 1 1 1 (7a)
b21 rbQTCI 1 1 2 2 2 2 0 (7b)
Solving together, we obtain the inter-temporal budget constraint of the private sector
C2 C1 1 , (10) 1 r1 where:
T2 Q 2 I 2 1brT 01 0 1 QI 1 1 (11) 1r1 1 r 1
Substituting in (11) the government inter-temporal budget constraint, the resource constraint of the economy becomes:
** G2 Q 2 I 2 1brG 01 0 1 QI 1 1 (12) 1r1 1 r 1
Using the later expression and (10) and (3), we obtain the economy inter-temporal budget constraint:
* * TB2 b0 1 r0 TB1 * 0 (13) 1 r1
This is the country’ solvency condition. It basically states that the present value of current and future TB plus the current NIIP shall be equal to zero.
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9.3 The Robinson Crusoe’ Problem
We start out with the case with no government ( G1 G 2 0 ), where production and consumption are undertaken by a sole agent (Robison Crusoe). We consider two alternative setups regarding the possibility of borrowing: first, we consider the case of a closed economy, i.e., an economy that cannot buy or sell asset broad. Then, we open the economy to international borrowing and lending.
As before, we consider the preferences of the representative household as follows
lnC UC ,C ln C 2 , (14) 1 2 1 1 where denotes for the rate of time preference. This utility function is somehow restrictive, but is what we need to obtain the main conclusions without too much complication.
9.3.1 Closed economy
Suppose that Robison Crusoe lives two periods, only. In the first period, Mr.
Robinson is endowed with a given amount of output, Q1 z1FK1 , and a stock of capital left from the current production cycle, K11 . The volume of production in period 2 depends on investment today.
Since there is no possibility of borrowing or lending, the CA must be zero. Hence, the equilibrium in goods market implies:
C1 Q1 I1 (4a)
C2 Q2 I 2 (4b)
Equations (4a) and (4b) stress the fact that in a closed economy any investment implies the sacrifice of contemporaneous consumption, on a one-to-one basis.
From (4a), (4b), (2a), (2b) and (1a), we obtain:
C2 z2 FK11 Q1 C1 K11 Q1 C1 1 (15)
Equation (15) is the Robinson Crusoe’ inter-temporal Production Possibilities Frontier. It gives the maximum possible consumption in period 2 that Mr. Robison can achieve, given the level of consumption in period 1. This frontier acts like a technology to
6 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) transfer income across time: it shows how Mr. Robinson can postpone or anticipate consumption through investment and disinvestment. The slope of the Production Possibilities Frontier is the Marginal Rate of Transformation:
dC2 MRT z2 FK 1 (16) dC1
This expression relates the amount of extra output that can be obtained in period 2 through investment to the cost of sacrificing one unit of output today. The extra output in period 2 is equal to the marginal product of capital (slope of the production function) plus what is left of this capital after being used in production and can be consumed.
The household maximizes its utility function (14) subject to the PPF (15). From the first order conditions of the maximization problem, one obtains the equality between the MRS and the MRT, that is:
C2 1z2 FK 1 (17) C1
This expression, together with (15), determines the optimal allocation of output in a closed economy. Figure 1 offers an illustration.
Figure 1 – Optimal Investment in a closed economy
C C2 2 1z2 FK 1 C1 zFK2 2 K 2 1
C2 Q 2 I 2
K2
C Q I 1 1 1 C1 Q1 K 1 1 K 2 Although we are assuming that Mr. Robinson is the only inhabitant of this economy,
a it is useful to introduce at this stage the shadow or “autarky interest rate”, r1 . This corresponds to the slope of the indifference curve - and of the PPF - in equilibrium (that is,
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a MRS MRT 1 r1 ). Thus, for instance, if the model was extended to include a government, that would be the interest rate at which Mr. Robinson would be willing to buy the government bonds. Similarly, if this economy was split into lending households and borrowing firms (as we will see later in this note), that would be equilibrium interest rate.
a Thus, r1 can be interpreted as the interest rate in the closed economy.
An important implication of (17) is that the optimal capital stock depends on preferences. Thus, for instance, when the rate of time preference increases, the point where the consumer’ indifference curve and the PPF are tangent moves to the right, along the PPF, implying more consumption today, less investment, and a higher interest rate. In a closed economy, the optimal investment depends on how patient households are. In other words, investment and consumption decisions are inter-related.
Another implication of being unable to borrow or lend abroad is that Mr. Robinson may find it optimal to invest, even with a very low return. Suppose for instance, that current output was Q1 100 , and that the optimal investment plan implied sacrificing 60 units of consumption today to obtain 40 units of output in period 2. That is, Robinson Crusoe was given the opportunity to choose between C1 100 and C2 0 , or C1 C2 40 . Even if the internal rate of return of the available investment opportunity was negative, (40-60)/20=- 33.3%, Robinson would still prefer to invest, because the alternative would be to consume nothing in period 2, having a utility level equal to minus infinity. Of course, Robinson would be very happy if he was given the opportunity to trade assets with the rest of the world at an interest rate higher than -33.3%. That would be the case in an open economy.
9.3.2 Open economy
Assume now that Robinson Crusoe could borrow or lend from abroad any quantity of
* output at the exogenous interest rate r1 . In this case, Mr. Robinson has two alternative avenues to transfer income over time: investment/disinvestment and lending/borrowing. In the open economy, the opportunity cost of investment is the international interest rate, that does not depend on the level of consumption.
With the possibility of borrowing or lending abroad, the equilibrium in the goods market each period becomes:
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Qt Ct It TBt (4c)
On the assets side, Mr. Robinson is now given the opportunity to buy an international
* * bond, b (liability when negative) offering a certain return r1 :
*** b11 rbQCI 0 0 1 1 1 (7aa)
In period 2, all assets must be paid off, so:
*** b21 rbQCI 1 1 2 2 2 0 (7ba)
Solving these two equations together, one obtains the household intertemporal budget constraint as (10), but where life-time wealth is defined as:
** Q2 I 2 1b 01 r 0 QI 1 1 * (12a) 1 r1
Given (2a), the expression for life-time wealth (12a) can be decomposed as follows:
* * 1 b0 1 r0 Q1 1 K1 V1 (18)
z2 FK2 K 2 1 V1 K 2 * (19) 1 r1
The first component in (18) – in brackets - corresponds to the total wealth that could be obtained if the household decided to uninstall all the existing capital stock in period 1. In that case, there would be no production in period 2. The second component, V1 , gives the increase (or decrease) in wealth arising from the investment opportunity. This is the Net Present Value of the capital stock K 4. 2
* When V1 0 , this means that K21 r 1 zFK 2 2 K 2 1 . In words, the value that would be obtained next year by purchasing international bonds in the amount K2 is less than what could be obtained if the same amount of resources was invested in the form of
4 Note that only in case 1 this will be the same as the net present value of investment.
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physical capital. Hence, buying physical capital is wealth-increasing. If, in alternative, V1 0 , then Mr. Robinson would be better off lending the amount K2 than employing the same amount in production. A negative Net Present Value (NPV) means that the corresponding investment destroys wealth.
The household chooses C1 , C2 and K2 to maximize the lifetime utility function (14) subject to (18) and (19). A key property of this problem is that consumption decisions and investment decisions are now separable. More precisely, the household can tackle the utility maximization problem in three steps: first, he chooses the level of investment I1 ( K2 ) that maximizes his lifetime wealth 1 ; second, given the resulting value of wealth, the household decides how much to consume today or to save; third, he borrows or lends to match his financing needs.
