Topic 8: Optimal Investment

Transcription

1 Topic 8: Optimal Investment Yulei Luo SEF of HKU November 22, 2013 Luo, Y. SEF of HKU) Macro Theory November 22, / 22

2 Demand for Investment The importance of investment. First, the combination of firms investment demand and households saving supply jointly determines how much of an economy s output is invested, which matters for long-run growth. Second, investment is highly volatile. That is, investment is important for understanding short-run aggregate fluctuations. A static investment model: Consider a firm that can rent capital at a price r K. The firm s profits are given by π K; X 1,, X 1 ) r K K, 1) where K is the desired capital stock and the X s are exogenous variables. e.g., X s include the output price and the costs of other inputs.) Assume that the revenue function π ) satisfies: π K > 0 and π KK < 0. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

3 Optimal Desired) Capital Stock The FOC for the profit-maximizing choice of K: π K K; X 1,, X 1 ) = r K, 2) which means that the firm rents capital up to the point where its marginal revenue equals its rental price. 2) implicitly defines the firm s desired capital stock as a function of r K and the X s: K = f r K ; X 1,, X 1 ). 3) The main problems of this investment problem: 1 Adjustments of capital stock are costless. In reality, capital stock adjusts gradually with respect to changes in the X s.) 2 Expectations do not affect the demand for investment. In realty, expectations about demand and costs are central to investment decisions.) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

4 Introducing Adjustment Costs To overcome the first problem, we assume that firms face costs of adjusting their capital stocks. Examples include the costs of installing the new capital and training workers to operate new machines. Specifically, assume that the adjustment costs, C i), where i is the rate of change of the firm s capital stock i.e., investment), satisfy C 0) = 0, C 0) = 0, C ) > 0, 4) which means that it is costly for a firm to change its capital stock, and that the marginal adjustment cost is increasing in the size of the adjustment. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

5 An Investment Model with Adjustment Costs The firm s objective function is to maximize the expected discounted present value of profits: ṽ t = max subject to [ s=t ) 1 s t A s F K s,, L s ) χ I 2 ) ] s w s L s I s, 1 + r 2 K s 5) K s+1 = 1 δ) K s + I s, s t 6) given K t 1. For simplicity, we may assume that δ = 0. I 2 s In this model, χ 2 K s is the cost of installation and is paid by the firm above the actual cost I s of purchasing the new capital goods. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

6 Optimal Conditions The Lagrangian is 1 L t = R s=t ) [ ) s t A s F K s, L s ) χ Is 2 2 K s w s L s I s +q s K s + I s K s+1 ) ], 7) where q s is the Lagrange multiplier. For simplicity, here we assume that w s and L s are given and F K s,, L s ) is linearly homogenous. The FOC for I s gives This condition can be rewritten as χi s K s 1 + q s = 0. 8) I s = q s 1 χ K s, 9) which means that investment is positive only if the shadow price q s of existing installed capital exceed 1, the price of new, uninstalled capital. Tobin s q theory.) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

7 The FOC for K s+1 gives [ q s = 1 R A s+1 F K K s+1 ) + χ 2 Is+1 K s+1 which can be regarded as the investment Euler equation. ) 2 + q s+1], 10) Implications: At an optimum for the firm, the time s shadow price q s of an extra unit of capital is the discounted sum of: 1 the capital s marginal product next period; 2 the capital s marginal contribution to lower installation costs next ) χ 2 period Is 2 K s ); 3 the shadow price of capital on the next period, s + 1. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

8 Intuitively, 10) means that at an optimum, the firm cannot increase profits by raising its installed capital at marginal cost: q s = 1 + χi s K s, 11) benefiting a higher marginal product and lower investment cost: Is+1 A s+1 F K s+1 ) + χ 2 K s+1 ) 2, 12) and then disinvesting the unit of capital at the end of s + 1 to reap a marginal revenue of q s+1 = 1 + χi s+1 K s+1. 13) Using the usual iterative substitution on 10), we have ) [ 1 s t q t = A s F K K s ) + χ ) ] 2 ) Is 1 T + lim q t+t. s=t+1 R 2 K s T R 14) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

9 If lim 1 ) T T R qt+t > 0, ) [ 1 s t q t > R s=t+1 A s F K K s ) + χ 2 Is K s ) 2 ], 15) where the RHS is the stream of earnings that a marginal unit of capital permanently in place will generate for the firm, and the LHS is the value to the firm of dismantling the marginal unit of capital and selling it on the market. That is, > says that the firm cannot be optimizing: its capital stock is too high since discounted profits can be raised by a permanent reduction in capital. A symmetric argument rules out the possibility that lim 1 ) T T R qt+t < 0. Therefore, lim 1 ) T T R qt+t = 0 and q t = s=t+1 1 R ) s t [ A s F K K s ) + χ 2 Is K s ) 2 ], 16) which means that the shadow price of installed capital equals its discounted stream of future marginal products plus the discounted stream of its marginal contributions to the reduction in future capital installation costs. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

