Lecture 2 The Centralized Economy

Lecture 2 The Centralized Economy Economics 5118 Macroeconomic Theory Kam Yu Winter 2013

Outline 1 Introduction 2 The Basic DGE Closed Economy 3 Golden Rule Solution 4 Optimal Solution The Euler Equation Interpretation Static Equilibrium Dynamics Algebraic Analysis 5 Real Business Cycle Dynamics Technology Shocks Golden Rule Revisited 6 Labour in the Basic Model 7 Investment q-theory Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 2 / 45

Introduction The Production Economy Recall a static production economy E = {(U i, e i, θ ij, Y j ) i I, j J } There are n goods and services. Each household i I has continuous, strongly increasing, and strictly quasi-concave utility function U i : R n + R + and is endowed with e i R n +. Each competitive firm j J has a production set Y j that is compact and strongly convex. θ ij is the share of household i in firm j. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 3 / 45

Introduction Really Nice Results from Microeconomic Theory 1 A Walrasian equilibrium exists: there is a price vector p such that d i (p, m i (p )) = y j (p ) + e i. i I j J i I That is, all n markets of goods and services clear. 2 FWTE: The Walrasian equilibrium allocation is Pareto efficient. 3 SWTE: Any desirable Pareto efficient allocation (x, y) can be achieved as a Walrasian equilibrium allocation after a suitable income transfer program between households. 4 With full information on preferences and technology, communism and capitalism achieve the same outcome in a static economy. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 4 / 45

Introduction To Make Things Really Simple... There are two goods, n = 2, called capital k and output y. The households only consume one good, effectively we can consider them as one big household ( I = 1) One aggregate competitive firm ( J = 1). Of course the ownership share become θ 11 = 1. This is a bit too simple. So instead of static equilibrium, we study this economy over time, t = 1, 2,... Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 5 / 45

The Basic DGE Closed Economy The Ramsey Model In period t, the endowment is the capital stock k t. The firm produces output y t using capital as input: y t = F (k t ) (2.3) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 6 / 45

The Basic DGE Closed Economy The Ramsey Model In period t, the endowment is the capital stock k t. The firm produces output y t using capital as input: y t = F (k t ) (2.3) Output y t is divided into two parts, consumption c t and investment i t : y t = c t + i t. (2.1) This is called the national income identity, or the resource constraint. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 6 / 45

The Basic DGE Closed Economy The Ramsey Model In period t, the endowment is the capital stock k t. The firm produces output y t using capital as input: y t = F (k t ) (2.3) Output y t is divided into two parts, consumption c t and investment i t : y t = c t + i t. (2.1) This is called the national income identity, or the resource constraint. Investment i t is saved as capital for next period. In production, the firm consumes only part of the capital stock, δk t, where δ is called the depreciation rate. Capital stock in period t + 1 is therefore k t+1 = k t+1 k t = i t δk t. (2.2) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 6 / 45

The Basic DGE Closed Economy Dynamic Resource Constraint The last three equations gives F (k t ) = c t + k t+1 + δk t. (2.4) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 7 / 45

The Basic DGE Closed Economy Dynamic Resource Constraint The last three equations gives F (k t ) = c t + k t+1 + δk t. (2.4) Like the static model, our objective is to maximize utility derived from consumption, not output (we are not communists!) But we have a problem. Should we maximize 1 utility in each period, treating every period as equally important, (golden rule) or, 2 the present value of total utility of all present and future periods, using an appropriate discount factor? (optimal solution) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 7 / 45

Golden Rule Solution Golden Rule The Steady State The dynamic resource constraint (2.4) can be written as c t = F (k t ) k t+1 + (1 δ)k t. (2.5) The steady state is attained when all variable are the same in all subsequent periods, i.e., c t = c and k t = k, t = 1, 2,.... Then (2.5) becomes c = F (k) δk. (2.6) The necessary condition for maximization is c k = F (k) δ = 0, (2.7) which means that marginal product is equal to the depreciation rate when consumption is maximized in the steady state. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 8 / 45

