Modèle de Solow

1 Di¤erence equations: stability analysis Consider xt+1 = f (xt ) The steady state (...xed point) is a value x such that x = f (x) Notice that the steady state cannot exist, it can be unique or there can be several points ...x. The stability analysis of the steady state tells us if then system converge towards it and in which way: monotonically or oscillatory etc. In order to appreciate the stability of the steady state we must study the derivative of the function f evaluated at the steady state i.e. f 0 (x) : We have the following results: 8 > if f 0 (x) 2 (0; 1) then the steady state is stable and the system converges toward it monotonically > < if f 0 (x) > 1 then the steady state is unstable and the system diverges from it monotonically 0 > > if f 0 (x) 2 ( 1; 0) then the steady state is stable and the system converges toward it in a oscillatory way : if f (x) < 0 -1hen the steady state is unstable and the system diverges from it in a oscillatory way 1.1 Examples Consider the linear di¤erence equation xt+1 = a + bxt We have that the steady state is unique and given by x = a + bx i.e. x= We have that f 0 (x) = a 1 b dxt+1 =b dxt Then we have 8 > if b 2 (0; 1) then the steady state is stable and the system converges toward it monotonically > < if b > 1 then the steady state is unstable and the system diverges from it monotonically > if b 2 ( 1; 0) then the steady state is stable and the system converges toward it in a oscillatory way > : if b < 0 -1hen the steady state is unstable and the system diverges from it in a oscillatory way Consider the example 1 xt+1 = 1 + xt 2 1 The steady state is 1 x=1+ x 2 i.e. x=2 We have f 0 (x) = dxt+1 1 = dxt 2 Then the system converges monotonically towards the steady state. Consider the example xt+1 = 1 + 2xt the steady state is x= 1 + 2x i.e. x=1 We have f 0 (x) = dxt+1 =2 dxt Then the system diverges monotonically from the steady state. Consider the example 1 xt+1 = 1 xt 2 The steady state is 1 x x=1 2 i.e. 2 x= 3 We have 1 dxt+1 = f 0 (x) = dxt 2 Then the system converges in a oscillatory way towards the steady state. Consider the example xt+1 = 1 2xt The steady state is x=1 i.e. x= We have f 0 (x) = 2x 1 3 dxt+1 = dxt 2 Then the system diverges in a oscillatory way from the steady state. 2 Consider the non-linear example xt+1 = xt There are two steady states if > 1. The ...rst one is x=0 meanwhile the second is x=1 We have dxt+1 =x dxt Consider ...rst the steady state x = 0. If 2 (0; 1) we have f 0 (x) = f 0 (0) = dxt+1 =0 dxt 1 1 = = +1 therefore the steady state x = 0 diverges monotonically from the steady state. If > 1 we have dxt+1 f 0 (0) = = x 1=0 dxt therefore the steady state is stable. If 2 ( 1; 0) we have f 0 (0) = dxt+1 =0 dxt 1 = 1 therefore the steady state x = 0 diverges in a oscillatory way from the steady state. If 2 ( 1; 1) we have f 0 (0) = dxt+1 =0 dxt 1 = 1 therefore the steady state x = 0 diverges in a oscillatory way from the steady state. Consider now the steady state x = 1: If 2 (0; 1) we have f 0 (1) = dxt+1 =1 dxt 1 = 2 (0; 1) therefore the steady state x = 1 converges monotonically towards the steady state. If > 1 we have f 0 (0) = dxt+1 =1 dxt 3 1 = >1 therefore the steady state diverges monotonically from the steady state. If 2 ( 1; 0) we have dxt+1 =1 dxt f 0 (0) = 1 = 2 ( 1; 0) therefore the steady state x = 1 converges in a oscillatory way towards the steady state. If 2 ( 1; 1) we have f 0 (0) = dxt+1 =1 dxt 1 = 2 ( 1; 1) therefore the steady state x = 1 diverges in a oscillatory way from the steady state. 2 Production function l: labor k : capital y : output y = F (k; l) Hypothesis 8 > Fk (k; l) > 0 > < Fl (k; l) > 0 > Fkk (k; l) < 0 > : Fll (k; l) < 0 Example Cobb-Douglas F (k; l) = k l1 with 2 (0; 1) Fk (k; l) = k Fl (k; l) = (1 Fkk (k; l) = Fll (k; l) = 2.1 11 l >0 )k l ( 1) k (1 >0 21 l )k l Returns to scale Increasing returns to scale: let >0 F ( k; l) > F (k; l) Decreasing returns to scale F ( k; l) < F (k; l) 4 <0 1 <0 Constant returns to scale F ( k; l) = F (k; l) Example Cobb-Douglas F (k; l) = k l + F ( k; l) = ( k ) ( l) = Increasing returns to scale: + + + + 3.1 kl = + F (k; l) > F (k; l) + kl = + F (k; l) < F (k; l) + kl = + F (k; l) = F (k; l) =1 F ( k; l) = ( k ) ( l) = 3 F (k; l) <1 F ( k; l) = ( k ) ( l) = Constant returns to scale: + >1 F ( k; l) = ( k ) ( l) = Decreasing returns to scale: kl = Pro...t maximization The case with one input Let consider a production function with labor l as unique input and let y be the output and > 0 : y = F (l) Increasing returns to scale: F ( l) > F (l) Decreasing returns to scale: F ( l) < F (l) Constant returns to scale: F ( l) = F (l) Example F (l) = l with > 0: Increasing returns to scale: >1 F ( l) = ( l) = Decreasing returns to scale: l>l <1 F ( l) = ( l) = 5 l1 ( 1) l 2 >0 It is not a maximum: the maximum is an in...nite production. Decreasing return to scale: < 1 ( 1) l 2 <0 2 =0 It is a maximum. The solution is interior. Constant return to scale: = 1 ( 1) l 6 In such a case the pro...t is pl wl = (p w) l The optimum if p > w is +1: The optimum is 0 if p w < 0: The optimum is whatever level of production and labor if p = w: Notice that when returns to scale are decreasing the labor demand and the supply of the good are a decreasing function of the real wage. Indeed l= y= 1 w p w p 1 and 1 When returns to scale are decreasing the pro...t is = pl " =p =p " =p =p =p =p " =p w p 1 1 w p 1 1 " w p w p w p =p 1 " 1 w p 1 1 7 1 +1 1+ 1 1 1 1 1 1 1 1 1 !