Lecture 10 Optimal Growth Endogenous Growth Noah Williams University of Wisconsin - Madison Economics 702 Spring 2018
Optimal Growth Path Recall we assume exogenous growth in population and productivity: A t N t = (1 + n)(1 + g)a t 1 N t 1 (1 + η)a t 1 N t 1 So now let s consider the optimal allocation: max β t U (C t ) {C t,k t+1 } t=0 subject to: C t = F(K t, N t ) K t+1 + (1 δ)k t, t, K 0 given This can be re-written as: max [β(1 + η) 1 σ ] t c1 σ t { c t, k t+1 } 1 σ t=0 subject to: c t = k α t (1 + η) k t+1 + (1 δ) k t
Characterizing the Optimal Growth Path Form the Lagrangian with β = β(1 + η) 1 σ : L = max ( { c t, k t+1 } t=0 ) β t c1 σ t 1 σ + λ t[ k t α (1 + η) k t+1 + (1 δ) k t c t ] First order conditions for any c t, and for k t+1, t > 0: (1 + η)λ t + λ t+1 [α k α 1 t+1 These imply the Euler equation: σ β t c t = λ t + 1 δ] = 0. (1 + η) c σ t = σ α 1 β c t+1 [α k t+1 + 1 δ]
Optimal Balanced Growth Look for a steady state of the transformed optimal allocation. (1 + η)( c ) σ = β( c ) σ [α( k ) α 1 + 1 δ] (1 + η) = β[α( k ) α 1 + 1 δ] Or, recalling that β = 1/(1 + θ): f ( k ) = = 1 + η β(1 + η) 1 σ + δ 1 (1 + θ) (1 + η) σ + δ 1 δ + θ + ση δ + θ + σ(n + g)
Optimal Balanced Growth Therefore we have capital per unit of effective labor in the balanced growth path: k = ( α δ 1 + (1 + θ)(1 + η) σ ( ) 1 α 1 α δ + θ + σ(n + g) ) 1 1 α This generalizes the solution we had for the optimal allocation without growth. As in the Solow model, along a balanced growth path all level variables are growing at rate η n + g. Unlike the Solow model, the steady state depends on the household preferences, as the savings rates are determined optimally.
Qualitative Dynamics We can analyze the qualitative dynamics just as we did without productivity growth. The key equations of the model are now: (1 + η)u ( c t ) = β(1 + η) 1 σ U ( c t+1 )[f ( k t+1 ) + 1 δ] (1 + η) k t+1 = (1 δ) k t + f ( k t ) c t The dynamics work in much the same way, only now they depend on η. So we can analyze the effects of a change in n or g which lead to a change in η. In steady state, c t+1 = 0, and f ( k ) δ + θ + ση Also in steady state k t+1 = 0, so: c = f ( k) (δ + η) k
Phase Diagram of Optimal Growth Model c c=0: f (k*)=δ+θ+ση c* k=0: f(k)-(δ+η)k k* Phase diagram of the optimal growth model k
Effect of an Increase in n or g c c* c0 c f (k )=δ +θ+ση f (k*)=δ +θ+ση f(k)-(δ+η)k f(k)-(δ+η )k k k* Phase diagram: An increase in the growth rate η to η. As before, initial effect depends on the slope of the saddle path. k
Endogenous Growth Models Now briefly discuss some models which try to explain sources of growth endogenous growth models. An active research topic initiated in late 1980s. Romer (1986, 1990), Lucas (1988) most influential: models of R & D, human capital. More recently Acemoglu et al: role of institutions in growth.
What s in TFP? Institutions & Geography Aside from innovations (which we ll turn to next), infrastructure, institutions, and geography are also important. Interesting comparison: experiences of former colonies. Acemoglu, Johnson and Robinson (2001). Small initial differences in income. Differences in settlers mortality influenced whether colony was run for extraction or whether colonists developed institutions. Those colonies where institutions took hold developed faster. Large differences in outcomes still today!
What s in TFP? Ideas and Human Capital Relatively new branch of economic theory: endogenous growth theory seeks to explain how technical change happens. Simple endogenous growth model (AK model): aggregate production function Y = AK. (Ignore labor and population growth, could think of this as per capita production.) Not subject to diminishing returns: MPK is constant F K = Y K = A. Idea: Aggregate capital K captures not just increases in physical capital but changes in the makeup of that capital.
Human Capital as a Source of Growth Human capital: knowledge, skills, and training of individuals. As economies become richer they invest in human capital in the same proportion, offsetting the diminishing marginal product of physical capital alone. Explicitly: production depends on human capital H, physical capital K: Y = zh θ K 1 θ Say H = hk, so that human capital is constant fraction of physical, then letting A = zh θ : [ Y = z(hk) θ K 1 θ = zh θ] K = AK
Other Interpretations Research and development programs are part of capital investment. They increase the stock of knowledge, which offsets diminishing marginal products of capital accumulation. Learning by doing: as economies produce more they learn better how to produce.
Implications of the Endogenous Growth Model Again savings constant fraction s of output. So: K = sak δk Since Y = AK, Ẏ Y = K K = sa δ Growth of output depends on the saving rate, even in the long run. No steady state. Higher savings more human capital, R&D, learning by doing. So higher savings leads to productivity improvements and higher growth. Important implication, some evidence that measured TFP does depend on savings, human capital.
A More Explicit Model of Human Capital Cobb-Douglas aggregate production function: Y = K α H β (AN ) 1 α β Again we have constant returns to scale now in (K, H, N ). Human capital and labor enter with different coefficients. Society accumulates human capital according to: Ḣ = s h Y δh Capital accumulation equation: Technological progress: K = s k Y δk Ȧ A = g > 0. Labor force grows at constant rate: Ṅ N = n > 0.