Golden Rule Solution Golden Rule Steady State Figure 2.1. The marginal product of capital. k # k y c # +δ k # F(k) δ k # max c = c # = F(k # ) δ k # δ k δ Figure 2.2. Total output, consumption, and replacement investment. k # k c t + k t + 1 Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 9 / 45

Optimal Solution The Optimization Problem In the optimal solution the present value of current (period t) and future (t + s) utility is maximized: max c t+s,k t+s+1 β s U(c t+s ) s=0 subject to F (k t+s ) = c t+s + k t+s+1 (1 δ)k t+s, where β = 1/(1 + θ) and θ > 0 is called the social discount rate. The Lagrangian is L t = s=0 { β s U(c t+s ) } + λ t+s [F (k t+s ) c t+s k t+s+1 + (1 δ)k t+s ], (2.8) where λ t+s is the Lagrange multiplier in period t + s. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 10 / 45

Optimal Solution Necessary Conditions The first-order conditions are L t c t+s = β s U (c t+s ) λ t+s = 0, s 0, (2.9) L t k t+s = λ t+s [F (k t+s ) + 1 δ] λ t+s 1 = 0, s 1, with the resource constraint F (k t+s ) = c t+s + k t+s+1 (1 δ)k t+s, s 0, and the transversality condition (2.10) lim s βs U (c t+s )k t+s+1 = 0. (2.11) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 11 / 45

Optimal Solution The Euler Equation The Euler Equation Eliminating λ t+s and λ t+s 1 in (2.10) using (2.9) gives β s U (c t+s )[F (k t+s ) + 1 δ] = β s 1 U (c t+s 1 ), s 0. For s = 1 this can be written as β U (c t+1 ) U (c t ) [ F (k t+1 ) + 1 δ ] = 1. (2.12) This is called the Euler equation, which is the corner stone of dynamic optimization problems in consumption. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 12 / 45

Optimal Solution Interpretation Interpretation of the Euler Equation The Euler equation reflects the intertemporal substitution of consumption between two consecutive periods. Consider periods t and t + 1: V t = U(c t ) + βu(c t+1 ). Using the implicit function theorem, the slope of the indifference curve in the (c t, c t+1 ) space is called the marginal rate of time preference: dc t+1 dc t = U (c t ) βu (c t+1 ). (2.13) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 13 / 45

Optimal Solution Interpretation Interpretation continued The budget constraint in period t and t + 1 can be written respectively as k t+1 = F (k t ) + (1 δ)k t c t, c t+1 = F (k t+1 ) k t+2 + (1 δ)k t+1. Using the chain rule to differentiate c t+1 with respect to c t, we get dc t+1 dc t = [F (k t+1 ) + 1 δ]. (2.14) This is the slope of the intertemporal production possibility frontier (IPPF). Equating (2.13) and (2.14) gives the Euler equation. That is, at the optimal point (ct, ct+1 ), the indifference curve of the household is tangent to the IPPF. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 14 / 45

Optimal Solution Interpretation aphical Representation of the Solution Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 15 / 45 mal Graphical Solution Interpretation c t + 1 max c t+1 c * t + 1 V t = U(c t ) + βu(c t + 1 ) c* t max c t 1 + r t + 1 Figure 2.4. A graphical solution based on the IPPF.

Optimal Solution Static Equilibrium Steady-State Solution In the steady state (long-run), c t = c and k t = k for all t. The Euler equation becomes or, with β = 1/(1 + θ), β U (c ) [ F U (c (k ) + 1 δ ] = 1, ) From the resource constraint we have F (k ) = 1/β + δ 1 = δ + θ. (2.21) c = F (k ) δk. (2.22) Comparing with the golden rule solution, where F (k # ) = δ, the long-run capital stock is at a lower level. That is, c < c # and k < k #. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 16 / 45

n is therefore Kam Yu (LU) different from Lecture that 2 The for Centralized the golden Economy rule, where F Winter (k) 2013 = 17 / 45 Optimal Solution Static Equilibrium k * k # k Comparing Figure Golden 2.5. Optimal Rulelong-run and Optimal capital. Solution y F(k) c # + δ k # c * + δ k * δ + θ δk δ k * δ k * k # Figure 2.6. Optimal long-run consumption. k