The first stage of the household’ problem - choosing the optimal capital stock - corresponds to the maximization of the corresponding net present value:
z2 FK2 K 2 1 Max V1 K 2 * 1 r1 K2
The first order condition to this problem implies:
z2 FK 1 * 1 . (20) 1 r1
Interpreting, equation (20) states that the value of having an additional unit of capital installed next year (left-hand side) must equal the marginal cost of investing one unit of output (ie, one unit of foregone consumption). The value of having an additional unit of capital installed next year consists in the increase in production resulting from one extra unit of installed capital at the beginning of period 2 (the marginal product of capital) plus 1
10 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) units of additional capital remaining at the end of period 2 that can be uninstalled and consumed. Usually, this expression is referred to as marginal q5:
z2 FK 1 q * (21) 1 r1
Rearranging (20), we get:
* 1 r1 z2 FK 1 MRT (20a)
Equation (20a) states that the maximum V is achieved when the Marginal Rate of Transformation is equal to the opportunity cost of current consumption in terms of future consumption in the market. Another way of writing (20) is:
* z2 FK r1 (20b)
This expression tells us that the net marginal product of capital (the marginal product of capital minus depreciation) shall be equal to the rate of return of the alternative asset in this economy (the international interest rate). Figure 2 illustrates the choice of the optimal capital stock in light of equation (20b).
5 You can interpret this term as the “sale value” of capital. In Box 2, the marginal q is recalculated, allowing one unit of capital to cost differently from one unit of output.
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Figure 2 – Optimal Capital Stock
Output
Q2 Q2=z 2 FK( 2 )
* max +r1 K 2
K2
* +r1
MPK z2FK
K K2 2
Note that the lower panel of figure 2 is not exactly an investment schedule. From (2a), the optimal investment is obtained as the difference between the optimal capital stock and the existing capital stock, that is: I1 K 2 K 1 1 . This gives a curve with the same properties as the one described in the lower panel of Figure 2, the difference being that the investment curve shall be located more to the left. Only in case 1 the two schedules will correspond exactly.
The second stage of the utility maximization problem corresponds to the maximization of (14) subject to (10) given the maximum level of wealth found above. The first order conditions imply the equality between the MRS and the relative price of current and future consumptions in the market:
C2 * MRS 1 1 r1 (22) C1
Solving together (22) and (10), the optimal current consumption is:
1 C1 1 (23) 2
Equation (20a) and equation (22) imply that condition (17) holds in the open economy, as it does in the closed economy. The big difference is that both the MRT and the
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MRS are now tied to the foreign interest rate, which is exogenous: in an open economy, the decision to invest does not depend on the decision to save and consume. Consumption and investment decisions are separate.
Of course, the level of investment affects wealth, and by then the amount consumed each period. But the optimal allocation between consumption today and in the future does not impact on investment decisions.
* Figure 3 – Optimal Investment in a open economy (b0 0 )
C 2 * zF2K 1 1 r 1 zF(K2 2 ) K 2 1
Q2 I 2
C2 * 1 1 r1 C1
C2 K2
C1 1 Q1 I1
Q1 K 1 1
V 1 Figure 3 illustrates the optimal allocation in an open economy. For simplicity we set
* b0 0 . First, given the initial endowment Q1 K 11 and the production function, we draw the PPF, as a positive function of future capital. Then, given the international interest rate, the optimal investment is determined such that the agent life-time wealth 1 is
* maximized. This corresponds to the point in the PPF where MRT 1 r1 (equation 20). The horizontal difference between the life-time wealth thereby achieved, 1 , and the initial endowment, Q1 K 11 , is the maximized Net Present Value of future capital, V1 . This gives the extra wealth made possible by investment.
The second stage of the household problem refers to consumption decisions. Given the maximized life-time wealth 1 , there is an inter-temporal budget constraint of the form (10), along which the agent can choose the optimal consumption pattern. This will be the
* point where MRS 1 r1 (equation 21).
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In Figure 3, the optimal choice involves a consumption today that is much higher than consumption tomorrow, reflecting a high degree of impatience ( ). This impatience however does not interfere with the investment decision: if you rotated the consumer’ income expansion path upwards or downwards, this would only change the optimal consumption basket along the optimized inter-temporal budget constraint. The optimal investment (as determined in figure 2) does not depend on preferences. This is the separation result between investment and consumption decisions that holds in a frictionless open economy.
Box 1: infinite horizon
Equation (20) was derived in the context of a two-period economy. The same condition can also be obtained assuming an infinite horizon. To see this, assume that the interest rate and the production function are both time-invariant. The life-time wealth becomes:
* * zFK2 I2 zFK3 I3 1 b0 1 r0 Q1 I1 * 2 ... (12b) 1 r 1 r*
Using (2), we know that K2 K11 I1 ,
2 3 2 K3 K2 1 I 2 K1 1 I11 I 2 , K4 K11 I11 I 2 1 I3 , etc. The optimal level of investment today shall be such that:
zF K zF K zF K 1 1 K 2 K 3 K 4 ... 0 * * 2 * 3 I1 1 r I1 1 r I1 1 r I1
zFK zFK zFK 2 1 * 2 1 3 1 ... 1 r 1 r* 1 r *
2 zF 1 1 zF 1 zF K K K * 1 * * ... * 1 1 r 1 r 1 r 1 r 1 r 1 * 1 r
… and equation (20b) follows.
Box 2: The Marginal q
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We have been assuming that one unit of output can be traded for one unit of capital. In the real world, however, the price of capital may vary over time, giving rise to capital gains or losses. To consider this, let’s assume that the price of one unit of capital is p K (in units of output), and that this price changes from period 1 to 2.
In this case, the Net present Value of the Capital stock is:
zFIK1 pIKK 1 1 K 2 1 1 2 1 1 V1 pIK 1 1 1 1 (19a) 1 r1
K In (19a), the capital employed in period 1 is valued at period 1 prices, p1 , and the
K non-eroded capital to be sold out in period 2 is valued at p2 .
Choosing I1 to maximize V1 , one obtains the following first order condition:
K V1 Kz2 FK p 2 1 K p1 p 1 q 1 0 I1 1 r1
Where we define:
K zFpK 21 1 r 1 q K (21a) p1
As before, the marginal q measures the value of having one additional unit of capital. Since this benefit is measured in units of foregone consumption current output, it must be
K divided by the cost of buying one unit of capital ( p1 ). The derivative of V1 reveals that investment shall be positive as long as q>1, that is, as long as the marginal benefit of installing capital exceeds its cost in terms of foregone consumption.
K K K Defining the capital gain related to the ownership of capital as p2 p1 1, At the optimum, q=1, implying zF pK1 r 1 K 1 . By approximation, this 2K 1 1 delivers the so-called Jorgensen formula:
K K zF2K p 1 r 1 (20d)
The difference relative to (20b) is that the user cost of capital (the term in the right- hand side) is now adjusted for eventual gains or losses incurred by carrying physical capital across periods.
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The q-ratio therefore contains the most relevant information regarding the profitability of investment: the marginal product of capital (which in turn depends on future technological developments) and the user cost of capital, that depends on the interest rate, the depreciation rate, and the eventual gains or losses relate to changes in the relative price of capital.
9.3.3 Savings and investment schedules
An alternative view of the economy’ equilibrium is offered by the savings and investment schedules. In this section, we summarize the previous results, extending the model to include a government.
The investment schedule is derived from the problem of maximizing the net present value of capital, which delivers equation (20). This equation implicitly defines a negative relation between investment and the interest rate, that is parametric on productivity, z2 , and on the initial capital stock. More generally, considering the discussion in box 2, one can write the investment function as:
K K I1 K 2. K 1 1 I 1 r 1 , z 2 , K 1 1 , p 1 , p 2 (24)
The investment function is depicted in the left panel of figure 4. It is negatively sloped, because when the interest rate increases, optimal investment declines. The investment function shifts upwards whenever the future price of capital rises relative to the current price of capital.
