Convex adjustment costs and the q-theory of investment

Transcription

1 Advanced Macroeconomics. Note Christian Groth Convex adjustment costs and the q-theory of investment 1 Introduction The neoclassical ( supply determined ) models considered so far (the Diamond OLG model, the Barro bequest model, the Blanchard OLG model) have ignored capital adjustment costs. In the closed economy aggregate investment was just a reflection of aggregate saving and appeared in a "passive" way in the models, that is, as just a residual of national income after households have decided their consumption. We now assume that associated with firm s investment are certain adjustment costs. By this we mean that in addition to the direct cost of buying new capital goods there are also costs of installation, costs of reorganizing the plant, costs of retraining etc. If the marginal adjustment cost is increasing in the level of investment (as is often assumed), then the presence of the adjustment costs implies that investment decisions are assigned an active role in the model. There will be both a well-defined saving decision and a well-defined investment decision, separate from each other. In the closed economy interest rates now have to adjust so that aggregate saving can be equal to aggregate investment at all points in time; or, what amounts to the same, so that the aggregate demand for goods, consumption plus investment, can be equal to the aggregate supply of goods. 1 This implies that even when ignoring uncertainty (i.e., assuming perfect foresight) the relationship between the (real) rate of interest and the (net) marginal product of capital becomes more loose in the short run. For a small open economy with perfect mobility of financial capital we avoid the counterfactual result that the capital stock changes instantaneously when the interest rate at the world capital market changes. As to the investment decision of the optimizing firm, it is essential that when faced with adjustment costs the firm has to take the whole future into account. Therefore, expectations 1 The model of debt dynamics and the short-term and long-term interest rates in Blanchard & Fischer, p , gave a hint of such a "flow determination" of the short-term interest rate. 1

2 become important. information arises. 2 Further, the firm will adjust its capital stock only gradually when new Themacroeconomictheoryoffirms capital investment when there are adjustment costs is called the q-theory of investment (after Tobin 1969). Under some conditions to be described below this theory gives a remarkable simple operational macroeconomic investment function, where the valuation of the firms by the share market (i.e., as given by share prices) relative to the replacement cost of the firms physical capital is the decisive factor. Compared to the exposition in B & S, p , the text below presents the theory in a slightly different(andinmyopinioneasier)way. 2 The representative firm The technology of the representative firm is Y = F (K, T N), where F is a neoclassical production function with CRS, and Y,K, and N are output, capital input, and labour input per time unit, respectively, while T is a technology factor growing at the constant rate g 0, that is, T t = T 0 e gt,t 0 > 0 (Harrod-neutral technical progress). 3 The increase per time unit in the firm s capital stock is given by K = I δk, δ 0, (1) where I is gross investment per time unit, and δ is the capital depreciation rate. 2.1 Convex capital adjustment costs Let J denote the capital adjustment costs (measured in units of output) per time unit. Assuming the price of investment goods is one (the same as that of output goods), then total investment costs are 1 I + J. The q-theory of investment assumes that the capital adjustment 2 Some empirical studies conclude that only a third of the difference between the current and the "desired" capital stock tends to be covered within a year (Clark 1979). Other explanations of sluggish capital adjustment focus on uncertainty and irreversibility (Zeira 1987) or financial problems due to bankruptcy costs (Nickell 1978, Chapter 8). 3 The reason that the technology factor is called T (and not A as in B & S) is that we want to use A for financial wealth, as usual. In B & S the rate of (Harrod-neutral) technical progress is called x insteadifourg. 2

3 J G(, IK) 0 I Figure 1: costs J is a strictly convex function of investment I and generally also depends negatively on the current capital stock, that is, J = G(I,K), where the "adjustment cost function" G satisfies G(0,K)=0,G I (0,K)=0,G II (I,K) > 0, and G K (I,K) 0. (2) For fixed K = K the graph is shown in Fig. 1. Also negative investment, i.e., sell off of capital equipment involves costs (to dismantling, reorganizing etc.).therefore G I < 0 for I<0. The important assumption, however, is that G II > 0 (strict convexity in I), implying that the marginal adjustment cost is increasing in the level of investment. If the firm wants to accomplish a given installation project in only half the time, then the installation costs are more than doubled (the risk of mistakes is larger, the problems with reorganizing work routines are larger etc.). It is convenient (and probably not unrealistic) to assume, in addition, that G K (I,K) < 0 for I = 0. This may reflect the hypothesis that a given amount of investment requires more reorganization in a small firm than in a large firm (size being measured by K) or that the more a firm has invested, historically, the more experienced it is now, and therefore its capital adjustment costs today, for a given I, are lower. The strictly convex graph in Fig. 1 illustrates the essence of the matter. Assume the firm wants to increase its capital stock by a given K and chooses the investment level Ī per time unit. Then, in view of (1), it takes approximately t = K/(Ī δk) units of time to accomplish the desired increase K. If, however, thefirm slows down the adjustment and 3