10 Investment Dynamics 9) can be rewritten in terms of capital: and 10) can be rewritten as K t+1 K t = q t 1 χ K t, 17) It+1 q t+1 q t = rq t AF K K t+1 ) χ ) 2, = 2 K t+1 q t+1 q t = rq t AF K 1 + q ) ) t 1 K t 1 χ 2χ q t+1 1) 18) 2 where we use the facts that K t+1 = 1 + q ) t 1 K t and χ It+1 K t+1 ) 2 ) qt =. χ In the steady state, both K t and q t remain constant over time. 17) and 18) clearly shows that q = 1 and K satisfies r = AF K K ). 19) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

11 The linearized two-equation system is K t+1 K t = K χ q t 1), 20) q t+1 q t = r AKF ) ) KK K ) q t 1) AF KK K Kt 21) K ). χ We can use the phase diagram to analyze the model s dynamics. The schedule, K t+1 = 0, means that it is horizontal at q t = q = 1. The schedule, q t+1 = 0 can be written as r AKF ) ) KK K ) q t 1) AF KK K Kt K ) = 0, χ which means that dq dk q t+1 = AF KK K ) r AKF KK K ) /χ < 0 22) because F KK K ) < 0. Therefore, the system is saddle-point-stable, i.e., for any starting capital stock K t, there is one and only one value of q t that places the firm on the stable adjustment path. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

12 Another Way to Check More compactly, [ ] K t+1 = ) q t+1 AF KK K 1 K χ R AK F KK K ) χ }{{} K [ K t q t ], 23) where K t = K t K and q t = q t 1. The characteristic roots b for the system satisfy trace = b 1 + b 2 = 1 + R AKF ) KK K > 2, 24) χ det = b 1 b 2 = R > 1. 25) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

13 conti.) Hence, the discriminant should be positive because = trace K ) 2 4 det K ) = 1 + R AKF ) ) 2 KK K 4R χ > R 1) 2 > 0 which means that both roots are real. Also, because det > 1 and trace > 2, the two roots must individually be positive. We can also judge the magnitudes of the two roots as follows: p b) = b b 1 ) b b 2 ) = 0 = p 1) = 1 b 1 ) 1 b 2 ) = 1 trace + det = AKF ) KK K χ < 0 This can only be true if one root say b 1 ) is less than 1 and the other root is greater than 1. We can then conclude and confirm the predictions of the PD) that the equilibrium is saddle-point. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

14 Marginal and Average q What is the relationship between q and the stock-market value of a unit of the firm s capital: v/k. In the deterministic model, 1 + r = d t+1 + v t+1 v t, 26) where v t+1 is the stock market value of the firm and d t+1 is the firm s dividend. 26) implies that at t, v t = d t r + v t r. 27) Continuing in this way, we have v t = ) 1 s t d s, R 28) s=t+1 provided a condition ruling out asset price bubbles: lim T 1 R ) T vt+t = 0. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

15 Multiplying K t+1 the investment Euler equation gives = 1 R = 1 R = 1 R q t K t+1 [ A t+1 F K K t+1, L t+1 ) K t+1 + χ 2 [ A t+1 F K K t+1, L t+1 ) K t+1 + χ 2 [ A t+1 F K K t+1, L t+1 ) K t+1 χ 2 It+1 2 ] + q t+1 K t+1 K t+1 It+1 2 ] + q t+1 K t+2 I t+1 ) K t+1 It+1 2 ] I t+1 + q t+1 K t+2 K t+1 where we use that fact that q t+1 = 1 + χi t+1 K t+1. Since the production function is linear homogenous, the forward iteration on q t K t+1 and using the Euler equation give q t K t+1 = ) 1 s t [ A s F K s, L s ) χ I 2 ] s w s L s I s v t. s=t+1 R 2 K s 29) Luo, Y. SEF of HKU) Macro Theory November 22, / 22

16 It means that q t = v t K t+1, 30) i.e., the shadow price of capital, q, equals the stock-market value of a unit of the firm s capital: v/k the marginal q equals the average q.) The asset-market equilibrium: [ At+1 F K K t+1, L t+1 ) χ I t+1 /K t+1 ) 2 ] /2 R = I t+1 /K t+1 + q t+1 K t+2 /K t+1 ), 31) q t equating the gross rate of return on a unit of the firm capital to R. The numerator on the RHS equals dividends paid out per unit of t + 1 capital plus capital gain. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