Figure 4 –Savings, investment and the current account
K K 1 r I1 I 1 r 1,,, z 2 p 1 p 2 1 1 r1 * CA S I S1 S 1 r 1,,,,,,,, Y 1 z 2 G 1 G 2 1 2 b 0 1 1 1
* 1 r1
a 1 r1
I ,S CA 1 1 CA 0 1 1
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The savings function is derived from optimal consumption, taking into account the government budget constraint. The optimal consumption is given by equation (23), with the private sector lifetime wealth defined as:
** G2 1brG 01 0 1 QK 1 1 1 Vrz 1 1 , 2 (18a) 1 r1
where V1 rz 1, 2 , is the net present value resulting from the investment decision. Given (8), private savings will be:
P** G 1 S1 rbd 0 0 0 QT 1 1 1 (8a) 1
Adding government savings, (9), we obtain the national savings function:
* * 1 S r b Q G S Y ,G ,G , ,r , z 1 0 0 1 1 2 1 1 1 1 2 1 2
The saving function slopes upwards in Figure 4. The savings schedule is parametric in current income, current and future government expenditures, the degree of impatience, and productivity. The right panel of Figure 4 shows the current account schedule, corresponding to the difference between national savings and investment:
CA1 Sr 1 1,... Irt 1 ,... (3b)
The CA schedule is positively sloped, reflecting the fact that investment depends negatively on the interest rate and savings depend positively on the interest rate. The autarky interest rate corresponds to the intersection of the saving and investment schedules. In that case the current account is zero.
In figure 4, we represent a case in which the autarky interest rate is lower than the international interest rate. This means that the home economy has comparative advantage in period-1 production (either because productivity at home is higher or because savings at home are lower). In a laissez fare, residents will find it beneficial to export current output, achieving a positive current account. In period 2, the international assets thereby accumulated will be used to buy additional units of future production, and consume ahead of production.
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9.4 Comparative statics
Using the framework above, we now examine adjustment to changes in the exogenous parameters, confronting the closed and open economy. For comparative purposes, we start out with a situation where it would be indifferent for the economy to be open or closed. This happens when the autarky interest rate is equal to the world interest rate. For simplicity, it is
* also assumed that 1, b0 0 , and there is no government. Since the economy is small, the international interest rate is given.
9.4.1 Fall in the world interest rate
In figures 5, we examine the implications of a decrease in the international interest rate. The initial equilibrium is described in the upper panel of figure 5 by the point P=C (the production point and the consumption point are the same). In the lower panel, we see that the
* initial interest rate r1 is such that the CA is zero.
* The fall in the world interest rate to r1 ' implies that the opportunity cost of investment decreases. Thus, projects with lower return will now become profitable. The production point moves from P to P’ in the upper panel, and investment expands along the investment schedule in the lower panel. On the other hand, the fall in interest rate implies that consumption today becomes relatively cheaper. Thus, the income expansion path (given by the Euler equation) rotates rightwards in the upper panel. The new consumption point will be C’, given by the intersection of the new income expansion path with the budget constraint passing in P’. The fact that both consumption and investment expanded implies a deficit in the CA, as shown in the bottom panel.
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Figure 5 –Fall in the world interest rate
C2
Q2=z 2 FK( 2 ) P‘ * MRS1 r1 ' ' Q2
C2= Q 2 C‘ P=C
' ' Q I C Q1 1 1 1 C1 C Q I 1 1 1
1 r 1 r 1 SS1 1 . 1
CA1 S1 I1
a * 1 r1 1 r1 * * 1 r1 ' 1 r1
I1 Irz 1 1, 2
I ,S 1 1 CA1
S1 I 1 0 CA 0 1
9.4.2 Temporary output contraction
We now examine the implications of a fall in current output ( Q1 0 ). Consider first the case of an open economy. From equation (20) we see that the optimal capital stock does
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not depend on Q1 : it depends only on the productivity of capital and on the international interest rate, which remain unchanged. Thus, the optimal capital stock is not be affected by the temporary shock. In the case of an open economy, a temporary output shock does not affect future production6.
In Figure 6, the fall in Q1 causes the PPF to shift leftwards. The optimal production point moves from P to P’, where the slope of the production possibility frontier is the same as in P=C.
* Figure 6 – Temporary fall in output, open economy ( 1 and b0 0)
C2
Q2 FK( 2 ) * MRS1 r 1 P=C ' P‘ Q2 Q 2
C‘2 C‘
Q‘ -K C‘ Q‘ C Q 1 2 1 1 1 1 C1
K' K 2 2 Of course, because current output has declined, the household is now poorer (with
1Q 1 0 ). Thus, the household optimally consumes less today and in the future. The new optimal consumption point is described by point C’, where the income expansion path (Euler equation) crosses the new intertemporal budget constraint. An external imbalance materializes, because production (P’) and consumption (C’) do not coincide.
6 Note that investment in period 1 will increase if the fall in Q1 is caused by some destruction of K1 . If the fall in Q1 is caused by a low materialization of z1 , investment in period 1 will not be impacted.
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* Figure 7 – Temporary output shock, closed economy ( 1 and b0 0 )
C a 2 MRS1 r1
Q2 =F(K2) * MRS1 r1 P=C Q2 P‘=C‘ Q‘2=C‘2
C‘1 Q‘ C Q 1 1 1 C1
K ' K 2 2 In Figure 7, we describe the adjustment to a temporary output shock in a closed economy. As before, the negative output shock implies a shift of the PPF to the left. Because the economy cannot borrow or lend, the new consumption point is doomed to be also the new production point (P’=C’): in the closed economy there is no separation of consumption and investment decisions.
As you may guess, the new autarky interest rate is higher than before, reflecting the scarcity of current consumption 7 . The desire to smooth consumption implies that the household will respond to the fall in current output by “borrowing from the future” in the only available manner: investing less. Thus, the marginal product of capital increases. In figure 7, the higher interest rate implies a rotation of the income expansion path to the left. All in all, because of the fall in investment, future output declines: in contrast to the case of the open economy, the temporary output shock has a persistent effect on production.
7 Note that at a constant interest rate, the optimal pattern would be as described by P’ and C’ in Figure 6; since in that case there would be an excess demand for current consumption (implying a trade deficit), the only way to clear the goods market in a closed economy will be with a higher price of current consumption.
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Figure 8 – Temporary output shock, savings and investment
1 r 1 1 r1 SS1 1 .
CA1 S1 I1
a 1 r1
* 1 r1
I1 Irz 1 1, 2
I1 ,S1 CA1 S1 I 1 0 CA 0 1
In figure 8, we summarize the implication of a negative output shock, referring to the savings and investment schedules. The fall in current output causes the savings schedule to shift leftwards: the household is poorer but the desire to smooth consumption results in lower savings today. Hence, the CA schedule also shifts to the left. In the open economy case, the interest rate is exogenous, so investment is not affected. In that case, a Current Account deficit emerges to finance the excess investment over savings. In the closed economy, foreign borrowing is not possible, and the scarcity of resources in period 1 implies an increase in the autarky interest rate.
9.4.3 Anticipated productivity change
We now examine the case where future productivity increases ( z2 0 ). In Figure 9, the productivity change is described by a rotation of the production function upwards (upper panel). In the bottom panel, this corresponds to a rightward shift of the marginal product of capital.
At an unchanged interest rate (open economy case), the new marginal product of capital at K2 exceeds the user cost of capital; looking in a different angle, the benefit arising from one more unit of capital installed exceeds the cost of acquiring that unit of capital (equation 20 does not hold in equality). Hence, it is optimal to expand the capital stock. Then, as the capital stock increases, the marginal product of capital declines back, returning to the
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' previous level. In the new optimum ( K2 ), equation (20) is verified again, with the marginal benefit of investment equalling to the cost of buying one unit of capital.
