Lecture Notes 1: Solow Growth Model Zhiwei Xu (xuzhiwei@sjtu.edu.cn) Solow model (Solow, 1959) is the starting point of the most dynamic macroeconomic theories. It introduces dynamics and transitions into aggregate economy. We rst list several key assumptions regarding the model s setup. To be consistent with future lectures, we assume that the time is discrete. The results from continuous-time model is similar. 1 Model Setup 1. Production Technology The rm produces nal goods by combining capital K t and labor L t : The production function is assumed to take the form Y t = F (K t ; A t L t ) ; (1) where F (:; :) is concave in K and L; A t is the neutral technology process. For simplicity, we assume that F (:; :) follows Cobb-Douglas form: K t (A t L t ) 1, where 0 < < 1: The concavity of F (K; L) ensures the uniqueness of balance growth path and the global convergency. 2. Labor Supply The labor input L t equals the population (full employment) and grows at a constant rate n > 0; that is 3. Technology Progress The neutral technology A t grows at a constant rate g > 0 : L t = (1 + n) L t 1 : (2) A t = (1 + g) A t 1 : (3) Note that Solow model introduces exogeneous growth by assuming the population and the technology are growing over time. 4. Consumption and Saving Households are representative. The saving, S t ; and consumption, C t ; decisions are static: S t = sy t ; (4) C t = (1 s) Y t ; (5)
2 where s is the constant saving rate. Note that the above decisions may not necessarily optimal for the household. Hence, Solow model does not consider the micro-level optimization decisions. This assumption will be relaxed in the Ramsey model and OLG model. 5. Capital Accumulation To make the model be dynamic, we need to introduce the dynamic process of capital accumulation: K t+1 = (1 ) K t + I t ; (6) where K t is the beginning-of-period capital stock. That is, K t is predetermined at period t 1, therefore K t is a state variable. 6. Capital Market Clearing I t = S t : (7) In the closed economy, aggregate investment is always equal to aggregate saving. However, in the open economy, two variables may not necessarily be the same. The di erence of saving and investment is the current account balance. 2 Dynamics 2.1 Dynamic System The full system is given by equations (1) to (7). From (4) to (7), we can simplify the system into one di erence equation K t+1 = (1 ) K t + s (K t ) (A t L t ) 1 : (8) To remove the non-stationary trend in the last equation, we need to detrend each variable. Divide the both sides of (8) by A t L t, and de ne new variable x t Rearranging the terms yields Xt A tl t ; we can rewrite (8) as k t+1 (1 + g) (1 + n) = (1 ) k t + sk t : (9) (1 + g) (1 + n) (k t+1 k t ) = sf (k t ) ( + g + n + g n) k t : (10) where f (k) = k : For simplicity, we ignore the terms with coe cient g n: We then obtain (1 + g + n) (k t+1 k t ) = sk t ( + g + n) k t : (11) The above di erence equation determines the process of e ective capital stock k t or K t A tl t. Before we discuss the dynamics implied by (11), we rst look at the growth in the long run, where k t = 0 or k t+1 = k t :
3 2.2 Balance growth path (BGP): BGP is the path on which all the variables of the model grow at a constant rate n + g; and the detrended variables (e.g. k t ) stay constant. On the balance growth path, k t is constant: k t = k ; for all t: It can be shown that under the Cobb-Dougals technology, there exists a unique k to solve the equation or s (k ) ( + g + n) k = 0; (12) k = s + g + n On the balance growth path, the captial stock, K t ; is given by 1 1 : (13) K B t = k A t L t : (14) Plugging K B t into (1) and (5), balance growth output, consumption and saving, are given by Y B t = (k ) A t L t ; (15) S B t = s (k ) A t L t ; (16) C B t = (1 s) (k ) A t L t : (17) The BGP indicates that the long-run growth does not depend on the capital accumulation. The growth rate is determined by the growth of technology and population. 2.3 Dynamics of k t The concavity of production technology F (:) ensures the global convercency of k t. To proof this, suppose k t < k ; the right hand side of (11) implies k t+1 k t > 0: Thus k t will monotonically increase until k t = k : For the case of k t > k ; similarly we have k t+1 < k t ; and k t will monotonically decrease until k t = k : Therefore, for any initial value of k t ; it will eventually converge to the steady-state
4 k : The phase digram below illustrates the dynamics of k t Figure 1. The dynamics of k t 2.4 Transition path of k t and y t To see how the capital k t transits from the initial state to the steady state k ; we do the following simulation. First, we set the values of deep parameters as: s = 0:4; n = 2%; g = 5%; = 0:5; = 0:1: We consider two scenarios: (1) k 0 = 0:8k ; (2) k 0 = 1:2k : The gure below shows that capital k t monotonically converge to its steady state k :
5 6 5.5 5 4.5 4 0 140 k =0.8k * 0 7 6.5 6 5.5 0 140 k =1.2k * 0 Figure 2. Transition paths of capital for di erent initial value (k 0 = 0:8k ; 1:2k ) 2.5 2.4 2.3 2.2 2.1 Output y t 0.01 0.005 0 0.005 0.01 Growth of y t Figure 3. Transition path of ouput (k 0 = 0:8k ) Suppose that poor country and rich country have the same economic structure (same parameter values), the above gure shows that poor country (low initial value) grows fast at the beginning, afterward the growth rate declines. Eventually the poor country catches up with the rich country, and converges to the unique steady state. The aggregate output (non-detrended) Y t will be on the balance growth path.