17 A Linear-Quadratic Version of Lucas and Prescott 1971) Consider an industry in which n identical competitive firms use a single input, capital, to produce a single output. The industry demand curve for output at t is p t = A 0 A 1 Y t + u t, 32) where A 0, A 1 > 0, p t is the price of output, Y t is industry output, and u t is a demand shock. The output of each firm is f 0 k t where k t is the firm s capital stock and f 0 > 1. Y t = nf 0 k t. The firms are competitive in the output and factor markets and thus are price takers with respect to the sequences of output prices {p t+j } j=0 and capital prices {q t+j } j=0. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

18 Optimizing Problem Suppose that the typical firm chooses a sequence of capital stocks, {k t+j } j=0, to maximize the following expected discounted present value: { [ ] v t = E t b j pt+j f 0 k t+j ) q t+j k t+j } k t 1+j ) 1 2 d k t+j k t 1+j ) 2, 33) j=0 given that k t 1 given and d > 0 determines the cost of adjustment. The Euler equation for this problem is E t [d k t k t 1 ) bd k t+1 k t )] = E t [p t f 0 q t + bq t+1 ], E t [k t ) k t + 1 ] b b k t 1 = 1 bd E t [p t f 0 q t + bq t+1 34) ] for any t. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

19 conti.) [ ) L + 1b ] b L2 [1 1b ) ] L 1 L) E t [k t+1 ] = 1 bd E t [p t f 0 q t + bq t+1 ], E t [k t+1 ] = 1 bd E t [p t f 0 q t + bq t+1 ], 1 L) E t [k t+1 ] = d 1 L 1 1 bl 1 E t [p t f 0 q t + bq t+1 ], 1 L) E t [k t+1 ] = d 1 Substituting 35) into 34), we can solve for k t : k t = k t bl 1 E t [p t+1 f 0 q t+1 + bq t+2 ].35) d 1 1 bl 1 E t [p t f 0 q t + bq t+1 ], 36) which gives the firm s rate of investment as a function of expected future values of the prices of output and capital. Expectation here plays a role in determining investment. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

20 Market Equilibrium While individual firms take the price of output as given when making decisions, the price is affected by the actions of all firms together since p t = A 0 A 1 nf 0 k t ) + u t. 37) We now seek an equilibrium pair of sequences {p t } t=0 and {k t } t=0 that satisfy the following two equilibrium conditions: 1 Given the typical firm s equilibrium capital sequence {k t } t=0, prices clear the market: p t = A 0 A 1 nf 0 k t ) + u t. 38) 2 When the firm faces the sequence {pt } t=0 as given, the sequence {kt } t=0 maximizes 33). That is, in the equilibrium the firm is on its demand curve for capital 36) and the market clears so that 38) is satisfied. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

21 Procedure to Determine the Equilibrium Substituting p t = A 0 A 1 nf 0 k t ) + u t into the Euler equation, E t [k t ) k t + 1 ] b b k t 1 = 1 bd E t [p t f 0 q t + bq t+1 ], 39) gives E t [k t b + A 1nf0 2 ) k t + 1 ] d b k t 1 = 1 bd E t [A 0 + u t ) f 0 q t + bq t+1 ], E t [1 λ 1 L) 1 λ 2 L) k t+1 ] = 1 bd E t [A 0 + u t ) f 0 q t + bq t+1 ], where λ 1 < 1 < 1 b < λ 2. Luo, Y. SEF of HKU) Macro Theory November 22, / 22

22 The solution is E t [kt+1] = λ 1 kt + λ 1d 1 1 λ2 1 E L 1 t [A 0 + u t+1 ) f 0 q t+1 + bq t+2 ], 40) for any t, which gives the equilibrium sequence of k t. Substituting 40) into 39), we can solve for the equilibrium k: k t = λ 1 k t 1 + λ 1d 1 = λ 1 k t 1 + λ 1 d 1 λ2 1 E L 1 t [A 0 + u t ) f 0 q t + bq t+1 ] ) 1 i E t [A 0 + u t+i ) f 0 q t+i + bq t+1+i ]. i=0 λ 2 Substituting the equilibrium k sequence into the market demand schedule gives p t = A 0 A 1 nf 0 k t ) + u t. 41) That is, by construction, we have generated sequences, {pt } t=0 and {kt } t=0 that satisfy the two equilibrium conditions. Luo, Y. SEF of HKU) Macro Theory November 22, / 22