Jean Monnet Chair in European Integration Studies Prof. PASQUALE TRIDICO Università Roma Tre

Transcription

1 Jean Monnet Chair in European Integration Studies Prof. PASQUAE TRIDICO Università Roma Tre

2 Two inputs,, A production function Cobb-Douglas Y= F, = 1 0 < < 1 Constant return to scale (decreasing marginal return for each factor) F h, h = hf(,)=hy F /, / = F/(/,/)=y=F() The Cobb-Douglas has constant returns to scale; if you double (halve) the amount of each input, you double (halve) output.

3 Substitutability between and Variation of different combinations of and I depends on r The hedge nife (Harrod instability) is solved through the variation of v= /Y allowed for by prices flexibility (prices of factors)

4 no government purchase of goods and services: Y = C + I for each t Hence saving S equals gross investment I Y- C = S = I for each t

5 Hp: the number of worers growth at the same rate of pop growth t = 0e nt Population growth and wor force growth exhibit exponential growth: ሶ t t = n

6 ሶ ሶ Output depends on and = I δ = sy δ is the change in capital stoc over time Income is equal to output Individuals save always the same fraction (with s<1, > 0 of Y) Aggregate saving = to gross I sy=s=i depreciates over time (with δ > 0, < 1 of )

7 ሶ Divide = I δ = sy δ by ሶ = s Y δ = s Y δ

8 if if if 0 0 0

9 perfect competition Price taers firms Y F(, ) 1 Profit maximizations solve the following problem Max F(,) r w,

10 F Y w (1 ) F Y r the share of Y for abour is 1 w Y the share of r Y Y for apital is

11 First ey equation of Solow y Y y

12 y = (GDP per capita)

13 ሶ ሶ ሶ = sy δ * (capital accumulation equation) About how capital accumulates The change of capital stoc is equal to the amount of gross investment sy less the amount of depreciation δ Worers/consumers save a constant fraction s of Y δ=5% (for instance) = d dt indicate the derivative with respect to time, or simply, t+1 t

14 n sy n y s n Y s n and Y s from ) ( ; *..... the second fundamental equation of Solow

15 . sy ( n ) The equation says that change in per worer is determined by 1. I per worers sy which increases 2. Depreciation which decreases 3. And population growth n which decreases In each period there are n new worers. If there were no new investments, capital per worer would decline because of the increase of labour force (by n)

16 y.. sy ( n )

17 . sy ( n ) n + δ

18 is the line where investment per worer is constant It represents both pop growth n and depreciation, as a fraction of that 0 ) ( * : 0 ) ( * :.. n sy if n sy if

19 For each value of, n indicates how much of Investment is needed in order to eep / constant Hence n is the rate of growth g of equilibrium The system tends towards n = g On the left of *, I are more of what is necessary in order to eep / constant / > / (n) On the right of *, I are insufficient to eep / constant / < / (n)

20 When = *: sy = (n + δ) /Y = v* When < * : sy > (n + δ) /Y < v* But in this case, excess supply of capital: r decreases It becomes more convenient to adopt techniques which use more capital:, /Y and converges to v* When > * : sy < (n + δ) /Y > v* But in this case, excess demand of capital: r increases It becomes more convenient to adopt techniques which use more labour:, /Y and converges to v*

21 Between 0 and before * we spea about capital deepening: the capital per worer ratio increases When capital per worer ratio is zero (it does not increases) we spea about capital widening (Capital is growing, but because of population growth / is constant

22 This is the solution of the Solow model in general terms. If 0 (the initial level of per worer), α, s, n and depreciation of capital, are nown, one can establishes whether capital per worer will grow or not (so the economy will grow or not). This has also very important policy implications as far as policy can influence s (and I), n and the other parameters

23

24 Is that path of growth that allows for s which represents the minimum share of total outcome and also the max possible consumption, or in another way: is the rate of savings which maximizes steady state level of growth and consumption

25 The SS rises from * to ** and the economy will be richer Before the new capital/worer ratio capital deepening occurs, and output per worer grows untill the economy reaches the new (higher) rate of steady state In ** income will be higher At the current value of capital stoc * investment per worer now (after the increase of s) exceeds the ammount required to eep the capital labour ratio constant and therefore a process of capital deepening restarts and the economy grows (GDP per capita grows) untill when sy reaches the new steady states in ** and is equal to n+d line

26

27

28 At the current value of capital stoc * investment per worer is now (after the increase of n) no longer high enough to eep the capital labour ratio constant in the face of a rising in the population. Therefore the capital labour ratio begins to fall untill the point at which investment (sy) is equal to the new n+depreciation in ** At this point the economy has less capital per worer than before and is therefore poorer. Per capita output (and icome) is lower after the increase of n The SS decreases from * to ** and the economy will be poorer

29

30

31