Optimal Solution Dynamics Linear Approximation So far we have established two dynamic relations between two consecutive periods, the Euler equation and the resource constraint: β U (c t+1 ) U (c t ) [ F (k t+1 ) + 1 δ ] = 1, k t+1 = F (k t ) δk t c t. (2.17) The relation between c t+1 and c t can be better seen by taking a first-order Taylor approximation of U (c t+1 ) about c t : U (c t+1 ) U (c t ) + U (c t ) c t+1. The Euler equation becomes, with U (c t )/U (c t ) 0, [ 1 c t+1 = U (c t ) U (c t ) 1 β[f (k t+1 ) + 1 δ] ]. (2.18) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 18 / 45

Optimal Solution Dynamics Dynamics of Consumption and Capital In the steady state, F (k t+1 ) = F (k ) = δ + θ and so c = U (c [ ] ) 1 U (c 1 = 0. ) β[δ + θ + 1 δ] Two conclusions: 1 When k > k, F (k) < F (k ) and by (2.18) c < 0. 2 When k < k, F (k) > F (k ) and by (2.18) c > 0. From the resource constraint (2.17), 1 When c t > F (k t ) δk t, then k < 0 2 When c t < F (k t ) δk t, then k > 0 Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 19 / 45

Optimal Solution Dynamics Phase Diagrams 2.4. Optimal 2. The Centralized Solution Economy c t + 1 = 0 k t + 1 < 0 c > F(k) δ k c t + k t + 1 c t + 1 > 0 c t + 1 < 0 c t + k t + 1 k t + 1 > 0 c < F(k) δ k k = 0 c = F(k) δ k Figure 2.8. Consumption dynamics. k * k t Figure 2.9. Capital dynamics. k that both equations are nonlinear. We therefore c consider t + k t + 1 S e., a solution that holds in the neighborhood of equilibugh linearizing the Euler equation by taking a Taylor c # series 1) about c t. This gives c * U (c t+1 ) U (c t ) + c t+1 U B (c t ). Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 20 / 45 A

ed by the economy, i.e., the parameters of the model, an Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 21 / 45 The Saddle Path Optimal Solution Dynamics Figure 2.9. Capital dynamics. k c t + k t + 1 S c # A * c B S k = 0 k * k # k Figure 2.10. Phase diagram.

Optimal Solution Algebraic Analysis Linear Approximation of the Euler Equation The Euler equation is a non-linear equation in c t+1, c t, and k t+1. For an algebraic solution we need to linearize it. So let f (x) = U (c t+1 ) [ F U (k t+1 ) + 1 δ ], (c t ) where x = [c t+1 c t k t+1 ] T. The first-order Taylor approximation of f about x = [c c k ] T is f (x) f (x ) + f (x ) T (x x ) = U (c ) U (c ) [F (k ) + 1 δ] + U (c ) U (c ) [F (k ) + 1 δ](c t+1 c ) U (c ) [U (c )] 2 U (c )[F (k ) + 1 δ](c t c ) + U (c ) U (c ) F (k )(k t+1 k ) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 22 / 45

Optimal Solution Algebraic Analysis f (x) F (k ) + 1 δ + U (c ) U (c ) [F (k ) + 1 δ](c t+1 c t ) + F (k )(k t+1 k ). It follows that the first-order Taylor approximation of the Euler equation is [ β F (k ) + 1 δ + U (c ) U (c ) [F (k ) + 1 δ] c t+1 ] + F (k )(k t+1 k ) 1. Using F (k ) = δ + θ (2.21) and rearranging gives (c t+1 c ) = (c t c ) F (k )U (c ) (1 + θ)u (c ) (k t+1 k ). (2.23) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 23 / 45