Figure 9: Impact of the productivity change on optimal investment
OutputQ2 Q2 =z'2F(K2 ) P‘
Q2 =z2F(K2)
P
K2
a +r1 * +r1
MPK z'2FK
MPK z2FK
' K K K2 2 2
In Figure 10, the same change is analysed in terms of current and future consumption. Initially, the economy is consuming and producing at P=C. Then, with the rotation of the PPF upwards, the optimal production point moves to P’, where the slope of the new PPF is equal to the international interest rate. Hence, production in period 2 increases and the household is now richer (his budget constraint moves to the right). Since the international interest rate remains unchanged, the income expansion path (OC’) remains unchanged: the optimal consumption basket moves along the income expansion path until it meets the new intertemporal budget constraint P’C’ in point C’. In the new equilibrium, the production point and the consumption point do not correspond, implying an external imbalance.
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* Figure 10 – Anticipated productivity shock, open economy ( 1 and b0 0 )
Figure 11– Anticipated productivity shock, savings and investment
1 r1 1 r1
CA1 S1 I1 SS1 1 .
a 1 r1
* 1 r1
I1 Irz 1 1, 2
I1 ,S1 CA1
S1 I 1 0 CA 0 1 In figure 11, we see the impact of the productivity change referring to the savings and investment schedules. The fact that capital becomes more productive shifts the investment schedule to the right (just like in figure 9). On the consumption side, the fact that future output is expected to be higher causes consumption to increase today. Thus, the saving schedule shifts leftwards. The impact on the current account is therefore reinforced: not only new financing is needed for the extra investment, but also for the decline in savings arising from the perception that life-time wealth has increased.
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In the closed economy case, in contrast, the market for loanable funds must balance domestically. Since the fall in savings and in the increase in investment cause an excess demand for loanable funds, the autarky interest rate must rise.
In Figure 12 we examine the closed economy case referring to the consumption space. As before, the shock implies a rotation of the PPF upwards. The fact that the autarky interest
a rates increases (to r1 ) implies that investment in the new equilibrium is lower than in the open economy case.
* Figure 12 – Anticipated productivity change, closed economy ( 1 and b0 0 , income and substitution effects cancelling out)
C a 2 MRS1 r1 P‘=C‘ MRS1 r* ‘ 1 Q2 =C‘2
C2=Q2 P=C
C1 Q 1 C1
K2
A different question is whether investment in the closed economy increases or decreases with the productivity change. In Figures 11 and 12, we depicted the case in which investment remains exactly the same as before. This is not however a general case. In the move from C to C’ in Figure 11, there are two opposing effects: on one hand, because the interest rate increases, current consumption becomes more expensive: thus, through a substitution effect, the household optimally reduces consumption today and increases consumption in the future. This is reflected in the rotation of the income expansion path to the left. On the other hand, with the productivity increase, the intertemporal budget constraint moves to the right. This positive wealth effect induces - all else equal - an increase in consumption today and in the future (along the new income expansion path). Whether the positive income effect outweighs or not the negative substitution effect, it depends on the properties of the utility function.
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Box 3. The Feldstein-Horioka puzzle
According to our findings, in an open economy investment decisions should be delinked from consumption decisions. For instance, in the face of a temporary output expansion, savings should increase reflecting the household’ desire to smooth consumption, but investment should remain invariant, as determined by productivity and the international interest rate. Thus, in an open economy, investment and savings should exhibit little correlation.
In a very influential study, Martin Feldstein and Charles Horioka, investigated the correlation between investment and savings in a sample of 16 OECD countries along the period from 1960 and 1974. Surprisingly, the authors found that the correlation between the two variables was very closed to one, as if these economies were closed8. This finding became known as the “Feldstein-Horioka puzzle”.
Many explanations have been proposed for the Feldstein-Horioka puzzle. The first is that capital mobility among OECD countries was not that high in the 1960s and early 1970s; most liberalization of capital flows occurred in the late 1980s and 1990s. Favouring this interpretation, there is some evidence that the correlation between investment and savings has declined in the last decades. Other possible explanation is that domestic and foreign assets are not exactly perfect substitutes: because of imperfect information and regulatory risk, agents prefer to buy home assets rather than foreign assets. This “home bias” implies that, when income and savings increase at home, more domestic resources are available to finance new investment at home.
Recently, Charles Horioka and a co-author proposed an alternative explanation for the puzzle9: there are barriers to trade in goods that force current accounts to vary less than implied by the model. A capital flow from one country to another can only materialize in
8 Feldstein, Martin; Horioka, Charles (1980), "Domestic Saving and International Capital Flows", Economic Journal, 90 (398): 314–329. 9 Ford, N and C Y Horioka, 2016. “The ‘real’ explanation of the Feldstein-Horioka puzzle,” Applied Economics Letters.
26 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) increased savings in the source country and increased investment in the recipient country if there is a current account imbalance: the recipient country must run a deficit, equal to the surplus in the saving country. In case there are strong barriers in international trade, or policy actions are such that current accounts evolve close to balance, the original capital inflow must give rise to a capital outflow in the opposite direction, implying that the net financial account remains equal to zero. That is, even if high capital mobility generates the potential for domestic savings and investment to decouple from each other, the materialization of this opportunity would require an equally high mobility of goods and services. To the extent that this is not the case, the high capital mobility translates into high gross capital flows across the world, but not necessarily to net capital flows and current account imbalances.
Box 4: The marginal q and installation costs
So far, we have been assuming that it is possible for the capital stock to jump to its optimal level instantaneously. If that was true, one should observe large swing in investment, from very high rates during seconds, to zero once the optimal capital stock was met. In real life, we know that investment does not drift up and down dramatically. It rather tends to spread over time. This pattern can be captured by the model, with a small amendment allowing for adjustment costs
A reason why the capital stock does not immediately adjust to its optimal level is that installing capital involves costs that are an increasing function of the speed at which capital ins installed. For instance, launching a new plant involves time-consuming actions like project evaluation, project design, building the infrastructure, placing the equipment, training workers, and so on. If one wants to build a plant in one month only, that will be more expensive than if the same plant is to be built in six months. This presence of adjustment costs slows down the adjustment of the capital stock.
To model this, we must assume that the costs of adjusting the capital stock increases more than proportionally with the amount of investment. Let the adjustment cost function, defined in units of the capital good, be as follows:
2 I K (25) t2 t t
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This equation states that installation costs are a positive function of net investment: (i.e, investment above what is necessary to replace depreciated capital). The simple replacement of depreciated capital involves no adjustment costs.
To see how adjustment costs impact on optimal investment, let’s return to our model with two periods10. To make the case more interesting, assume that one unit of capital costs p K units of output. Equation (19a) becomes:
K 2 zFIK2 1 11 pIK 2 1 1 1 1 VpIK IK K 1 1 1 1 1 1 1 2 1 r1 (19b)
Choosing I1 to maximize V1 , one obtains the following first order condition:
K V1 K zFK p2 1 p11 IK 1 1 0 I11 r 1
Implying
z F pK 1 2K 2 1 I K (20d) K 1 1 1 r1 p 1
As in (20), this equation implies that the benefit arising from getting one additional unit of capital installed next year must equal the marginal cost of buying that unit of capital. The novelty in (20d) is that the marginal cost of buying one unit of capital includes installation costs. These installation costs introduce a wedge between the marginal benefit of capital and the cost of acquiring one unit of capital in terms of foregone consumption. This wedge explains why q may depart from the value of 1.
