Investment. Investment expenditures are one of the most volatile elements of aggregate economy.

Transcription

1 Investment Investment expenditures are one of the most volatile elements of aggregate economy. Critical to productivity and growth. Key issue for policy intervention: investment tax credits. accelerated depreciation allowances.

2 General Problem Model of a plant. The manager maximises the flow of profits by choosing: the inputs in the production process, x. the optimal stock of capital, K. Subject to: factor prices: w. technology shock: A. depreciation of capital: δ. costs of adjustment of capital stock. Profit function: Program of the manager: Π(A, K) = max R(Â, K, x) wx. x V (A, K, p) = max K Π(A, K) C(K, A, K) p(k (1 δ)k) +βe A AV (A, K, p ) K = K(1 δ) + I

3 No Adjustment Costs C(K, A, K) = 0, so the program simplifies to: V (A, K, p) = max K Π(A, K) p(k (1 δ)k)+βe A AV (A, K, p ) First order condition for optimal investment: βe A,p A,pV k (A, K, p ) = p with V k () = V ()/ K Using the Bellman equation above: βe (A,p A,p)[Π k (A, K ) + (1 δ)p ] }{{} = p marginal return on capital The cost of an additional unit of capital today (p) is equated to the marginal return on capital. This marginal return has two pieces: the marginal profits from the capital (Π k (A, K )) and the resale value of undepreciated capital at the future price ((1 δ)p ). Substituting for the future price of capital and iterating forward, we find: p t = β [β(1 δ)] τ E At+τ A t Π K (K t+τ+1, A t+τ+1 ) τ=0

4 No Adjustment Costs Simple model. Easily estimated from the first order conditions with data on K and p. Model does not fit the data well: model implies excessive sensitivity of investment to variations in profitability = investment too volatile. do not match inaction periods at plant level.

5 Convex Adjustment Costs We assume that C(K, A, K) is a strictly increasing, strictly convex function of future capital, K First order condition for optimal investment: C K (K, A, K) + p = βe (A,p A,p)V K (A, K, p ). The left side of this condition is a measure of the marginal cost of capital accumulation and includes the direct cost of new capital as well as the marginal adjustment cost. The right side of this expression measures the expected marginal gains of more capital through the derivative of the value function. marginal Q : q = βe (A,p A,p)V K (A, K, p ). First order condition for optimal investment: C K (K, A, K)+p = βe (A,p A,p){Π K (K, A )+p (1 δ) C K (K, A, K )}.

6 Empirical Strategy Two approaches: Q theory Approximate V K (A, K, p ): Q models. use information in financial markets to approximate V K (A, K, p ) by Ṽ /K: average Q. Old tradition, dates back to Keynes (1936), Tobin (1969). Requires a number of assumptions when replacing marginal by average q (Hayashi (1982)): product and factor markets are competitive. production and adjustment cost technologies are linear homogeneous. capital is homogeneous. investment decisions are separated from other real and financial decisions. advantage: financial market data already incorporates expectation of future variables. No need to specify the information set. Euler equation.

7 Q Theory: Models Following Hayashi (1982): Profit function is proportional to K. Cost of adjustment is quadratic. Optimal Program: V (A, K) = max AK γ ( K ) 2 (1 δ)k K K 2 K p(k (1 δ)k) + βe A AV (A, K ) Suppose that A = ρa + ɛ where ρ < 1, ε white noise investment rate i I/K: i = 1 γ (βe A AV K (A, K ) p). E A AV K (A, K ): marginal Q

8 Solving the Dynamic Programming Model Suppose V (A, K) = φ(a)k. Expected marginal Q is a function of A: E A AV K (A, K ) = E A Aφ(A ) φ(a). Note that in this case V K (A, K ) = V (A, K )/K Replace marginal Q by its expression to get: i = 1 γ (β φ(a) p) z(a). The investment rate is independent of the current level of capital stock. Replace investment policy function into the original value function: φ(a)k = AK γ 2 (z(a))2 K pz(a)k +β φ(a)k[(1 δ)+z(a)] Note that our initial guess was right. The value is expressed as a the product of K and a function of the technology shock.

9 Q Theory: Evidence Empirical specification: (I/K) it = a i0 + β γ E q it+1 + a 2 (X it /K it ) + υ it X it : other conditioning variables (example: financial variables (profits, cash-flow). ). In theory a 2 = 0. Very limited success. Hayashi (1982): ˆγ 25, Gilchrist and Himmelberg (1995) ˆγ 30. Implausibly large cost of adjustment. Capital would react much too slowly to technology shocks. a 2 0. Profits, cash-flows predict investment rate. Imperfect capital markets? Serial correlation in the residual. Lagged average Q or investment rates are also significant. Difficulty in measuring average Q. Instability of financial markets. Bubbles.

10 Euler Equation Approach Same approach as in consumption literature. Euler equation: i t = 1 [β[e t (π K (A t+1, K t+1 ) + p t+1 (1 δ) + γ ] γ 2 i2 t+1 + γ(1 δ)i t+1 ] p t. Define ε t+1 as: ε t+1 = i t 1 γ [β[(π K(A t+1, K t+1 ) + p t+1 (1 δ) + γ ] 2 i2 t+1 + γ(1 δ)i t+1 )] p t. Moment restrictions: Estimation using GMM. Eε t+1 = 0 and Eε t+1.z t = 0

11 Borrowing Restrictions Often motivated by failure of Q theory and significance of financial variables. Model: V (A, K) = max K Γ(A,K) AKα γ ( K ) 2 (1 δ)k K 2 K p(k (1 δ)k) + βe A AV (A, K ) Γ(A, K) restricts the choice set for the future capital stock. example: investment financed out of current profits: K (1 δ)k AK α Γ(A, K) = [0, AK α + (1 δ)k] unresolved issues have limited research in this area: What are the Γ(A, K) functions suggested by theory? For what Γ(A, K) functions is there a wedge between average and marginal Q?