4 invests only half of Ī per time unit, then it takes approximately 2 t units of time to accomplish K. The total cost of the two approaches are approximately G(Ī, K) t 1 and G( 2Ī, K)2 t, respectively (ignoring discounting). It is up to the reader to show, using Fig. 1, that the last-mentioned cost is smaller than the first-mentioned. On the other hand, in reality there is discounting, and changes in the firm s environment take place continually, so that it is not advantageous to postpone the investment too much. 2.2 The decision problem of the firm Let cash flow (before interest payments) at time t be denoted R t.then R t F (K t,t t N t ) w t N t I t G(I t,k t ), (3) where w t isthe(real)wageattimet. The adjustment cost G(I t,k t ) implies that a part of output Y t is used up in transforming investment goods into installed capital, and therefore less output is available for sale. Assume the firm is a price taker and that there is no uncertainty. Then the decision problem, as seen from time 0, is to choose a plan (N t,i t ) t=0 to maximize the value of the firm, V 0 = R t e t rsds 0 dt = [F (K t,t t N t ) w t N t I t G(I t,k t )] e t rsds 0 dt s.t. (4) 0 0 N t 0, no restriction on I t, (5) K t = I t δk t, k 0 given, (6) K t 0 for all t. (7) Instead of a specific terminal condition we have posited the natural feasibility condition (7) that the firm can never have a negative capital stock. To solve the problem we use the Pontryagin maximum principle. The problem has two control variables, N t and I t, and one state variable, K t. The procedure is: a) Set up the current-value Hamiltonian: H(N t,i t,k t,q t,t) F (K t,t t N t ) w t N t I t G(I t,k t )+q t (I t δk t ), (8) where q t (to be interpreted below) is the adjoint variable associated with the dynamic constraint (6). 4

5 H N t b) Partially differentiate H w.r.t. the control variables and set equal to zero: = F 2 (K t,t t N t ) w t =0, i.e., F 2 (K t,t t N t )T t = w t ; and (9) H I t = 1 G I (I t,k t )+q t =0, i.e., 1+G I (I t,k t )=q t. (10) c) Partially differentiate H w.r.t. the state variable and set equal to q t + r t q t,sincer t is the discount rate in (4): H K = F K(K t,t t N t ) G K (I t,k t ) δq t = q t + r t q t. (11) d) Now, use the Pontryagin Theorem, saying (in the present case): For an interior optimal path (N t,i t,k t ) 4 there exists an adjoint variable q t such that for all t 0 the conditions (9), (10), and (11) are valid, and the transversality condition is satisfied. lim q te t rsds 0 K t =0. (12) t The optimality condition (9) is the usual employment condition equalizing the marginal product of labour to the real wage. The right-hand side of (10) gives the cost of acquiring one extra unit of installed capital at time t (both the cost of buying the marginal investment good and the cost of its installation). Therefore, the right-hand side of (10) is the marginal cost, MC, of increasing the capital stock. Since (10) is a necessary condition for optimality, the left-hand side of (10) must be the marginal benefit, MB, by increasing the capital stock. Hence, q t represents the value to the firm of having one more unit of (installed) capital at time t. To put it differently: the adjoint variable q t can be interpreted as the shadow price (measured in current output units) of capital along the optimal path. 5 This interpretation is confirmed when we solve (integrates) the differential equation (11), see below. The condition (12) is a transversality condition saying that, if lim t K t > 0, then lim q te t rsds 0 > 0 t 4 The path (N t,i t,k t ) is called interior, if, for all t>0, N t > 0 and K t > 0. 5 We remember that a shadow price, measured in some unit of account, of a good is the number of units of account that the optimizing agent is just willing to offer for one extra unit of the good. 5

6 is impossible since the firm would otherwise prefer to decrease its ultimate capital stock (acquire more capital earlier and less later). By that alternative time profile of the capital stock, implying better utilization of the potentialities of capital before eternity, a higher value of the criterion function (4) would be obtained. Multiplying with e t (rs+δ)ds 0 on both sides of (11), we get by integration and application of the transversality condition (12), 6 q t = t [F K (K τ,t τ N τ ) G K (I τ,k τ )] e τ (rs+δ)ds t dτ. (13) The right-hand side of (13) is the PDV (as seen from time t) of expected future increases of the firm s cash-flow that would result if one extra unit of capital were installed at time t. Indeed, F K (K τ,t τ N τ ) is the direct contribution to output of one extra unit of capital, while G K (I τ,k τ ) 0 is the adjustment cost reduction of future investment projects (capital adjustment costs are decreased by G K,whenK τ is one unit larger). However, future increases of cash-flow should be discounted at a rate equal to the rate of interest plus the capital depreciation rate (because, from one extra unit of capital at time t only e δ(τ t) unitsareleftattime τ). To check our interpretation of q as representing the value to the firm of having one extra unit of capital at time t, let us perform a thought experiment. Assume that a extra units of (installed) capital at time t drops down from the sky. At time τ >tthere are a e e(τ t) units of these still in operation. Now, replace t by τ in (3) and consider the firm s cash-flow R τ as a function of K τ,n τ,i τ,t,τ, and a. Wefind R τ =[F K (K τ,t τ N τ ) G K (I τ,k τ )] e δ(τ t). a a=0 Let the value of the firm, i.e., the value of the integral (4), as seen from a fixed point, t, in time, be called V t. Then, V t a a=0 = t [F K (K τ,t τ N τ ) G K (I τ,k τ )]e τ t (rs+δ)ds dτ = q t, (14) when the firm moves along the optimal path, the second equality sign coming from (13). 7 This 6 Integrate from 0 to t 1, let t 1, use (12) with t replaced by t 1. Finally, replace t and 0 by τ and t, respectively. 7 Not only is the current value of the adjoint variable, q t, equal to the marginal contribution to the value of the firm, as seen from "now", of a hypothetical injection of capital "now", but the discounted value of the adjoint variable, t q t exp( r τ dτ), measures, in optimum, the marginal contribution of such an injection to V 0 (see Lèonard 1987). 6 0