Solving for I1 and borrowing from (21a) the definition of “marginal q”, we get the optimal investment in period 1 as a function of q:
10 We assume that the sale of the non-eroded capital in period 2 involves no “uninstallation costs”.
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1 I K q 1 (24a) 1 1
The parameter governs how sensitive net investment is to q. Thus, whenever q>1 the optimal net investment will be positive. Because of installation costs, however, the gap between the marginal benefit of capital and the cost of acquiring one unit of capital will be only partially eliminated. The parameter explains why q is not necessarily equal to 1.
Because in this economy there are only two periods, the dynamics cannot be explored. In Appendix 1, we show that, in a model with multiple periods, the capital stock engages in a slow adjustment process which speed depends on parameter . The presence of installation costs slows down investment, preventing the optimal capital stock to me met at once. In the long run, q=1 and optimal investment is zero.
9.5 Households and firms and the debt-equity mix
We now enrich the model, separating the economy into a large number of households and firms. For simplicity, we assume that there is no government. To distinguish labour and capital incomes, we augment the production function with labour. Assume that the production function takes the form:
Qt zt FKt , Nt (1b)
This production function is assumed to exhibit constant returns to scale (CRS).
In this economy, households supply labour and loans to firms, and are also the shareholders of firms. We abstract from consumption leisure decisions, postulating an inelastic labour supply, N. We also assume that wages are flexible, implying that full will always hold.
Financial assets are of three types: foreign bonds, b* , corporate bonds, bF and shares of the representative firm. The quantity of shares is denoted by and the market price of each share is denoted by a . The household net asset position at the end of period t is equal to:
* F bt b t b t tt a (28)
Shares pay a dividend each year. The value of shares may change over time. The shares’ value is defined at the end of period (that is, after the period’ dividend is paid).
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9.5.1 Households
Households carry from period zero a given stock of financial wealth
* F b0 b 0 b 0 0 a 0 . The change in the individual’ net worth from period 1 to period 2 is:
b b b** b bF b F a a a . (29) 10 10 10 101 010
The term in square brackets in the right-hand side corresponds to the household’ net acquisition of financial assets in period 1. This component must match individual savings. The second component corresponds to valuation changes: increases or decreases in the value of the household’ wealth due to changes in share prices. This second component involves no new purchase of assets and henceforth is not included in the definition of savings.
* At the beginning of period 1, last year’ bonds pay an interest r0 ( r0 if the economy is open). Shares acquired in period 0 pay a dividend, 1 , and worth a1 at the end of period. The household’ net worth at the end of period 1 will be:
**F F bbb11111 a 1 rbb 000 011 a wNC 111 . (30)
The household’ savings in period 1 are:
H* F Srbb1 0 0 0 0 1 wNC 1 1 1 . (31)
The correspondent to (30) in period 2 is:
* F b21 rbb 1 1 1 1 2 wNC 2 2 2 . (32)
As we know, there is a transversality condition stating that at the optimal plan, the household will not under accumulate or over accumulate assets. In the two-period economy, this means that the individual net financial position must be zero at end of period 2. Setting
* F b2 b 2 b 2 2 a 2 0 and solving together (31) and (32), we get a new version of the household’ lifetime budget constraint, consisting in equation (10) plus:
* F N2 w 2 2 100011b b 1 r N w * 0111 a a 1 1r1 1 r 1
The last term in brackets corresponds to the benefit of buying the quantity 1 of shares in the first period. The household maximizes the value of its portfolio choosing the 1
30 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) that delivers a higher lifetime wealth. As you may easily check, at the optimum the following condition must hold:
2 a1 (33) 1 r1
The interpretation of (33) is straightforward: since in this model there is no uncertainty, shares and bonds are perfect substitutes. Hence, absence of arbitrage opportunities implies that the corresponding returns must be equal. Given (33). the consumer life-time wealth simplifies to:
* F N2 w 2 1bb 0 0 1 r 0 0 1 aNw 1 1 1 * (12c) 1 r1
In equation (12c), the household life-time wealth appears as the sum of wealth generated by bonds, shares, and human wealth. The wealth generated by shares, corresponding to the discounted sum of dividends, will be labelled the value of the firm:
2 0 1a 1 0 1 (35) 1 r1
Households maximizes utility (14) subject to (10) and (12c). As before, the optimal consumption is given by equation (33), with the difference that life-time wealth is now given by (12c). The main question to check is whether the life-time wealth generated in an economy with households and firms (12c) differs from that of the Robinson Crusoe (12a).
9.5.2 Firms
From the firm’ point of view, bonds bF are liabilities. These, together with equity are used to finance the stock of capital (left side of the firm’ balance sheet). The firm’ liabilities accumulate according to:
F F b10 b1 r 0 Q 111101110 I N w a (36)
Another way of writing (36) is stressing the fact that bonds and shares are issued to match the financial gap arising from firm’ investment and savings:
FF F b1 b 0 a 1 1 0 I 1 S 1 (36a)
Where we used
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F F S1 Q 1 rbo 0 Nw 1 1 0 1 (37)
The firms’ savings are retained earnings, which are given by the difference between profits and dividends paid.
The analogue to (36) in period 2 is:
F F bb2 11 r 1 QINw 2 2 2 2 1 2 0 (38)
F Setting b2 0 , using (33), and solving together, we obtain an equation for the life- time payments flow generated by the firm:
F Q2 I 2 Nw 2 2 b01 r 0 0 1 aQINw 1 1 1 1 1 (39) 1 r1
Using (2a) and (2b) in (39), this gives an expression for the value of the firm:
0 1a 1
zFKN , K (1 ) Nw QNwK 1 bF 1 r K 2 2 2 2 2 2 1 1 1 1 0 0 2 1 r1 (35a)
In the right-hand side of (35a) we have two terms in brackets. The first one corresponds to the value the firm as if it was sold out today. The second term is the net present value of having K2 units of capital installed.
As long as there are no agency problems - that is, the manager of the firm is aligned with the shareholder’ interests - the manager will choose the capital stock that maximizes the value of the firm, or – which is the same – the life-time payoff to shareholders. In (35a), the first term in brackets is pre-determined. Hence, then manager of the firm maximizes the second term in brackets.
Taking the derivative of (35a) in order to K2 and equalling to zero, one obtains equation (20): the optimal investment rule is the same as in the Robinson Crusoe case.
Taking the derivative in order to N2 , one obtains the optimal demand for labour:
zFN2N 2 wN 2 2 (40)
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At this stage, it is important to remember the assumption that the labour supply is inelastic and wages are flexible. This assumption ensures that the quantity of labour used will the exactly the one Robinson Crusoe would be willing to supply, and therefore future output
Q2 will be the same.
9.5.3 Fisher separation theorem
Given the optimized values of Q2 and K2 , replacing (39) in (12c), we obtain exactly (12a). This is a very important result, because it implies that private wealth does not change with the separation of the economy into households and firms.
The conclusion of this exercise is that it doesn’t matter whether the household engages directly in production (auto-investment) or it owns firms that maximize profits. If the firm maximizes the life-time pay-off to shareholders (the market value of the firm), the later need to know nothing about production, and firms need to know nothing about preferences: the equilibrium will be the same as if households engaged directly in production. Separating the economy into households and firms does not produce real effects. This is the Fisher separation theorem.
The Fisher separation theorem presumes that the financial market is just a veil: if there were any financial market frictions (uncertainty, different lending and borrowing interest rates) or if there were agency problems (the firm’ manager could have other interests), then the equivalence would no longer hold.
9.5.4 Dividend policy
A corollary of what we just saw is the irrelevance of the dividends policy. To see this, suppose that the firm increases dividends today. The decision to pay more dividends upfront does not impact on optimal investment, because the optimal investment (20) does not depend on how investment is financed). The firm can always borrow to cover a financing gap.
