Population growth and technological progress in the optimal growth model

Transcription

1 Quantitative Methods in Economics Econ 600 Fall 2016 Handout # 5 Readings: SLP Sections , pages 55-87; A Ch 6 Population growth and technological progress in the optimal growth model In the optimal growth model there is no growth in the steady state. When there is a steady state that exhibits growth we say that the model reaches a balanced growth path. In what follows, technological progress generates a balanced growth path in our model. Sequece problem SP: The central planner CP weighs the utility of each period by the number of people existing in each period. The SP of the CP is: max β t N t u c t, 1 h t, β 0, 1 {k t+1,h t} N t c t + i t = N t F k t, γ t h t, γ 1, θ 0, 1 ; N t+1 k t+1 = 1 δ N t k t + N t i t N t+1 = ηn t, η 1; h t 0, 1 ; k 0 given, and N 0 = 1, where c t, i t,h t, k t are all per capita variables and N t is the population in period t. The problem can be easily reduced to a per capita problem: max βη t u c t, 1 h t, β 0, 1 {k t+1,h t} c t + i t = F k t, γ t h t, γ 1, θ 0, 1 ; ηk t+1 = 1 δ k t + i t h t 0, 1 ; k 0 given. 1

2 The SP can be recursively formulated as V k, t = max {u c, 1 h + β η V k, t } {k,h} c + i = F k, γ t h, k = 1 δ k + i η η, t = t + 1, h 0, 1. This problem reduces itself to: { V k, t = max u F k, γ t h ηk 1 δ k, 1 h + β η V k, t } {k,h} t = t + 1. Important: Note that it is necessary to describe the law of motion of t since it is not already substituted in the Bellman equation. The first order condition with respect to leisure yields u 2 c, 1 h u 1 c, 1 h = γt F 2 k, γ t h which simply states that the MRS between the leisure and consumption must equal to the relative price of leisure in terms of consumption foregone. Show by combining the foc with respect to k and the Envelope condition that u 1 c, 1 h βu 1 c, 1 h = F 1 k, γ t+1 h + 1 δ, that is, the MRS between present consumption and future consumption equals the relative price of present consumption in terms of future consumption foregone try to argue and understand this interpretation. 2

3 Recall that the resource constraint is c + ηk = F k, γ t h + 1 δ k Thus an optimal sequence {c t, h t, k t+1 } 0 must satisfy 1-2 below. and u 2 c t, 1 h t u 1 c t, 1 h t = γt F 2 kt, γ t h t, 1 u 1 c t, 1 h t βu 1 c t+1, 1 h t+1 = F 1 kt+1, γ t+1 h t δ, 2 c t + ηk t+1 = F k t, γ t h t + 1 δ kt 3 It is important to note that these 3 equations are not suffi cient for the optimality of the sequence {c t, h t, k t+1 } 0. Note that the Euler equation is a second order difference equation in k t, k t+1, and k t+2, we also need a transversality condition to pin down the path for k t Furthermore, this is not a stationary economy. Note that we are dealing with two state variables k, t. This is obvious because our model exhibits balanced growth i.e., growth in the steady state. However, things become easier if we can map this economy into a stationary model because 1 we can get rid of state variable t and 2 we already know how to deal with a stationary economy. The first step is to figure out the growth rate of this economy along the balanced growth path. This is easily done using the resource constraint and the "real world" feature that h, k, i, c exhibit a horizontal trend, y y y once we abstract from fluctuations. In other words, if we filter out the fluctuations, k, c, i, and y recall they are all per-capita variables grow on an average at the same growth rate g. Hence, along the balanced growth path c t = g t c, k t = g t k, it = g t ī Additionally, h t = h. If we substitute these into the resource constraint 3, we get g t c + g t+1 η k = F g t k, γ t h + 1 δ g t k 3

4 Note that the only way this equation can be satisfied if both RHS and the LHS grow at the same rate. But, this is only possible if g = γ. Because then we can have Hence g = γ. g t c + gη k = g t F k, h + 1 δ k This is the reason behind having a labor-augmenting technology. If the technological progress was not labor augmenting we could not have reached this result. In other words, we could not have taken advantage of the fact that F. is homogenous of degree one to factor out g = γ from the production function. Hence, for balanced growth, technological progress must be labor-augmenting. However, the equations that characterize the optimal sequence that is 1-2 still have to hold under balanced growth. With g = γ, then the following must hold u 2 γt c, 1 h u 1 γt c, 1 h = γt F 2 k, h, 4 u 1 γt c, 1 h βu 1 γ t+1 c, 1 h = F 1 k, h + 1 δ, 5 c + γ η k = F k, h + 1 δ k. 6 Equation 6 clearly holds. However, nothing guarantees that equations 4 and 5 also hold. See that u 2 γt c, 1 h γ t u 1 γt c, 1 h }{{} Not necessarily constant u 1 γt c, 1 h βu 1 γ t+1 c, 1 h }{{} Not necessarily constant = F 2 k, h, 7 }{{} constant = F 1 k, h + 1 δ }{{} constant Then, what conditions on the utility function guarantee that 7 and 8 also hold? 8 4