7 confirms that q t is the shadow price of capital at time t, andthatq t represents marginal benefit by increasing the capital stock. On this background it becomes understandable that the control variables at any point in time should bo chosen so that the Hamiltonian function is maximized. This amounts to maximizing the sum of the direct contribution to the criterion function in the current time interval and the indirect contribution, the benefit of having a higher capital stock in the future (as measured approximately by q t K t ). In the same way we now understand the last equality sign in (11); a condition for optimality must be that the firm acquires capital up to the point where the value of the "net marginal product of capital", F K G K δq t, is equal to the "net capital costs", r t q t q t. Herewelookatq t as the the "overall" price at which the firm can buy and sell the marginal unit of installed capital. Continuing along this line of thought, by reordering in (11) we get the optimality condition F K G K δq + q = r, (15) q saying that, the rate of return on the marginal unit of capital must be equal to the (real) interest rate. As is well-known the Pontryagin maximum principle gives only necessary conditions for an optimal path. We use the principle as a tool for finding candidates for a solution. Having found in this way a candidate, it is a solution, if the Hamiltonian function is concave in the endogenous variables (here K, L og I, cf. (8). 8 In our context we get concavity by assuming G(I,K) is a convex function in (I,K), whereby G(I, K) is concave. 2.3 The implied investment function From condition (10) we can derive an investment function. First (10) can be written G I (I t,k t )=q t 1 (16) along the optimal path. function we see that Combining this with the assumptions (2) on the adjustment cost I t 0 for q t 1, respectively. (17) Indeed, in view of G II > 0, (16) implicitly defines optimal investment, I t, as a function of the shadow price q t on K t : I t = M(q t,k t ), where M q =1/G II > 0. (18) 8 Strict concavity ensures uniqueness. 7

8 It follows that optimal investment is an increasing function of the shadow price of installed capital, and since G I (0,K)=0,wehaveM(1,K)=0. Not surprisingly, the investment rule is: Invest now, if and only if the marginal value to the firm of installed capital is larger than the price, 1, of the capital good (excluding adjustment costs). At the same time, however, (16) and say (18) that, because of the convex adjustment costs, invest only up to the point where the marginal adjustment cost, G I (I t,k t ), becomes equal to the difference between q t and 1. Condition (18) shows the remarkable information content that the shadow price q t has. As soon as q t is known (along with the current capital stock K t ),thefirm can decide the optimal level of investment through knowledge of the adjustment cost function G alone (since, when G is known, so is its inverse, the investment function M). All the information about the production function, output and input prices, the rate of interest now and in the future that is relevant to the investment decision is summarized in one number, q t, and do not affect the form of the investment function M. This will turn out to be a very useful property in empirical analysis, as we shall see soon. 2.4 A convenient special case: An adjustment cost function which is homogeneous of degree one We now introduce the convenient case where the installation function G is homogeneous of degree one so that we can write J = G(I,K)=G( I K, 1)K g( I )K, K or (19) J K = g( I K ), where g(0) G(0, 1) = G(0,K)=0, g = G I og g = KG II > 0 (for K>0). 9 The graph of g(i/k) is qualitatively the same as that in Fig 1 (imagine we have K =1inthat graph). 10 The homogeneity assumption implies that the adjustment cost relative to the existing capital stock is a strictly convex function of the "investment ratio", I/K. An example: J = G(I,K)= 1 2 βi2 /K is convex in (I,K) and gives J/K = 1 2 β(i/k)2. Since g (I/K)=G I (I,K) we can now write (16) as g (I/K)=q 1. (20) 9 Proof:: G I = g 1 K K = g and G II = g 1 K. 10 The relationship between our function g and the function φ introduced in B & S, p , is commented on in Appendix A. Essentially, the two expositions are equivalent. 8