To simplify the algebra, assume for a moment that number of shares is constant at
1 0 1. From equation (37), an increase in dividends paid in the first period translates
F into a decrease of firm’ savings: S1 1 0. Since investment is constant, the lower
F saving must be financed with new borrowing, b1 1 0 (eq. 36). Because of the higher
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F ** debt, dividends in period 2 will decrease by the amount 2br 11 1 1 1 r 1 .
* Since 1 21 r 1 0, the value of the firm – and the household life-time wealth – remain unchanged.
Household’ savings are however impacted: from (31), if consumption is constant,
H then S1 1 . The increase in households’ savings corresponds to the decrease in firm’s savings. Hence, there is only a change in the composition of private savings. Adding (37) and (31), we see that the dividend policy does not alter private savings:
P H F * * S1 S1 S1 r0 b0 Q1 C1 . (8b)
Summing up, when current dividends increase by one unit of output, the firm will borrow one unit of output to keep the optimal investment unchanged. By the same token, the household will respond saving exactly one unit of output, to keep the consumption level unchanged. This extra saving by the household materializes in the purchase of the firm bond, without any other impact in the economy.
We conclude that it will be irrelevant from the shareholder point of view whether the new investment is financed hiring new debt or with retained earnings. As long as there are no financial market frictions, the optimal investment level is determined by the interest rate only, and the way the firm finances investment - through corporate debt or with shareholders money - has no real consequences.
The reasoning underlying the irrelevance of dividends policy is similar to that in the Ricardian equivalence: as long as the interest rates faced by the firm and by the household are the same, anticipating or postponing dividends does not alter the household wealth and henceforth consumer decisions. Just like in the Ricardian equivalence, the proposition breaks
34 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) out in the presence of market frictions. If, for instance, firms were paying a higher interest rate on borrowing than households obtained on lending, the irrelevance would disappear11.
9.5.5 Value of the firm and capital structure
The remaining question to address relates to the value of shares a1 . To make the
K K case more interesting, we now allow the relative price of capital to be p1 in period 1 and p2 in period 2 (see box 2). In that case, dividends in period 2 will be:
K F 1 2QwN 2 2 2 pIb 2 2 11 r 1 (38a)
At this stage it is important to remember that the production function has constant returns to scale. An important property of CRS (linear homogeneity) is that:
zFKN , zFNN zFK K . Since profit maximization implies that wages must equal the marginal product of labour, this implies: Q2 wN 2 2 zFKN 2 2, 2 wN 2 2 zFK 2K 2 . Using (40) in (38a), the optimized dividends in period 2 will be:
K F 1 2zFKpK 2K 2 2 21 b 1 1 r 1 (38b)
Replacing this in the value of shares, one obtains:
1 2 K F 1a 1 qp 1 K 2 b 1 (33a) 1 r1
Where q is defined as in (21a). This equation reveals that buying shares is like purchasing fractions of the capital stock that is not already committed to the firm’ creditors. The market value of shares is obtained in is independent of the number of shares: if more shares are issued (1 ), then the value of each share ( a1 ) must decrease proportionally, for the
11 This is an illustration of the Modigliani-Miller theorem, that states that the value of the firm is unaffected by the debt-equity mix. The theory of corporate finance is dedicated to analysing the reasons for the Modigliani-Miller theorem to fail. These reasons include the differential tax-treatment of debt vs equity and bankruptcy risk [Modigliani, F.; Miller, M. (1958). "The Cost of Capital, Corporation Finance and the Theory of Investment". American Economic Review. 48 (3): 261–297].
35 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) equality to be maintained. Issuing more shares only dilutes capital without altering the market capitalization of the firm.
Equation (33a) implies that the value of shares is lowered by the amount of the firm liabilities on a one-to-one basis. To illustrate, consider the decision of an investor who wants to buy a leveraged versus an unleveraged firm. If the investor buys a firm that already
F K F borrowed b1 , he will pay 1a 1 qp 1 K 2 b 1 . If the investor buys an unleveraged firm, he
K will pay a higher value: 1a 1 qp 1 K 2 . From the investor point of view, buying the leveraged
F firm or buying the unleveraged firm and borrowing b1 from the bank to finance the gap would deliver exactly the same value. As long as the cost of borrowing is the same for the investor and for the firm, the capital structure of the firm will not matter. This illustrates the proposition that the total value of the firm (value of its shares plus liabilities) is unaffected by the debt-equity mix.
9.5.6 Average q
Solving (33a) for q, one obtains:
F 1a 1 b 1 q K (21b) p1 K 2
In this new incarnation, the q-ratio appears as the ratio between the market value of a firm (equity plus liabilities) and the cost of acquiring the corresponding capital stock in the market (replacement cost). Because it relates the total value of the firm to the total replacement cost of capital, this version of the q-ratio is also labelled as “average-q”.
The “marginal-q”, given by (21a), relates the marginal benefit of investing to the cost of buying one extra unit of capital; the “average q” (21b) relates the market value of the all
36 26/09/2021 Politicas macroeconomicas, handout, Miguel Lebre de Freitas ([email protected]) the installed capital (measured by the market value of equity and bonds that represent this capital) with the cost of acquiring that capital (replacement cost)12.
The average q offers a theory for investment relating the decision to buy new equipment to the valuation of shares in stock market13. For different reasons, the market valuation of a firm’ shares and liabilities may exceed the replacement cost of the installed capital that these shares and liabilities represent. According to our model. when this is the case, the q value will be greater than 1, implying that it will be profitable for firms to buy new capital.
9.6 Taking stock
In an open economy, the interest rate is given. Given the interest rate, the optimal savings and investment are determined, and by then the current account. In the closed economy, a current account imbalance cannot occur. Hence the domestic (autarky) interest rate adjusts to ensure the equality between saving and investment. Since in a closed economy investment is the only way of transferring income over time, the decision to invest depends on how impatient consumers are. In an open economy, the optimal is independent of preferences.
In an open economy, temporary output changes do not affect the optimal investment, and therefore do not impact on future output: the household can borrow or lend to reach the optimal level of investment and then decides consumption and savings. In a closed economy, temporary output shocks alter the optimal choice between consumption and investment, as well as the
12 In our model, the marginal-q and the average-q are the same. This happens because we are assuming perfect competition and a CRS production function. The conditions under which the marginal q and the average q are the same were stated by Hayashi, F. 1982, Tobin's Marginal q and Average q: A Neoclassical Interpretation, Econometrica, 50, issue 1, p. 213-24. 13 Tobin, J., 1969. A General Equilibrium Approach to Monetary Theory. Journal of Money, Credit and Banking, 1, 15-29.
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autarky interest rate. In a closed economy, a temporary output shock impacts on future production.
An anticipated productivity shift causes the optimal capital stock to change at a given interest rate. In an open economy, investment responds accordingly, and the implied wealth effects causes current and future consumption to move in the same direction. In a closed economy, the adjustment to a productivity shift involves a balance between less consumption today and more consumption tomorrow. Hence, the response in terms of investment is much more limited than in the case of the open economy.
A current account imbalance can be interpreted as the materialisation of comparative advantages: whenever the home economy has comparative advantages in future output – either because households are impatient or because productivity of investment at home is high - there will be a current account deficit. In theory, financial openness not only delinks savings from investment decisions, it also allows for a better allocation of the world capital across different economies.
In the basic formulation of the model, it is assumed that the capital stock jumps to the new optimum instantaneously. Accounting for adjustment costs, we found that the ideal stock of capital will not, in general, be reached at once. Installation costs explain why q is not always equal to unity. Installation costs also provide an explanation for why market value of a firm shares plus liabilities may exceed the price of acquiring the capital stock that these shares and liabilities represent. In light of this theory, periods of stock market valuation should be associated to higher investment rates.