Optimal Solution Algebraic Analysis Linear Approximation of the Resource Constraint Recall the resource constraint k t+1 = F (k t ) δk t c t. (2.17) Use a first-order Taylor approximation for F (k t ), (2.17) becomes k t+1 k t F (k ) + F (k )(k t k ) δk t c t. Using (2.21), c = F (k ) δk (2.22), and rearranging gives k t+1 k F (k ) + (δ + θ)(k t k ) δk t c t k + k t = F (k ) δk + δk t + θk t θk δk t c t k + k t = (c t c ) + (1 + θ)(k t k ). (2.24) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 24 / 45

Optimal Solution Algebraic Analysis Back to the Euler Equation Substitute (2.24) into (2.23), we have (c t+1 c ) = (c t c ) F (k )U (c ) (1 + θ)u (c ) [(1 + θ)(k t k ) (c t c )] or (c t+1 c ) = [ 1 + F (k )U (c ] ) (1 + θ)u (c (c t c ) ) F (k )U (c ) U (c (k t k ) (2.23a) ) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 25 / 45

Optimal Solution Algebraic Analysis Linear Dynamical System The linearized Euler equation (2.23a) and resource constraint (2.24) can be expressed in matrix form as [ ct+1 c ] ] [1 + F U k t+1 k = (1+θ)U F U [ct U c ] 1 1 + θ k t k. This is a two-dimensional linear dynamical system x t+1 = Ax t. with x t = (c t c, k t k ) T. The system converges to the steady-state if the absolute values of the two eigenvalues of the matrix A are both less than 1. See Devaney (2003, p. 173 179) for details. In particular, the optimal solution will give the saddle path depicted in Figure 2.10. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 26 / 45

Real Business Cycle Dynamics The Business Cycle An economy is constantly impacted by shocks. Shocks can be temporary or permanent, anticipated or unanticipated. Real business cycle theory focuses on technology shocks (innovations). After a shock the economy follows the saddle path and converges to the new steady-state equilibrium. During the adjustment periods the optimality assumption is maintained. Stabilization policy may be useful in the presence of market imperfections. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 27 / 45

Real Business Cycle Dynamics Technology Shocks Adjustment Process for a Positive Technology Shocks For a permanent shock, 1 Marginal product shifts from F 0 to F 1. 2 Optimal long-run capital stock raised from k 0 to k 1. Equilibrium point moved from A to B. 3 At time t = 0, capital is fixed at k0. Consumption jumps from c 0 to c 1, i.e., from point A to C on the new saddle path. 4 Consumption and capital stock converge over time to the new steady-state equilibrium at point B. 5 Results: consumption and capital both increase. For a temporary shock, k remains the same. Consumption is temporary adjusted to absorb the shock. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 28 / 45

Real Business Cycle Dynamics Technology Shocks Effects of a Positive Technology Shock A F 1 ' F 0 ' k* k * 0 1 k Figure 2. The 2.11. Centralized The effect Economy on capital of a positive technology sh F'(k) c t + k t + 1 c = 0 B δ + θ A B F 1 ' F 0 ' ' c 1 ' c 0 C A k = 0 * k 1 * k k 0 k* k * 0 1 k The effect on capital of a positive Figure 2.12. technology The effect shock. on consumption of a positive technology + k t + 1 2.5.2 Permanent Technology Shocks A positive technology shock increases the marginal product of ca Kam Yu (LU) depicted Lecture in figure 2 The 2.11 Centralized as Economy a shift from F 0 to FWinter 1.As 2013 δ + θ 29 is / 45 un

Real Business Cycle Dynamics Dynamics of the Golden Rule c t + k t + 1 Golden Rule Revisited We looked at the steady-state equilibrium of the golden rule: F (k # ) = δ, c > F(k) δ k k t + 1 > 0 c < F(k) δ k c # = F (k # ) δk #. What is the dynamics from the initial stage to the steady state? Figure 2.9. Capital dynamics. k = 0 c = F(k) δ k k Take the optimal solution model and set θ 0. Then β 1 in the Euler equation. Resource constraint unchanged. Point B approaches A. c t + k t + 1 c # * c S B S A k * k # k Figure 2.10. Phase diagram. k = 0 is determined by the economy, i.e., the parameters of the model, and c Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 30 / 45