9 IK / mq () δ + n+ g 1 q* q Figure 2: This equation defines the investment ratio, I/K, as an implicit function, m, of q 1: I t K t = m(q 1) m(q t ), where m(1) = m(0) = 0 and m = m =1/g > 0. (21) In this case, q encompasses all information that is of relevance to the decision about the investment ratio I/K. Fig. 2 illustrates the implied investment function m. It may be non-linear, but using the example above, where g(i/k)= 1 2 β(i/k)2, (20) gives I/K =(q 1)/β m(q), a linear function. Toseehowtheshadowpriceq changes over time we rewrite (11) as In the special case (19) we have q t =(r t + δ)q t F K (K t,n t )+G K (I t,k t ). (22) G K (I,K) = J K = g( I K )K K from (21) and (20). Inserting this into (22) gives = g( I K )+Kg ( I K ) I K 2 = g( I K ) I K g ( I )=g(m(q)) m(q)(q 1) K q t =(r t + δ)q t F K (K t,n t )+g(m(q t )) m(q t )(q t 1). (23) 9

10 This differential equation turns out to be very useful in macroeconomic analysis, where, given full employment and exogenous labour supply L t,wecanreplaceemploymentn t by labour supply L t (as we shall soon do). In a macroeconomic context, for steady state to obtain (gross) investment must be large enough to compensate not only for capital depreciation, but also for growth in the effective labour input (T N). That is, the (gross) investment ratio, I/K, must be equal to the sum of the depreciation rate, the growth rate of the labour force, and the rate of technical progress, i.e., δ + n + γ. That level of q whichisrequiredtomotivatesuchaninvestmentrationiscalled q in Fig. 2. We return to this later. 3 Marginal q and average q Our q above, determining investment, should be distinguished from what is usually called Tobin s q or average q. Letp It denote the purchase price (in terms of output units) per unit of the investment good. Then Tobin s q or average q is defined as qt A V t, p It K t that is, Tobin s q is the ratio of the total value of the firm to the replacement cost of its total capital stock (the top index "A" stands for "average"). In our simplified context context we have p It 1 (the price of the investment good is the same as that of the output good), and therefore Tobin s q can be written qt A V t, (24) K t In general, this is different from our (real) shadow price on capital, our q above. In the language of the q-theory of investment our q is called marginal q, representingthevalueto the firm of one extra unit of capital relative to the replacement cost (the purchase price of the investment good which in our simplified context is 1). Indeed, the term marginal q is natural in view of condition (14) saying that along the optimal path we have q t =( V t / K t )/p It qt M ("M" for "marginal"). Since we have p It 1 we can now write q M t = q t = V t / K t. (25) The two concepts, average q and marginal q, have not always been clearly distinguished in the literature. What is relevant to the investment decision is marginal q, since the analysis above shows that optimal investment is an increasing function of marginal q, q M. Further, the 10

11 analysis showed that a "critical" value of q M is 1 so that only if q M investment warranted. > 1, is positive (gross) The importance of average q, q A, is that it easier to measure empirically (as the ratio of the sum of the share market value of the firm and its debt to the replacement cost of its total capital). Since q M is harder to measure than q A it is important to know how q M is related to q A. Fortunately, we have a simple theorem giving conditions under which q M = q A. Indeed, Hayashi s theorem (Hayashi 1982) gives a sufficient (and generally also necessary) condition for this equality. The Hayashi theorem. Assume the firm is a price taker, that the production function F is concave in (K, N), and that the adjustment cost function G is convex in (I, K). Then, along the optimal path we have: (i) qt M = qt A for all t 0, if F and G are homogeneous of degree 1. (ii) qt M <qt A for all t, if F is strictly concave in (K, L) and/or G is strictly convex in (I, K). Proof. See Appendix B. The assumption that the firm is a price taker is, of course, critical. For firm setting its own price, facing a downward-sloping demand curve, we would normally have q M <q A along the optimal path. To sell the extra output made possible by larger productive capacity the firm would have to decrease its output price. When q M is approximately equal to q A, the theory gives a remarkable simple operational macro investment function, I = m(q A )K, cf. (21). Here q A is the market valuation of the firms relative to the replacement cost of their total capital stock. This market valuation can be considered as a (more or less reliable) indicator of the future earnings potential of the firms. Under the conditions in (i) of the Hayashi theorem the market valuation also indicates the marginal earnings potential of the firms, hence, it becomes a determinant of their investment. This establishment of a relationship between the share market and firms aggregate investment is the basic point in Tobin s q-theory of investment. 11