In our model, we ruled out information failures and financial market imperfections. In this well-functioning economy, the separation of the economy into households and firms does not change the nature of the equilibrium, and the dividends policy is irrelevant.
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Further reading
Carlin, W., Soskice, D., 2015. Macroeconomics: Institutions, instability, and the financial system. Oxford University Press.
Boileau, Martin (no date), Two period Economies: a review, mimeo, University of Colorado.
Burda and Wyploz, Macroeconomics, chapters 7 and 8.
Obstfeld, M., Rogoff, K., 1996. Foundations of International Macroeconomics. MIT press. Chapter 2.5.
Schmitt-Grohe, S., Uribe, M., Woodford, M., International macroeconomics, 2016. mimeo. Chapters 4 and 6.
Appendix 1: Tobin’s q, infinite horizon
In this appendix, we extend the model with infinite horizon described in Box 1 to account for installation costs. To make the case simple, we postulate that productivity remains constant over time, as well as the interest rate. Adapting (12b) for the presence of installation costs (25) and maximizing, we get the first order condition:
2 1 K zFK zFK1 zF K 1 p1 IKt t ... 0 I1 r* **2 3 1 1r 1 r
That is,
zF K 1 I K (20d) r* pK t t
Similarly, to (20), this equation states that the profits stream generated by a unitary increase in the capital stock today must equal the marginal cost of investing. The later is equal to the cost of buying one unit of capital plus the marginal installation cost. Equation (20b) corresponds to the case with 0 and pK 1.
Solving for the optimal rate of investment, we get the same expression as (24a), with marginal q now taking the form:
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zF q K . (21c) pK r*
Figure A1 offers a graphical illustration of the dynamics of investment in this model. In the figure, the upward sloping curve describes the marginal cost of investing, in units of the consumption good. The downward sloping curves describe different q-functions, depending on the level of K. Along a q-function, the marginal return of investment decreases with the amount of investment, due to diminishing returns on capital.
Suppose that the economy lies initially in point A, where q=1. This means that the marginal return of installed capital is exactly equal to the cost of buying one unit of capital. Thus, the economy is operating with the ideal capital stock, and net investment is zero, meaning that gross investment is equal to depreciation only.
Figure A1 – Optimal investment with adjustment costs, infinite horizon
q
1I K B qB q C C zFK r q A pK 1
q 1 qC 1 B I K Now assume that some shock tilted the productivity of capital (z) to a new level, once-and-for all. With a higher productivity of capital the curve describing the Tobin q shifts to the right (the numerator in (21c) increases). Given the current capital stock, the return of new capital will exceed the replacement cost of capital, implying that a positive net investment is profitable. Due to installation costs, the capital stock does not jump immediately to the new optimum. The optimal net investment in period 1 occurs in point B, where the marginal benefit of investment is equal to the marginal cost of investing. In point B, q is still greater than one (the value of an additional unit of capital installed exceeds its acquisition cost).
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Once the investment in period 1 materializes and the capital stock expands, the downward sloping curve describing the Tobin-q shifts to the left (due to diminishing returns). This means that the marginal return of new investment in period 2 is now lower than what it was in period 1. Still, as long as it is greater than the acquisition costs of capital (q>1), optimal investment will be positive (point C). Hence, there will be a new increase in the capital stock and the curve describing the Tobin q will shift again to the left. This process will continue, implying lower and lower q-values - and lower and lower investment rates – until the original point A is met. In that equilibrium, the marginal benefit of net investment is again equal to the cos of acquiring one unit of capital and hence only replacement investment takes place.
Review questions and exercises
Review questions
9.1. Explain how financial openness may improve the efficiency in the world allocation of capital.
9.2. Explain what the Fisher separation theorem states. Is it expected to hold equally in closed and in open frictionless economies? And what about in the real world?
9.3. Consider an economy where initially savings are equal to investment. Examine graphically the effects of an increase in government spending today, distinguishing the cases in which: (a) the economy is open to capital flows; (b) the economy is closed to capital flows.
Exercises
9.4. (Small economy with production: closed versus open). Consider an economy
where the preferences of the representative consumer are given by U C1C2 and the 1 2 production function is given by Q2 10 K 2 , with capital depreciating fully each year. Further assume that there are no initial debts or liabilities, and that current output is
Q1 48 . a) Assume first that the economy is closed to capital flows. Find out the optimal consumption and investment patterns. Represent in a graph. b) Now assume that the economy was open to capital flows and that the international interest rate was r * 0 . Find out the equilibrium levels of investment, consumption, national expenditure and the current account.
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c) Describe this economy in the YA and IS diagrams.
9.5. (Temporary shock, open and closed). Consider a one-good economy with no government and no initial assets or debt. The representative consumer lives for two
periods and has a life-time utility function given by: U lnC1 lnC2 . In period 1,
there is a pre-determined amount of output, equal to Q1 300 . As for the second period, there are investment opportunities, as described by the following production 0.5 function, Q2 22K 2 where K depreciates fully after one period. Further assume that the economy is open, and the international interest rate is 10%. a) (Equilibrium): Compute: the optimal investment function, the corresponding Net Present Value, optimal consumption, and the current account in period 1. Represent graphically. b) (Temporary shock) Now assume that the economy was hit by a temporary output
shock, so that current output turned out to be Q1 192 . Would consumption be impacted proportionally? What about investment? Compute the new equilibrium, as well as the current account balances in period 1 and 2. Describe the adjustment in the IS diagram. c) (Closed economy) If the economy was closed, how would the new equilibrium look like? In particular, find out the impact of the shock on investment, consumption today, production in the future. d) (Theory): Compare in a graph the cases b) and c) and conclude.
9.6. (Productivity shock, open and closed). Consider a one-good two-period economy with no government, where NIIP is initially zero. The representative consumer has a
lifetime utility function given by: U C1C2 . Production is a pre-determined in period 0.5 1, Q1 192 , and a function of the capital stock in period 2, Q2 16K , where K depreciates fully after one period. Initially, the economy is open, and the international interest rate is zero. a) Compute the optimal investment plan as a function of the interest rate, as well as the corresponding Net Present Value. Given the interest rate, describe in a graph the impact of the investment opportunity on the economy’ intertemporal budget constraint. b) Find out the optimal consumption and saving as a function of the interest rate. Describe the consumer optimum when the international interest rate is zero. c) Describe this economy using the SI diagram. d) Now assume that the economy was blessed with a productivity surge, so that the 0.5 production function shifted to Q2 20K . (d1) Describe the impact of this shock in the SI diagram. Find out then new: (d2) optimal investment; (d3) consumption in period 1 and 2; (d4) Current account in periods 1 and 2. e) Finally, consider the case where the economy was closed to financial trade. If the productivity term increased from z=16 to z=20, what would be the implications? Find out the optimal values of consumption and of investment after the productivity shock and compare to d). Explain with the help of a graph. f) Which of the two cases, (d) or (e), is more in accordance to the findings of Feldstein and Horioka?