12 Nonconvex Adjustment Costs Frequent periods of inactivity at plant level. Bursts of investment in some periods. Cost only incurred if investment, independent of magnitude.

13 Descriptive Statistics, LRD, US Variable LRD Average Investment Rate 12.2% Inaction Rate: Investment 8.1% Fraction of Observations with Negative Investment 10.4% Spike Rate: Positive Investment 18% Spike Rate: Negative Investment 1.4% inaction: investment rate less than spike: investment rate > 20%.

14 Model with Nonconvex Adjustment Costs Value of the firm: V (A, K, p) with technology shock A. stock of capital K. price p. Discrete action either adjust stock of capital. remain inactive. V (A, K, p) = max{v i (A, K, p), V a (A, K, p)} for all (A, K, p) Value of inaction: V i (A, K, p) = Π(A, K) + βe A,p A,pV (A, K(1 δ), p ) Value of adjustment: V a (A, K, p) = max K Π(A, K)λ F K p(k (1 δ)k)+βe A,p A,pV (A, K, p ). 1 λ : loss of production. F : fixed cost (adjusted for size of firm).

15 Remarks Two types of costs have implications for cyclical properties of investment: if λ < 1, more expensive to invest in periods of high profitability. if F 0, invest when profitability is high. Investment can be either pro or countercyclical. Depends also on persistence in profitability shocks A.

16 Machine Replacement Example Cooper and Haltiwanger (1993). No choice of the size of the machine. Old machines are scrapped and replaced by new ones at net price p. V (A, K) = max{v i (A, K), V a (A, K)} with: V i (A, K) = Π(A, K) + βe A AV (A, K(1 δ)) V a (A, K) = Π(A, 1)λ p + βe A AV (A, (1 δ)). Solution: there is a critical size of the capital stock (K (A)) such that machine replacement occurs if and only if K < K (A). To see why this policy is optimal, note that by our timing assumption, V a (A, K) is in fact independent of K. Clearly V i (A, K) is increasing in K. Thus there is a unique crossing of these two functions at K (A). Replacement cycle.

17 Aggregate Implications of Machine Replacement Distribution of capital across plants in period t: f t (K) Shock A t = a t ε t a t : aggregate shock. ε t : plant specific shock, iid. Policy function: z(a t, ε t, K t ) z(a t, ε t, K t ) = 1: action. z(a t, ε t, K t ) = 0: inaction. Define: H(a t, K) = ε z(a t, ε t, K)dG t (ε) where G t (ε) is the cumulative distribution of plant specific shocks. H(a t, K) is the hazard of replacement, the probability of adjustment, given a t and K t, but unconditional on ε t. H(a t, K t ) [0, 1]

18 Aggregate Implications of Machine Replacement Aggregate Investment: I(a t ; f t (K)) = K H(a t, K)f t (K). Thus total investment reflects the interaction between the average adjustment hazard and the cross sectional distribution of capital holdings. The evolution of the cross sectional distribution of capital is given by: g t+1 ((1 δ)k) = (1 H(a t, K))g t (K)

19 Irreversibility Some degree of irreversibility in investment. Buying price > selling price. Frictions. Specifications so far do not distinguish between buying and selling. Value of buying: V (A, K) = max{v b (A, K), V s (A, K), V }{{}}{{} i (A, K) }{{} Buy Sell Inaction V b (A, K) = max Π(A, K) I + βe A AV (A, K(1 δ) + I) I Value of selling: V s (A, K) = max R Π(A, K) + p sr + βe A AV (A, K(1 δ) R) Value of inaction: V i (A, K) = Π(A, K) + βe A AV (A, K(1 δ)) if p s < 1, inaction may be optimal.

20 Example: Rich Model of Adjustment Costs Cooper and Haltiwanger (2000) Program of the plant: V (A, K) = max{v b (A, K), V s (A, K), V i (A, K)} with V b (A, K) = max I Π(A, K) F K I γ 2 [I/K]2 K +βe A AV (A, K(1 δ) + I), V s (A, K) = max R Π(A, K) + p s R F K γ 2 [R/K]2 K +βe A AV (A, K(1 δ) R) V i (A, K) = Π(A, K) + βe A AV (A, K(1 δ)). Estimate three parameters Θ = {F, γ, p s }, fix β = 0.95 and δ = Profit function: Π(A, K) = AK θ, θ = 0.5.

21 Example: Rich Model of Adjustment Costs Estimation: Indirect inference. Auxiliary model: i it = α i + ψ 0 + ψ 1 a it + ψ 2 (a it ) 2 + u it where a it : log of profitability shock. α i : fixed effect. Results: Spec. Structural parameters parm. est. Estimates (s.e.) for aux. model γ F p s ψ 0 ψ 1 ψ 2 LRD all.043 (0.002).00039(.00005).967(.00112) F only (.00002) γ only.125(.0001) p s only (.0003) Both convex and non convex costs of adjustment. Fixed cost at about 0.04% of capital = large inaction.