12 4 Applications 4.1 Convex capital adjustment costs in a Ramsey model for a closed economy It is straightforward to allow for convex capital adjustment costs in the Ramsey model, but the analysis is cumbersome (see Abel and Blanchard, 1983). Among the insights are that investment decisions attain an active role in the economy. Expectations become important for firms investment. Expected future changes in taxation and depreciation allowance rules tend to change firms investment today. Further, the relationship between the (real) rate of interest and the (net) marginal product of capital becomes less direct in the short run, since the interest rate has to adjust so that aggregate saving can be equal to aggregate investment at all points in time. 4.2 A small open economy with convex capital adjustment costs and stationary "effective" labour force A more simple set-up is (as usual) that of a small open economy (henceforth SOE) with perfect capital mobility. Introducing convex capital adjustment costs we avoid the counterfactual result that the capital stock changes instantaneously when the interest rate at the world capital market changes. We assume: 1. Perfect mobility across borders of financial capital. 2. Domestic and foreign financial claims are perfect substitutes. 3. No mobility across borders of labour. 4. Labour supply is inelastic and constant. 5. The capital adjustment cost function G(I,K) is homogeneous of degree 1. Therefore, the SOE faces an exogenous (real) rate of interest, r t, given from the world capital market. For convenience, it is assumed that r t = r>0 for all t 0, 12

13 where r is a constant. Let L >0 denote the constant labour supply in our SOE, that is, we ignore, for simplicity, population growth; further, let the technology level be a constant T 0 =1, that is, we ignore technical progress, i.e., n = g =0. 11 Then, assuming clearing on the labour market under perfect competition we have N t = L for all t 0 and w t = F 2 (K t, 1 L) w(k t ). (26) At any time t, K t is predetermined so that (26) determines the marker real wage w t. Since the capital adjustment cost function G(I,K) is assumed homogeneous of degree 1, the analysis of Section 2.4 applies, and we can write (23) as q t =(r + δ)q t F K (K t, L)+g(m(q t )) m(q t )(q t 1). (27) Since r and L are exogenous this is a differential equation with the capital stock, K, and its shadow price, q, as the only endogenous variables. Another differential equation with these two variables as the only endogenous ones can be obtained by inserting (21) into (6) to get K t =(m(q t ) δ)k t. (28) Fig. 3 shows the phase diagram for these two coupled differential equations. We have (suppressing, for convenience, the explicit time indexes) K =0 for m(q) =δ, i.e., for q = q, defining q by the requirement m(q )=δ. Notice, that when δ > 0, we get q > 1. This is so because also mere reinvestment, to offset capital depreciation, requires an incentive, namely that the marginal value to the firm of replacing worn-out capital is larger than the purchase price of the investment good (since the installation cost must also be compensated). We see that K 0 for m(q) δ, respectively, i.e., for q q, respectively, cf. the horizontal arrows in Fig. 3. From (27) we have q =0for 0=(r + δ)q F K (K, L)+g(m(q)) m(q)(q 1). (29) 11 In the next section, however, we allow n>0 and/or g>0,and there, the general specification of the decision problem of the representative firm in Section 2.2 above will be useful. 13

14 q B * q E K = 0 * K 0 K q = 0 K Figure 3: If, in addition K =0(hence, q = q and m(q) =m(q )=δ), thisgives 0=(r + δ)q F K (K, L)+g(δ) δ(q 1), (30) where the right-hand-side is increasing in K (in view of F KK < 0). Hence, there exists at most one value of K such that the steady state condition (30) is satisfied 12 ;thisvalueiscalledk, corresponding to the steady state point E in Fig. 3. The question now is: What is the slope of the q =0locus? In Appendix C it is shown that at least in a neighbourhood of the steady state point E this slope is negative in view of the assumption r>0. From (27) we see that q 0 for points to the left and to the right, respectively, of the q =0locus, since F KK (K t, L) < 0. The vertical arrows in Fig. 3 show these directions of movement. Altogether the phase diagram shows that the steady state E is a saddle point, and since there is one predetermined variable, K, and one jump variable, q, this steady state is saddlepoint stable. Usually we can exclude the divergent paths by appealing to the transversality condition and the No-Ponzi-Game condition of the households. Here we have instead the firms transversality and feasibility conditions. Indeed, paths starting on the vertical line through K = K 0 in Fig. 3, but above the saddle path, can be shown to violate the transversality 12 And assuming that F satisfies the Inada conditions, (30) shows that such a value does exist. 14

15 q * q E ' E K = 0 new q = 0 B q = 0 K * ' * K K Figure 4: condition (12). And paths starting below the saddle path can be shown to violate the feasibility condition K t 0 for all t 0. Hence, a movement along the saddle path towards the steady state is the unique solution to the model. The effect of a shift in the interest rate. Assume that until time 0 the economy has been in the steady state E in Fig. 3. Then, an unexpected shift in the interest rate r at the world capital market occurs so that the new interest rate is a constant r > 0 (and the interest rate is expected to remain at this level forever in the future). From (28) we see that q is not affected by this shift, hence, the K =0 locusisnotaffected. But the q =0locus shifts to the left, in view of F KK (K, L) < 0. Fig. 4 illustrates the situation for t 0. At time t =0the shadow price q jumps down to a level corresponding to the point B in Fig. 4. This is because of the more heavy discounting of the future gains that the marginal unit of capital can provide. As a result the incentive to invest is diminished, and gross investment does not even compensate for the depreciation of capital. hence, the capital stock decreases gradually. But this implies increasing marginal productivity of capital, hence, increasing q, and the economy moves along the new saddle path and approaches the new steady state E as time goes by. Suppose that for some reason such a decrease in the capital stock is not desirable from a 15

