Econ 7110 slides Growth models: Solow, Diamond, Malthus. January 8, 2017

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1 Econ 7110 slides Growth models: Solow, Diamond, Malthus January 8, 2017

2 Production functions Standard setting: Y =output, K (capital) and L (labor) K, L referred to as inputs Usually assumed to satisfy: Y = F (K; L) (1) (1) Positive marginal products: F K () > 0; F L () > 0 (2) (2) Diminishing marginal products: F KK () < 0; F LL () < 0 (3)

3 (3) The Inada condition: for Z = K; L lim Z() Z!0 = 1 (4) lim Z() Z!1 = 0 (4) Constant Returns to Scale (CRS): for all > 0 F (K; L) = F (K; L) (5)

4 Intensive-form production function Let lower-case variables denote per-worker levels CRS implies y = Y L = F (K; L) L = F K L ; 1 = F (k; 1) f(k) (6) Assumptions (1)-(3) imply: f 0 (k) > 0 (7) f 00 (k) < 0 lim (k) k!0 = 1 lim (k) k!1 = 0

5 Also, using l Hôpital s Rule: f(k) lim k!0 k = lim f 0 (k) k!0 1 = 1 (8)

6 Factor prices Atomistic rms take factor prices as given when maximizing pro ts: maxf (K; L) K wl rk (9) K;L w=wage rate, r=real interest rate, =depreciation rate F (K; L) K=net output (output net of capital depreciation) Rewrite F (K; L) = Lf(k) = Lf( K L ) Exercise: show that w = F L (K; L) = f(k) f 0 (k)k (10) r = F K (K; L) = f 0 (k)

7 Parametric examples of production functions Cobb-Douglas: F (K; L) = K L 1 (11) f(k) = k CES (various formulations): F (K; L) = [K + (1 )L ] 1 (12) f(k) = [k + (1 )] 1 where 2 ( 1; 1], 6= 0 Note: CES does not always satisfy Inada

8 The Solow Growth Model Discrete time setting: the time variable t is a (non-negative) integer: t 2 f0; 1; 2; :::g Notation: K t = capital in period t = depreciation rate s = rate of saving/investment out of income, Y t Evolution of capital stock: K t+1 = sy t + (1 )K t (13)

9 L t = population/labor force in period t n = (net) growth rate of population L t+1 = (1 + n)l t (14) Assume n > 0, 2 (0; 1]; s 2 (0; 1]

10 Y t = F (K t ; L t ) (15) Task: nd di erence equation for k t = K t =L t, on the form: k t+1 = (k t ) Use (13) to (15), and y t = Y t =L t = F (k t ; 1) = f(k t ), to get K t+1 L = K t+1 L t+1 t L t+1 L = k t t+1 (1 + n) = sy t+(1 )K (16) t L = sy t t + (1 )k t = sf(k t ) + (1 )k t Or: k t+1 = sf(k t) + (1 1 + n )k t (k t ) (17)

11 0 (k t ) = 1 Properties of (k t ) 1+n + s 1+n f 0 (k t ) > 0 00 (k t ) = lim k t!0 0 (k t ) = 1 s 1+n f 00 (k t ) < 0 1+n + 1+n s lim k t!0 f 0 (k t ) = 1 (18) lim k t!1 0 (k t ) = 1 1+n + 1+n s lim f 0 (k t ) = 1 k t!1 1+n < 1 Together these guarantee: existence, uniqueness, and stability of steady state (Uniqueness except for the trivial one where k t = 0) Illustrate in 45 -diagram

12 Parametric example: Cobb-Douglas production Y t = ZK t L 1 t (19) y t = Zk t k t+1 = szk t + (1 1 + n Steady state capital stock per worker: k = Steady state output per worker: y = Z 1 1 sz n + s n + )k t (k t ) (20) 1 1 : (21) 1 : (22)

13 Taking stock In this parametric example, what is e ect on output per worker from rise in productivity, Z? A rise in n? Steady state and transition? Implications for testing on cross-country data?