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9.7. (Small open economy, durable capital). Consider an economy open to capital flows, facing a world interest rate equal to r* 0.25 . In this economy, the production 0.5 function is given by Qt 16Kt , the current capital stock is K1 400 , and the depreciation rate is equal to 0.15 . Finally, assume that consumers live only two periods, and the preferences of the representative consumer are given by ln C U ln C 2 , with the rate of time preference equal to 0.25. 1 1 a) Find out the investment function. Given the interest rate, find out the optimal investment [60]. b) Given the investment function, find out the representative agent’ life-time wealth as a function of the interest rate [788]. c) Describe this economy in the SI diagram. d) Find out the values of: the current account in periods 1 and 2; production in periods 1 and 2; income in periods 1 and 2; current account in periods 1 and 2. e) Finally, assume that this economy started out with no capital at all (that is,
K1 Q1 0 . Would the optimal investment change? What would be the new values of consumption today and in the future? [71]
9.8. (Open economy, infinite horizon) Consider a small open economy where consumers have an infinite live, and maximize their welfare when consumption is constant over 0.5 time. The production function is given by Qt 14 K t . The depreciation rate is zero ( 0 ) and r* 0.2 . This economy starts out with an initial capital stock * corresponding to K1 900 and foreign assets amounting to b0 155 / 1.2 . g) Determine: (a1) the optimal investment in period 1; (a2) the production pattern in periods 1 and 2; (a3) life-time wealth. h) Find out: (b1) the optimal consumption path; (b2) the trade balance in periods 1 and 2; (b3) NIIP in period 1; (b4) GNE in period 1; (b5) NFIA in period 2; (b6) GNI in period 2; (b7) CA in period 2.
9.9. (Sudden stop, debt structure) Consider an economy where the preferences of the
representative consumer are given by U ln C1 ln C2 and the production function is 1 2 given by Q2 10 K 2 . The economy initial Net International Investment position is B* 52 , Q 100 , and the international interest rate is constant at r * 0 . 0 1 a) Considering the case where the economy is open to capital flows, find out the optimal capital stock, as well as the corresponding NPV and wealth (after
investment). [ 1 73]. b) Find out the current consumption and the current account, sticking with the assumption of openness [CA=38.5]. c) Assume that all initial liabilities matured at the beginning of period 1. Describe the equilibrium in case the economy lost access to in international financial markets. In particular, describe the impact of the “sudden stop” on consumption, interest rate an in the current account. Compare with b) using the IS diagram. [1+r=1.25]
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d) In alternative, suppose all external debt matured at the beginning of period 2. In that case, what would be the implications of the sudden stop. Compare with c) and conclude.
9.10. (Fiscal policy in a production economy): Consider an economy where the
preferences of the representative consumer are given by U C1C2 and the production 1 2 function is given by Q2 20K2 , with capital depreciating fully each year. Further
assume that there are no initial debts or liabilities, and Q1 300 . a) Assume first that this economy was closed to capital flows. Referring to period 1, what would be the optimal consumption, investment and savings in this case? And the interest rate? Describe the equilibrium in the C1C2 space. [A: 200, 100, 100, 0%]. b) If the economy open to capital flows and the interest rate abroad was r * 0 , would the equilibrium change? Why? Describe the equilibrium using the SI diagram.
c) Now suppose that the government decided to launch G1 T1 108
holdingG2 T2 0 . c1) Assuming full access to foreign credit, what would be the impacts on: investment, consumption, private savings, government savings, trade balance. Describe the change in equilibrium using the YA and the IS diagrams. In this case, which agent would hold foreign liabilities in period 1? c2) Would it make a difference if the government decided to finance the expenditure in the first period with debt? [A: 100, 146, 46, -54]. d) Assuming that no agent in this economy had access to credit abroad, what would be the impact of the policy shift on: investment, consumption, private savings, government savings, interest rate. Would it make a difference if government expenditures in the first period were financed with debt? [A: 64, 128, 64, 0, 25%].
9.11. (Temporary shock, productivity shock, Bailout). Consider an economy
where the preferences of the representative consumer are given by U C1C2 and the 1 2 production function is given by Q2 z2 K2 . Further assume that there are no initial
debts or liabilities, that z2 10 and that current output is Q1 75 . a) Assume first that the economy is closed to capital flows. Find out the optimal consumption and investment patterns, as well as the autarky interest rate. [A: 50; 25; 0%]. b) Now assume that the economy was open to capital flows and that the international interest rate was r * 0 . Describe the impact of trade openness on current consumption, investment and trade balance. Describe this economy in the SI graph.
c) Productivity shock: examine the implications of a productivity shock from z2 10 to
z2 8 when: (c1) the economy is open [A: +13.5]; (c2) there is no access to external credit [A: -20%]. (c2) capital is mobile. (c3) Compare the two cases referring to the SI graph
d) Temporary shock: Return to the case with z2 10 . With all remaining parameters equal, analyse the impact of a temporary output shock that drives down current GDP
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to Q1 48 , when: (d1) The economy is open to financial trade; [A: -13.5]; (d2) the private sector cannot borrow or lend abroad. [A: 25%]. Compare the two cases using the SI diagram. e) (Bailout) (e1) Departing from d2) assume that the government decided to launch a 13.5 lump sum subsidy today, financed with a external loan at r * 0 . Will this scheme have any impact on investment? (e2) What is the policy was financed with a domestic loan?
9.12. (Productivity change, average-q). Consider a small open economy, where
the preferences of the representative consumer are given by U CC1 2 . Production is undertaken by households who are simultaneously investors and the suppliers of
labour. In this economy, the supply side is described by Q1 192 and 1/2 1 2 Q2 zNK 2 2 2 where z2 refers to productivity, K2 refers to the capital stock (that
depreciates fully each year), and N2 100 is the labour supply assumed inelastic. The * world interest rate is 1r1 1.25.
a) Suppose that agent in this economy anticipate z2 2.0 . Find out: a1) investment in period as a function of the interest rate; a2) the consumer life- time wealth; a3) the savings function; a4) the consumption pattern; a5) life- time utility. a6) Represent the equilibrium in a graph. b) Now, suppose that, before period 2 materialized, people revised the
expectation regarding productivity in period 2, to exactly z2 2.5 . b1) show what happens to the savings and investment functions. Find out: b2) optimal investment and consumption in each period; b3) trade balance in periods 1 and 2; b4) National income in period 2. [To abstract from uncertainty in utility
maximization, assume that households believed z2 2.5 to be the only possible scenario]. c) Assume now that the economy is divided into households and profit- maximizing firms. The households’ wealth is composed by foreign assets and shares. Firms cannot borrow or lend. Wages are flexible. (c1) Show how the circular flow of income looks like in (a) and in (b). (c2) Find out what happened to the value of shares when expected productivity changed from E z2 2.0 to z2 2.5 . d) Departing from b), suppose that, after investment and consumption decisions in period t=1 were made, people realized that productivity was smaller than
anticipated (that is, productivity ended up being z2 20 ). In that case, how much will be: d1) Output in period 2; (d2) National income in period 2; d3) Consumption in period 2; d4) life-time utility. d5) Describe this equilibrium in a graph and conclude.
9.13. (Two-country world with production) Consider a world with two large economies, H and F. In both economies, the preferences of the representative
consumer are given by U lnC1 lnC2 1.5 and the production function is
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1 2 Q2 15K 2 , where capital depreciates fully after use. Further assume that there are no
initial debts or liabilities and that Q1 is pre-determined. a) Closed economy: Find out the optimal investment and the optimal consumption H pattern as a function of current output. Solve for the case with Q 1 100 [A: r=50%]. b) Saving and investment schedules: find out the optimal investment, the optimal current consumption, and the optimal savings for each level of the interest rate and of current output. Use these tools to confirm your answer in (a).
F c) In the foreign country, current output is Q 1 188 . Explain why, without capital movements, the interest rate has to be lower there. d) Free capital movements: Find out the equilibrium interest rate under perfect capital mobility. Which country will be running a deficit? Why? Describe graphically the adjustment process in the two countries [A: 25%; -17.6]. e) Under free capital movements, find out the optimal consumption, investment, and current account in periods 1 and 2 in the home economy [81.8; 36; -17.6, +17.6]. f) Is the equilibrium with free capital movements the first best outcome for the world as a whole? And from the home country perspective? Explain, with the help of a graph, how a tax on capital controls could improve the home country’ welfare.
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