16 social point of view. 13 Then the government could decide to implement an investment subsidy σ, 0 < σ < 1, so that to attain an investment level I purchasing the investment goods involves acostof(1 σ)i. Assuming the subsidy is financed by some tax not affecting firm s behaviour, investment is increased again, and the economy might in the long run end up in the old steady state, E, again A growing small open economy with convex capital adjustment costs The basic assumptions are the same as in the previous section except that now labour supply (still exogenous) grows at the rate n 0, while there is technical progress at the rate g 0. Assuming full employment, as before, we have N = L for all t 0. Now we need the intensive production function f defined through Y = F (K, TL) =F ( K, 1)TL f(ˆk)tl, TL where ˆk K/(TL) is the "effective" capital intensity, and f satisfies f > 0,f < 0. The market-clearing real wage at time t is determined as w t = F 2 (K t,t t L t )T t = f(ˆk t ) ˆk t f (ˆk t ) T t ŵ(ˆk t )T t, (31) where both ˆk t and T t are predetermined. By log-differentiation of ˆk K/(TL) we get ˆk t ˆk t = K t K t (g + n) =m(q t ) (δ + g + n), from (28), so that ˆk t =[m(q t ) (δ + g + n)] ˆk t. (32) The change in the shadow price of capital is now described by from (23). q t =(r + δ)q t f (ˆk t )+g(m(q t )) m(q t )(q t 1), (33) 13 This could be because of positive external effects of investment, e.g., learning by doing. 14 In fact, the q-theory framework has proved very useful for the study of effects on investment in capital as well as housing of different kinds of subsidies and different taxation and depreciation allowance rules, see, e.g., Summers (1981). 16

17 q q = 0 new q = 0 B * q E E ' ˆ k = 0 ˆk * *' ˆk ˆk Figure 5: The differential equations (32) and (33) constitute our new dynamic system. Fig. 5 shows the phase diagram, which is, qualitatively, similar to that in Fig. 3. We have ˆk =0 for m(q) =δ + g + n, i.e., for q = q, where q now is defined by the requirement m(q )=δ + g + n. Notice, that since g + n>0, we get a larger steady state value q than in the previous section. This is so because now a higher investment ratio is required for a steady state to be possible. From (33) we see that q =0now requires If, in addition 0=(r + δ)q f (ˆk)+g(m(q)) m(q)(q 1). ˆk =0(hence, q = q and m(q) =m(q )=δ + g + n), thisgives 0=(r + δ)q f (ˆk)+g(δ + g + n) (δ + g + n)(q 1), where the right-hand-side is increasing in ˆk (in view of f (ˆk) < 0). Hence, the steady state value ˆk of the capital intensity is unique, cf. the steady state point E in Fig. 5. Assuming r>g+ n we have, at least in a neighbourhood of E in Fig. 5, that the q =0 17

18 locus is negatively sloped (see Appendix C). 15 Again the steady state is a saddle point, and the economy moves along the saddle path towards the steady state. In Fig. 5 it is assumed that until time 0 the economy was in the steady state E. Then, an unexpected shift upwards in the interest rate r takes place. The q =0locus is shifted to the left, in view of f < 0. At time t =0the shadow price q jumps up to a level corresponding to the point B in Fig. 5. The economy moves along the new saddle path and approaches the new steady state E with a higher capital intensity as time goes by. 5 Concluding remarks How has the q theory of investment done empirically? At the aggregate level not entirely well, unfortunately. Movements in q A, even taking account of changes in taxation, seem capable of explaining only a minor fraction of the movements in investment. And the estimated equations relating investment to q A typically give strong auto-correlation in the residuals. Other variables such as changes in aggregate output, the degree of capacity utilization and corporate profits often seem to have explanatory power independently of q A (see Abel 1990). It follows that there is reason to be somewhat sceptical toward the hypothesis that al information of relevance for the investment decision is reflected by the share market valuation of the firms. contained, er indfanget af aktiemarkedets vurdering af virksomheden. In view of the restrictive preconditions behind part (i) of Hayashi s theorem (perfect competition and homogeneity of degree one for both the production function and the adjustment cost function), this is not at all surprising. And going outside the model, so to say, there are further circumstances relaxing the link between q A and investment. In the real world physical capital is heterogeneous. If for example a sharp increase in the price of energy takes place, a firm with energy-intensive technology will fall in market value. At the same time it has an incentive to invest in energy saving capital equipment. Hence, we might observe a fall in q A at the same time as investment increases. Also imperfections on markets for financial capital may disturb the relationship between q A and investment. In fact, this may be the decisive factor behind the observed positive correlation between investment and corporate profits. Finally, we might also be a little sceptical as to the validity of the basic assumption behind 15 In our perfect foresight model we in fact have to assume r>g+n for the firm s decision problem in Section 2.2tobewell-defined. If instead r g + n, the market value of the representative firm would be infinite, and maximization would loose its meaning. 18