5 Note that with technological progress the MPL or you can say the real wage γ t F 2 k, h increases with time. This has two effects Wealth or income effects: since leisure is a normal good then consumption of leisure rises, that is, hours worked fall Substitution effect: Relative price of leisure rises, consumption of leisure falls, hours worked rise Recall that along the balanced growth path h t = h. For balanced growth, the wealth effect and substitution effects of the increase in real wage that takes place in each period must cancel or offset each other. For this to occur, a utility function of the following form is necessary and suffi cient: u c, h = { c 1 σ φ 1 h if σ 1 1 σ ln c + φ 1 h if σ = 1. To check this simply use this utility function c1 σ φ 1 h to verify if 1 σ 7 and 8 hold: u 2 γt c, 1 h γ t u 1 γt c, 1 h = cφ 1 h 1 σ φ 1 h = constant, u 1 γt c, 1 h γσ = βu 1 γ t+1 c, 1 h β = constant. check what it looks like for the log case. Hence, under balanced growth path, we have the following system of three equations in 3 unknowns: cφ 1 h = F 2 k, h, 1 σ φ 1 h γ σ = F 1 k, h + 1 δ, β c + γ η k = F k, h + 1 δ k. Once we solve for k, c, and h, we can characterize the economy along the balanced growth path: 5

6 c t = γ t c, k t = γ t k, ht = h, and y t = F k t, γ t h t = F γ t k, γ t h = γ t F k, h = γtȳ. Specific functional forms: a common choice of utility functions that satisfies balanced growth as well as other desired properties is c α 1 h 1 α 1 σ 1 u c, h =. 1 σ Note that according to this function the intratemporal elasticity of substitution between consumption and leisure is unity. If we choose σ = 1 the utility function becomes u c, h = α ln c + 1 α ln 1 h. We can always take any monotonic transformation of u where B = 1 α α. u c, h = ln c + B ln 1 h, Production function: to choose an appropriate production function we know that it must satisfy Constant returns to scale homogeneity of degree one Labor-augmenting technological progress A Cobb-Douglas production function that satisfies both requirements is y t = F k t, γ t h t = A k θ γ t h t 1 θ. There are two advantages of this function. The labor augmenting technological progress becomes neutral and so we need not take a stand with respect to technological progress. This is useful specially if we are not convinced that technological progress is labor augmenting: empirically, labor-augmenting technological progress is a strong assumption.: y t = Aγ }{{ t1 θ k } θ h 1 θ t. Neutral tech. prog. 6

7 The function exhibits constant shares of capital and labor in total output, replicating a stylized fact observed in real data: w t h t y t = 1 θ, r t k t y t = θ, which follows from the profit maximization problem of a firm in the decentralized economy: π t = y t w t h t r t k t, where w t and r t are wage and capital rental rate that firms take as given. 0.1 Mapping into a stationary economy In what follows we will assume that u c, h = ln c + B ln 1 h. F k, γ t h = k θ γ t h t 1 θ Up until now we have been dealing with an economy that is not stationary. Again, note that we have been dealing with two state variables k and t. The first step is to find the growth rate of the economy along the balance growth path. We already did this and found that g = γ. The second step is to redefine variables by detrending them, that is, let ˆk t = k t γ t, ĉ t = c t γ t, î t = i t γ t, ĥ t = h t. We can write the SP as max {ˆk,ĥt} βη t ln γ t ĉ t + B ln 1 ĥt, βη 0, 1 9 t+1 ĉ t + î t = ˆk t θ ĥ1 θ t, ηγˆk t+1 = 1 δ ˆk t + î t k 0 given. 7

8 Note that 9 reduces to 1 max {ˆk,ĥt} βη t βη ln ĉ t + B ln 1 ĥt + ln γ 2, βη 0, 1 t+1 1 βη and we can ignore the constant term for our problem. Thus the above SP problem can be recursively written as ˆk { ˆkθ V = max ln ĥ 1 θ ηγˆk + 1 δ ˆk + B ln 1 ĥ ˆk,h + βηv ˆk } The two focs and Envelope condition yield 1 θ ˆk θ ĥ θ ĉ ηγ ĉ = B 1 ĥ ˆk θ 1 ĥ 1 θ θ + 1 δ = βη ĉ In stead state a variable ˆx = ˆx = x. Then we have B 1 θ 1 θ k h θ =, 10a h ĉ γ = β θ k θ 1 h1 θ + 1 δ, 10b c + ī = k θ h1 θ, 10c ηγ k = 1 δ k + ī 10d Thus, there are 4 equations in 4 unknowns: k, h, c, and ī. 0.2 Calibrating the model To calibrate the model means to give k, h, c, and ī values that fit the real data. Since k, h, c, and ī depend on the parameters of the model see 10a -10d, to calibrate means to first give those parameters some values that are consistent with the data. 1 For a < 1, a t = 1 1 a ta t 1 1 = 1 a 2 ta t = a 1 a 2. 8

9 From the data we compute average values of k y, i y, and c y = 1 i y. Labor share: 1 θ = wh y labor compensation = GDP η = 1+ net growth rate of population γ = 1+ net growth rate of per capita output h = fraction of substitutable time spent working in the market. From 10a: From 10b: From 10c: B c 1 h y = 1 θ h γ = β [θ y ] k + 1 δ ηγ + δ 1 = i k = i y k y Three equations in three unknowns B, β, and δ. 9