6 2.4.1 The e ect of saving rate (s) on the long-run growth To see how the saving rate a ects the steady state k, we take logs of both sides of (13) and take derivative w.r.t s; For the steady-state output, we have @ log k @ log s = 1 1 : (18) @ log y @ log s = 1 : (19) Higher capital share, larger e ect of saving rate on output. As saving rate has positive e ect on capital k, more capital intensity in the economy implies that raising saving rate will increase the output more. As the consumption is given by c = (1 @ log c @ log s = s) (k ) ; we further have s 1 s + 1 : (20) Golden Rule: the consumption c achieves the optimal level, which implies that s = : @ log c @ log s = 0 or The following gure plots the transition path from the old steady state (s = 0:4) to the new steady state (s = 0:5). 8 6 Capital k t 3 2.5 2 Output y t 0.02 0.01 0 0.01 Growth of y t 1.5 1 Consumption c t Figure 4. Transition path to the high saving rate steady state
7 2.5 Short-run dynamics The above discussions are related to the long-run dynamics. I.e., we study the dynamic path of the economy transits from the initial state to the steady state. Now, we discuss the short-run dynamics around the steady state. We rst linearize the equation (11) around the steady state k : (k t+1 k ) (k t k ) = s (k ) 1 ( + g + n) (k t k ) 1 + g + n (1 ) ( + g + n) = (k t k ) : (21) 1 + g + n The second line is due to the relationship s (k ) = ( + g + n) k : De ne the convergent speed as (1 )(+g+n) 1+g+n : Equation (21) implies k t k = (k 0 k ) (1 ) t : (22) Last equation describes the evolution of capital around the steady state. Regarding the output, similarly we have y t y = (y 0 y ) (1 ) t : (23) Given our previous calibration, is equal to 0:08, implying that it takes approximate 8 years ln 0:5 (' ln(1 ) ) to get the halfway of the balance-growth-path value. 3 Extensions 3.1 Natural Resources and Land We extend the benchmark Solow model by augmenting production function with two additional inputs: natural resources R t and land T t. Speci cally, we assume that the production function is Y t = K t R t T t (A tl t ) 1 ; (24) where > 0; > 0; > 0; + + < 1: The process of natural resource R t is assumed to follow R t = (1 b) R t 1 ; (25) and the land is assumed to be constant, i.e., T t = 0: On the balance growth path, it can be shown that capital and output have the same growth rate. This is because (8) implies On BGP, Y t =K t is constant or the growth rates satisfy g Y = g K : K t+1 K t = (1 ) + s Y t K t : (26)
8 or To compute the growth rate of output on BGP, we take logs of both sides of (24) and get g Y = g Y b + (1 ) (g + n) ; (27) g Y = 1 b + 1 (g + n) : (28) 1 De ne ~g Y = g + n the growth rate where capital and labor are the only inputs. Since 1 1 < 1 and 1 b < 0; the growth rate of output in the model with natural resource g Y is less than that in baseline model ~g Y : That is, the natural resources (limited supply) reduce the BGP growth rate. The gap between g Y and ~g Y is called the "growth drag". 3.2 Growth Trap: Production with Fixed Cost So far there is no friction in the Solow model. We now introduce one friction xed production cost. You may see that with this minor extension, the dynamics of the model will change a lot. Suppose that each period, in order to make production the producer has to pay a xed amount of cost : In reality, this cost may be due to rental rate of land, loan payment, etc. The production function for y t (Y t =A t =L t ) takes the form of y t = max fkt ; 0g : (29) The di erence equation (11) now becomes (1 + g + n) (k t+1 k t ) = s max fkt ; 0g ( + g + n) k t : (30) It can be shown that there exist two k such that s max fk ; 0g = ( + g + n) k: (31) Denote them as k and k ; where k < k : As shown in Figure 5, k is stable in the sense that for any capital k t > k ; k t would eventually converge to k : While k is not stable because for any k t < k ; k t would converge to the zero point.
9 Figure 5. Growth Trap 3.3 Application: Middel Income Trap The middle income trap is an economic development situation, where a country which attains a certain income (due to given advantages) will get stuck at that level. production cost may capture the main idea. Consider two dynamic systems Solow model with xed (1 + g + n) (k t+1 k t ) = sz l kt ( + g + n) k t ; (32) n o (1 + g + n) (k t+1 k t ) = s max z h kt ; 0 ( + g + n) k t ; (33) where z l < z h : The economy starts with a low level K will trap at the K mid equilibrium. To pull out the economy from middle income trap, the government need to either reduce the friction or raise the aggregate capital K: Figure 6 provides a graphical illustration.
Figure 6. Middle Income Trap 10