14 The Diamond Overlapping Generations Model Agents live in two periods: working age, retirement L t = number young (working) agents in period t; L t+1 = (1 + n)l t c 1;t = consumption of working agent in period t c 2;t = consumption of retired agent in period t s t = saving of working agent in period t R t+1 = 1 + r t+1 = gross interest rate on savings held from period t to t + 1 w t = period-t wage rate

15 Consider agent young/working in period t Budget constraints c 1;t = w t s t (23) c 2;t+1 = R t+1 s t (24) Utility: U t = U(c 1;t ; c 2;t+1 ) (25) Optimal savings decision given by s(w t ; R t+1 ), de ned from s(w t ; R t+1 ) = arg maxu(w t s t ; R t+1 s t ) (26) s t 2[0;w t ]

16 Capital accumulation: Total savings in period t = total capital stock in period t + 1 s(w t ; R t+1 )L t = K t+1 (27) Recall: L t+1 = (1 + n)l t K t+1 L t = K t+1 L t+1 L t+1 L t = k t+1 (1 + n) = s(w t ; R t+1 ) (28) Factor prices (recall) given by marginal products w t = f(k t ) f 0 (k t )k t w(k t ) (29)

17 R t+1 = f 0 (k t+1 ) + 1 R(k t+1 ) (30) Thus: k t+1 = (k t ) (31) where (k t ) is de ned from (k t ) = sfw(k t); R((k t ))g 1 + n (32) Cannot solve explicitly for (k t )

18 Parametric example: logarithmic utility, Cobb-Douglas production Log utility: First-order condition: U t = (1 ) ln(c 1;t ) + ln(c 2;t+1 ) (33) Solving for s t : (1 )(w t s t ) 1 + s 1 t = 0 (34) s t = w t (35) Cobb-Douglas production: Y t = ZKt L 1 t (36) y t = Zkt w t = (1 )Zk t

19 k t+1 = (1 )Zk t 1 + n Steady state capital stock per worker: = (k t ) (37) k = Steady state output per worker: " (1 )Z 1 + n # 1 1 (38) y = Z " (1 )Z 1 + n # 1 = Z 1 1 " (1 ) 1 + n # 1 (39) Illustrate dynamics, steady state in 45 -diagram

20 Taking stock In this parametric example, what is the e ect on output per worker from rise in productivity, Z? A rise in n? Steady state and transition? Implications for testing on cross-country data? What assumptions turns Solow into Diamond in this parametric case?

21 The Malthus Model Agents live in two periods: children, adults L t = number adult (working) agents in period t c t = consumption of adult agent in period t q = cost per child (can be interpreted as consumption per child); exogenous n t = number of children per adult L t+1 = n t L t (40) Note: notation is di erent from above!! n t 1 is here net growth rate of L t

22 y t = period-t income per adult Y t = total output Production function: land, labor as inputs Y t = F (X; L t ) (41) X=land (here constant) Adult s budget constraint: c t = y t qn t (42) Utility U t = U(c t ; n t ) (43)

23 Parametric example: logarithmic utility, Cobb-Douglas production U t = (1 ) ln(c t ) + ln(n t ) (44) Utility maximization gives n t = q! y t (45) Y t = Z(X) (L t ) 1 (46) y t = Y t = Z X! L t L t Dynamics equation for (adult) population: L t+1 = n t L t = q! Z X L t! L t = q! ZX Lt 1 (47)

24 Illustrate dynamics, steady state in 45 -diagram Steady state population: L = "! # 1 " ZX = q q! Z# 1 X (48) Steady state output per worker: y = q (49)

25 Taking stock In this parametric example, what is the e ect on output per worker from rise in productivity, Z? E ect on total population? Steady state and transition? In Solow and Diamond, we asked about e ects of changes in n. questions about the e ects of n t not meaningful here? Why are How would we test predictions of the Malthus model on data?