19 the q theory of investment, namely that the capital adjustment costs have the hypothesized convex form. There is no reason to doubt that there exist costs associated with installation, reorganizing and retraining etc., when new capital equipment is procured. But is it true that they are strictly convex in the volume of investment? To think about this, let os for a while ignore the role of the existing capital stock for the adjustment costs. Hence, we write total adjustment costs J = G(I) with G(0) = 0. It does not seem problematic to assume G (I) > 0 for I>0. The question concerns the assumption G (I) > 0. According to this assumption the average adjustment cost G(I)/I must be increasing in I. 16 But against this speaks the fact that capital installation may involve indivisibilities, fixed costs, acquisition of new information etc. all these features tending to imply decreasing average costs. In any case, at least at the microeconomic level one should expect unevenness in the capital adjustment process rather than the above smooth adjustment. 6 Appendix 6.1 A. Extensions and comments on the literature The simple relationship we have found between I and q can easily be generalized to the case where the purchase price on the investment good, p It, is allowed to differ from 1 (its value above), and the capital adjustment cost is p It G(I t,k t ).Inthiscaseitisconvenienttoreplace q in the Hamiltonian function by, say, λ. Then the first order condition (10) becomes p It + p It G I (I t,k t )=λ t, implying G I (I t,k t )= λ t 1, p It and we can proceed, defining q t by q t λ t /p It. Sometimes in the literature you will find adjustment costs J appearing in a slightly different way compared to the above exposition. For example, Romer (2001, p. 371 ff.) assumes the capital adjustment costs J to depend only on I so that G K 0. This simplification (independence of scale) may be doubtful from an empirical point of view, and the approach is not well suited to a context with economic growth (n >0 and/or g>0) Indeed, for I = 0we have d[g(i)/i]/di =[IG (I) G(I)]/I 2 > 0, when G is strictly convex and G(0) = Reading these pages in Romer it is important to realize that he assumes δ =0, so that the steady state value of q becomes 1. When δ > 0, a steady state requires some positive gross investment to compensate for capital depreciation and therefore the steady state value of q in our models above is larger than 1. 19

20 Barro and Sala-i-Martin (1995, p ) introduce a function, φ, representing capital adjustment costs per unit of investment as a function of the investment ratio. That is, total adjustment cost is J = φ(i/k)i, where φ(0) = 0, φ > 0. This implies that J/K = φ(i/k)(i/k). The right-hand side of this equation may be called g(i/k), and then we are back at our formulation in Section 2.4 above. Indeed, defining x I/K we have g(x) =φ(x)x so that φ(0) = 0 g(0) = 0, and g (x) =φ(x)+xφ (x) 0 for x 0, respectively, by the assumptions φ(0) = 0, φ > 0. Further, g (x) =2φ (x)+xφ (x) > 0, by the assumption in footnote 20 on p. 120 in Barro and Sala-i-Martin. In fact, what is needed for the theory to work is that g (x) > 0, and the assumptions φ(0) = 0, φ > 0, and φ 0, mentioned by B & S on p. 120 and again on p. 121, are not sufficient for this (since x<0 is possible), contrary to the claim in the middle of p And it is certainly not true that assuming 2φ (x)+xφ (x) > 0 is weaker, as claimed by B & S in footnote 20. The example, J = G(I,K)= 1 2 βi2 /K, considered in Section 2.4 above, corresponds to the linear specification (3.49) in B & S, p The example leads to J/K = 1 2 β(i/k)2, and from the general optimality condition G I (I,K)=q 1we get βi/k = q 1 or I/K =(q 1)/β. Other authors let the capital adjustment cost G(I, K) appear, not as a reduction in output, but as a reduction in capital formation so that K = I δk G(I,K). This approach is used by Hayashi (1982) and Heijdra & Ploeg (2002, p. 573 ff.). Indeed, the "capital installation function" ϕ(i/k) introduced by Heijdra & Ploeg can be seen as defined by ϕ(i/k) [I G(I,K)] /K = I/K g(i/k), where the last equality comes from assuming G is homogeneous of degree 1. We get K/K = ϕ(i/k) δ, cf. (T5.1) on p. 573 in Heijdra & Ploeg. In one-sector models, as we usually consider in this course, this changes nothing of importance. In more general models this installation function approach may have some analytical advantages. What gives the best fit empirically is an open question. 6.2 B.AproofofHayashi stheorem For convenience we repeat: 20