Real Business Cycle Dynamics Golden Rule Revisited Stability of the Golden Rule The book says the golden rule has a unstable equilibrium. But... Golden rule is a special case of optimal solution, with social discount rate θ = 0. There is no inherent instability in the golden rule, unless we insist on setting c # = F (k # ) δk # in all time. Shocks can be accommodated by adjusting consumption to be on the saddle path. The true value of θ is an empirical question. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 31 / 45

Labour in the Basic Model Work and Leisure Assumptions: Households choose between labour time, n t and leisure time l t. Total time normalized to one: n t + l t = 1. The utility function U(c t, l t ) is increasing and concave, with U c > 0, U l > 0, U cc 0, U ll 0, U cl = 0. The production function F (k t, n t ) satisfies the Inada conditions. That is, F k > 0, F kk 0, F n > 0, F nn 0, F kn 0, lim k F k = 0, lim k 0 F k =, lim n F n = 0, lim n 0 F n =. Resource constraint: F (k t, n t ) = c t + k t+1 (1 δ)k t. Labour constraint: n t + l t = 1. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 32 / 45

Labour in the Basic Model Optimization The Lagrangian is L t = s=0 { β s U(c t+s, l t+s ) + λ t+s [F (k t+s, n t+s ) c t+s k t+s+1 + (1 δ)k t+s ] } + µ t+s [1 n t+s l t+s ]. The first-order conditions are L t = β s U c,t+s λ t+s = 0, c t+s s 0, (2.25) L t = β s U l,t+s µ t+s = 0, l t+s s 0, (2.26) L t = λ t+s F n,t+s µ t+s = 0, n t+s s 0, (2.27) L t = λ t+s [F k,t+s + 1 δ] λ t+s 1 = 0, k t+s s 1, (2.28) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 33 / 45

Labour in the Basic Model Key Results 1 Euler Equation: β U c,t+1 U c,t [F k,t+1 + 1 δ] = 1. (2.29) 2 Eliminating λ t+s and µ t+s from the first three first-order conditions gives, for s = 0, U l,t = U c,t F n,t. (2.30) This means that if the household provides an extra unit of working time, marginal product is F n,t. Marginal utility gain of consuming this extra output is U c,t F n,t. This should be equal to the marginal utility of leisure. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 34 / 45

Labour in the Basic Model Steady-State Equilibrium 1 Setting U c,t+1 = U c,t = U c and F k,t+1 = F k in the Euler equation gives F k = θ + δ. 2 Consumption c, labour n, and leisure l can be solved by the resource constraint, labour constraint, and (2.30). 3 The short-run solutions for c t and k t are the same as before. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 35 / 45

Labour in the Basic Model Wage Rate and Rate of Return to Capital Assume that technology exhibits constant returns to scale. Then the production function F (k t, n t ) is linearly homogeneous. Applying Euler theorem to F gives F (k t, n t ) = F n,t n t + F k,t k t. (2.31) With two factors of production, this means national product is equal to national incomes. Real wage rate w t and return to capital r t are therefore given by w t = F n,t, r t = F k,t δ. Equation (2.31) can be written as The wage rate is given by F (k t, n t ) = w t n t + (r t + δ)k t. w t = F (k t, n t ) (r t + δ)k t n t. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 36 / 45

Investment Time to Build and Installation Costs of Capital Capital stock takes time to adjust, but so far we have assumed that investment is instantaneous with no installation cost. In practice, capital investment takes time to build and additional resources are needed for design and installation. Capital investment like wind turbines needs time and resources to design and build. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 37 / 45

Investment q-theory Costs of Installation Suppose that installation cost of each unit of capital is 1 2 φi t/k t, where φ 0. The resource constraint becomes (abstract from labour and leisure) ( F (k t ) = c t + 1 + φi ) t i t, φ 0. (2.32) 2k t The Lagrangian of the optimization problem is L t = s=0 { β s U(c t+s ) [ + λ t+s F (k t+s ) c t+s i t+s φi t+s 2 ] 2k t+s } + µ t+s [i t+s k t+s+1 + (1 δ)k t+s ]. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 38 / 45