Analyzing the Model Dividing the production function by AN : ỹ = k α hβ Decreasing returns to scale in per efficiency units. The evolution of inputs is determined by: k = s k kα hβ (n + g + δ) k h = s h kα hβ (n + g + δ) h System of two differential equations determining k, h.
Balanced Growth Path To find the BGP equate both equations to zero: From first equation: s k k α h β (n + g + δ) k = 0 s h k α h β (n + g + δ) h = 0 ( (n + g + δ) h = s k Plugging it in the second equation k 1 α ) 1 β ( (n + g + δ) n + g + δ s h k α s k 1 α (n + g + δ) k s k ( ) 1 s h n + g + δ s k = k 1 α β k s k k 1 α ) 1 β = 0
Finding the Balanced Growth Path 1 α 1 k β s h s k k = = ( n + g + δ 1 α β k β s k = s k s h k 1 α ) 1 β ( (n + g + δ s k ) 1 β ( k = ( h = s 1 β k s β h n+g+δ s α k s1 α h n+g+δ ) 1 1 α β ) 1 1 α β
The Balanced Growth Path Using the production function: ỹ = Y AN = k α hβ = ( s 1 β k s β ) α ( 1 α β h sk αs1 α h n + g + δ n + g + δ ) β 1 α β y = Y ( s 1 β N = k s β ) α ( 1 α β h s α ) β k s1 α 1 α β h A n + g + δ n + g + δ Given some initial value of technology A 0 we have: y = ( s 1 β k s β ) α ( 1 α β h s α ) β k s1 α 1 α β h A 0 e gt n + g + δ n + g + δ
Evaluating the Model Taking logs: log y = log A 0 + gt α + β log (n + g + δ) + 1 α β α + 1 α β log s β k + 1 α β log s h What if we have a lot of countries i = 1,..., n? We can assume that log A 0 = a + ε i Also assume that g and δ are constant across countries.
TFP growth rate, 1965-95 (Labor share=0.65, no returns to education).04.02 0 -.02 -.04 IDN IRL PRT MUS TUN CHL BWACOG MYS ZWE ITA ISR IND KEN COL URY BRA BEL SWESP NLD AUS GRC AUT DNK PAK PRY MAR USAGBR FRA ECU EGY GHA GTMLKA UGA BGD DOM SYRTUR CAN TTO ZAF DZA PAN MLI PHL ARG PER JAMEX BFA BOL NZL CHE ETH SLV CIV NPL HND MWI VEN SEN JOR CRI RWA BDI TZA BENCMR PNG MDG CAF MRT TGO NGA NER ZMB AGO MOZ ZAR NIC HKG KOR THA JPN FIN NOR 0.2.4 Saving rate, 1965-95 Figure 1: Relation of TFP growth to saving rate SGP TFP growth rate, 1965-95 (Labor share=0.65, no returns to education).04.02 0 -.02 -.04 THA IDN PRT NOR TUN MUS ZWE BWA ITAMYS CHL COG JPNISR KEN BRAIND COL URYGRCAUT SWE PAK MAR PRY ECU GBR FRA ESP BEL FIN AUSDNK NLD USA GTM UGA BGD GHA DOM TUR LKA SYR EGY CAN DZA PAN ZAF ARGMEX PER TTO BFA MLI PHL JAM MWI CIV NPL HND BOL CHE NZL BDI TZA ETH SLV VEN SEN CRI RWA BEN PNG CMR MDG CAF MRT NGA NER ZMB TGO AGO MOZ ZAR NIC 0.05.1.15 Human capital investment rate, 1965-95 Figure 2: Relation of TFP growth to schooling rate HKG SGP KOR IRL JOR 45
TFP growth rate, 1965-95 (Labor share=0.65, no returns to education).04.02 0 -.02 -.04 PRT NOR IRL THA IDN ITA JPN MUS CHL COG ISR TUN MYS ZWE BWA BEL FIN URY AUT GRC SWE DNK GBR FRA ESPNLD AUS IND BRA COL KEN USA MAR PAK ECU PRY CAN LKA EGY ARG CHE NZL TTO BGD GHA GTM TUR PAN DOM SYR JAM ZAF UGA PER BFA MLI PHL MEX DZA BOL NPL MWI HND SLV VENCIV ETH SEN CRI BDI TZA RWA PNG CMRBEN MDG CAF MRT NGA TGO AGO NER ZMB MOZ HKG KOR 0.02.04.06 Labor force growth rate, 1965-95 Figure 3: Relation of TFP growth to labor force growth rate SGP ZAR NIC JOR TFP growth rate, 1965-95 (Labor share=0.65, 7% return to education).04.02 0 -.02 -.04 IRL IDN COG PRT MUSCHL ITA IND KEN TUN MYS COL URY BWA BRA BEL ZWEAUS DNK AUT ISR GTMLKA PAK PRY USAGBR SWESPGRC NLD FRA ECU UGA GHA BGDDOM CAN MLI EGY BOL TTO ZAF SYRTUR DZA ARGJAM PAN MWIPHL PERNZL MEX CHE SEN SLV NPL HND TZA CRI VEN RWA BENCMR PNG JOR MOZ CAF TGO NER ZAR NIC ZMB HKG KOR THA 0.2.4 Saving rate, 1965-95 Figure 4: Relation of TFP growth to saving rate JPN FIN NOR SGP 46