21 The Hayashi theorem. Assume the firm is a price taker, that the production function F is concave in (K, N), and that the adjustment cost function G is convex in (I, K). Then, along the optimal path we have: (i) qt M = qt A for all t 0, if F and G are homogeneous of degree 1. (ii) qt M <qt A for all t, if F is strictly concave in (K, L) and/or G is strictly convex in (I, K). Proof. We introduce the functions A = A(K, N) F (K, T N) F 1 (K, T N)K F 2 (K, T N)TN, and B = B(I,K) G 1 (I,K)I + G 2 (I,K)K G(I,K). Then the cash-flow of the firm, given by (3), at time τ can be written R τ = F τ F 2 TN G τ I = A τ + F 1 K τ + B τ G 2 K τ G 1 I τ I τ, wherewehaveusedfirst (9) and then the definitions of A and B above. The value of the firm as seen from time t is now V t = = t t (A τ + B τ )e τ t rsds dτ + (A τ + B τ )e τ t rsds dτ + q t K t, t [(F 1 G 2 )K τ (G 1 +1)I τ ]e τ t rsds dτ (34) when moving along the optimal path, cf. Remark 1 below. It follows that q M t q t = V t K t 1 K t t [A(K τ,n τ )+B(I τ,k τ )]e τ t rsds dτ. (35) Since F is concave, we have for all K and N, A(K, N) 0 with equality sign, if and only if F is homogeneous of degree one. Similarly, since G is convex, we have for all I and K, B(I,K) 0 with equality sign, if and only if G is homogeneous of degree one. Now the conclusions (i) and (ii) follow from (35) and the definition of q A in (24). Adifferent way of understanding (i) in the Hayashi theorem is the following. Let x t I t /K t. Then K t /K t = x t δ, implying K τ = K t e τ t (xs δ)ds. Hence, when F and G are homogeneous of degree one, not only are A and B equal to zero, but K t can be put outside the integral in (34). We get V t = K t [f (k τ ) g(x τ ) x τ ]e τ t (rs xs+δ)ds dτ, t 21

22 along the optimal path. In this expression, f is the intensive production function, k τ is the effective capital intensity ( K τ /(T τn τ )), determined by the market real wage w τ,and,finally, g(x) G(x, 1). In view of (21), with t replaced by τ, the optimal investment ratio x τ depends, for all τ, only on q τ, not on K τ, hence not on K t. Hence, V t / K t = t [f (k τ ) g(x τ ) x τ ]e τ t (rs xs+δ)ds dτ = V t /K t, and the conclusion q M t = q A t follows from (25) and (24). Remark 1. Here we prove that the last integral in (34) is equal to q t K t, when investment follow the optimal path. Keeping t fixed and using z as our varying time variable we have (F 1 G 2 )K z (G 1 +1)I z =[(r z + δ)q z q z ]K z (G 1 +1)I z =[(r z + δ)q z q z ]K z q z ( K z + qk z )=r z q z K z ( q z K z + q z K z )=r z u z u z, where we have used (11), (10), (6), and the definition u z = q z K z, respectively. We look at this as a differential equation: u z r z u z = ϕ z,whereϕ z [(F 1 G 2 )K z (G 1 +1)I z ].The solution of this linear differential equation is u z = u t e z z t rsds + ϕ τ e z τ rsds dτ, t implying, by multiplying through by e z t rsds, reordering, and inserting the definitions of u and ϕ, z t [(F 1 G 2 )K τ (G 1 +1)I τ ]e τ t r sds dτ = q t K t q z K z e z t rsds q t K t for z, from the transversality condition (12) with t replaced by z and 0 replaced by t. Remark 2. We have assumed throughout that G is strictly convex in I. This does not imply that G is strictly convex in (I,K). For example, the function G(I,K) =I 2 /K is strictly convex in I (since G 11 =2/K > 0). ButatthesametimethisfunctionhasB(I,K)=0and is therefore homogeneous of degree one. Hence, it is not strictly convex in (I,K). 22

23 6.3 C. The slope of the q =0locus We want to calculate the sign of the slope of the q =0locus in the case g + n =0, considered in Fig. 3. Totally differentiating in (29) w.r.t. K and q gives 0 = F KK (K, L)dK + {r + δ + g (m(q))m (q) [m(q)+(q 1)m (q)]} dq = Therefore = F KK (K, L)dK +[r + δ m(q)] dq, (since g (m(q)) = q 1, by (20)). dq dk q=0 = F KK(K, L) r + δ m(q) for r + δ = m(q). From this it is not possible to sign dq/dk at all points along the q = 0 locus. But in a neighbourhood of the steady state we have m(q) δ, hence r + δ m(q) r>0. And since F KK < 0, this implies that at least in a neighbourhood of E in Fig. 3 the q =0locus is negatively sloped. Similarly, for the case g + n>0, considered in Fig. 5, we get dq dˆk q=0 = f (ˆk ) r + δ m(q) for r + δ = m(q). From this it is not possible to sign dq/dˆk at all points along the q = 0 locus. But in a neighbourhood of the steady state we have m(q) δ + g + n, hence r + δ m(q) r g n. Since f < 0, then, at least in a neighbourhood of E in Fig. 5, the q =0locusisnegatively sloped, if r>g+ n. 23