Investment q-theory First-Order Conditions L t = β s U (c t+s ) λ t+s = 0, s 0, c t+s ( L t = λ t+s 1 + φi ) t+s + µ t+s = 0, s 0, i t+s k t+s [ L t = λ t+s F (k t+s ) + φ ( ) ] 2 it+s k t+s 2 k t+s µ t+s 1 + (1 δ)µ t+s = 0, s 1, Define the Tobin s q in period t as q t = µ t /λ t. Then the second equation can be written as i t+s = 1 φ (q t+s 1)k t+s, s 0. (2.33) Therefore investment takes place only if q t+s > 1. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 39 / 45

Investment q-theory Tobin s q Tobin s q can be interpreted as the ratio of market value of one unit of investment to its cost. (Exercise) Using the first-order conditions and setting s = 1, we get F (k t+1 ) = U (c t ) βu (c t+1 ) q t (1 δ)q t+1 1 2φ (q t+1 1) 2. (2.35) Four equations, (2.32), (2.33), (2.35), and the capital accumulation equation k t+1 = i t + (1 δ)k t (2.32a) can be used to solve for four unknowns, c t, k t, i t, and q t. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 40 / 45

Investment q-theory Long-Run Solution In the steady state, (2.32a) implies that i = δk. Also, (2.33) implies that Therefore or Equation (2.35) gives (exercise) i = 1 (q 1)k. φ 1 (q 1) = δ, φ F (k) = θ + δ + φδ q = 1 + φδ 1. (2.36) ( θ + δ ) θ + δ. (2.37) 2 No installation cost means φ = 0 so that q = 1 and F (k) = θ + δ as before. With φ > 0, the steady-state capital stock given by (2.37) is lower. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 41 / 45

Investment q-theory Short-Run Dynamics To see the dynamics of the model we need to linearize (2.35) about the steady-state solution with the following steps (exercise): 1 Take the first-order Taylor approximation of (q t+1 1) 2 about the steady-state value q. 2 Put the result in (2.35). 3 Let c t+1 = c t in the steady state. 4 Use (2.36) and (2.37) to show that 5 Combining the results to get δ + φδ2 2 = F (k) θq. q t q = β(q t+1 q) + β[f (k t+1 ) F (k)]. (2.38) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 42 / 45

Investment q-theory Solution Equation (2.38) is a first-order difference equation in q t q. The forward solution is q t q = β s [F (k t+s ) F (k)]. s=1 Therefore q t can be seen as the present value of future marginal products of the investment in period t. Eliminating i t using (2.32a) and (2.33) gives or, using q = 1 + φδ, 1 φ (q t 1)k t = k t+1 (1 δ)k t, (q t q + φ)k t = φk t+1. (2.40a) This equation with (2.38) form the dynamic interaction between k t and q t. Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 43 / 45

Investment q-theory Dynamics of q t and k t The linearized versions of equations (2.38) and (2.40a) are (1 β)(q t q) βf (k)(k t k) = β q t+1 + βf (k) k t+1, (2.39) k t (q t q) = φ k t+1. (2.40) In the steady state q t+1 = k t+1 = 0, (2.39) becomes k t k = Since F (k) < 0, k t is negatively related to q t. θ F (k) (q t q) (2.41) Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 44 / 45

Investment q-theory A Permanent Positive Productivity Shock The line q 36 = 0 is from 2. The Centr (2.41). The line k = 0 is from (2.40). The two lines original intersects at A. Productivity shock shifts the q = 0 line. Economy jumps to B on the saddle path and converges to C. q 1 + φδ B A C q = 0 k = 0 Figure 2.13. Phase diagram for q. where k is the steady-state level of k t. Thus, as F kk < 0, in st negatively related to q t through θ Kam Yu (LU) Lecture 2 The Centralized Economy Winter 2013 45 / 45 k