# 1 1 1 1 1 1 1 # !# 1 1 # # 1 1 1 1 1 1 w p 1 1 # w p 1 1+1 1 1 1 1 1 1 1 w p 1 1 1 1 w p 1 1 1 1 1 1 1 w p 1 w p " w 1 " " w p 1 w p 1 w p " =p w p wl = p 1 # !# !# Since < 1 this means that =p " 1 1 > 1: Therefore w p 1 1 1 1 1 1 > 0 and so 1 # >0 Result: when the returns to scale are decreasing the pro...t is positive. When returns to scale are increasing the pro...t is in...nite. When returns to scale are constant: if p > w the supply of the good is in...nite and then = (p w) l = +1 If p < w the supply of the good is zero and so =0 If p = w the supply of the good is whatever and so =0 3.2 The case with two inputs Let p be the price of the good, w the nominal wage and r the interest rate (the cost of capital). The pro...t maximization is max pF (k; l) wl k;l rk The ...rst order conditions are pFk (k; l) r=0 pFl (k; l) w=0 Notice that Fk (k; l) is the marginal productivity of capital and pFl (k; l) is the marginal productivity of labor. We can write Fk (k; l) = r p this means that the marginal productivity of capital must equalize the real interest rate. w Fl (k; l) = p This means that the marginal productivity of labor must equalize the real wage. Example Cobb-Douglas: F (k; l) = k l We have Fk (k; l) = k 8 1 l 1 Fl (k; l) = k l Therefore we have 1 k r p l= and 1 kl = w p The pro...t is then = py wl rk this means = pk l wl rk =p k l w l p r k p 1 k which can be written as i.e. =p k l kl l 1 lk this means =p k l kl kl i.e. =p k l ( + )k l If returns to scale are constant, i.e. + =p k l = 1 then the pro...t is kl =0 Result: when returns to scale are constant the pro...t is zero and the supply of the good is whatever. If returns to scale are decreasing, i.e. + < 1 then the pro...t is =p k l ( + )k l >0 Result: when returns to scale are decreasing the pro...t is positive. When returns to scale are increasing one can show that second order conditions are not satis...ed and pro...t is in...nite. 3.3 The real wage and the real interest rate Consider a constant returns to scale technology F (K; L) = Lf (k ) where k= 9 K L Pro...t maximization implies K L max Lf K;L rK wL The ...rst order conditions are Lf 0 (k ) 1 =r L and Lf 0 (k ) f (k ) K =w L2 i.e. f 0 (k ) = r and f (k ) f 0 (k ) k = w Since pro...ts are zero we have Lf (k ) = f 0 (k ) K + wL i.e. f (k ) = f 0 (k ) L K +w L L i.e. f (k ) = f 0 (k ) k + w i.e. f (k ) f 0 (k ) k w = + f (k ) f (k ) f (k ) i.e. f 0 (k ) k =1 f (k ) Therefore, since w f (k ) w f (k ) 2 (0; 1) we have f 0 (k ) k <1 f (k ) This result will be very useful in the sequel. 4 The neoclassical synthesis In order to be more clear, we provide the reader with a speci...c example regarding the fundamentals of the model. However, the results obtained can be extended to whatever fundamentals'speci...cation. There is one representative 10 consumer and one representative ...rm. Labor l is the unique input and c denotes consumption. The utility function of the representative consumer is 12 l 2 u (c; l) = c The budget constraint is pc = wl + where w is the real wage, p the price of the good and the pro...t of the ...rm whose owner is the consumer that take it as given. The maximization problem is therefore maxc;l c 1 l2 2 s:t: pc = wl + From the budget constraint we can solve for c : c= w l+ p p Then we have the unconstrained maximization problem: max l The ...rst order condition is w l+ p p w p 12 l 2 l=0 The second order condition is 1<0 which ensures us that it is a maximum. From the ...rst order equation we can derive the labor supply: w ls = p which is a increasing function of the real wage. Consumption is therefore c= ws ww l+ = += p p pp p w p 2 + p The ...rm has the following production function: 1 F (l) = 2l 2 which exhibits decreasing returns to scale. The pro...t maximization is: 1 max p2l 2 l The ...rst order condition is pl 1 2 wl w=0 11 The second order condition is 1 pl 2 3 2 <0 which ensures us that it is a maximum. From the ...rst order equation we can write 1 w l 2= p and therefore the labor demand is 2 w p ld = which is a decreasing function of the real wage. The pro...t of the ...rm is therefore w p 1 = 2w 12 = p2 w w p w 2 12 i.e. p p i.e. p2 p2 p2 = >0 w w w which is a decreasing function of the real wage. Equilibrium in the labor market implies =2 ls = ld i.e. w = p w p i.e. 3 w p which implies 2 =1 w =1 p i.e. w = p: Equilibrium labor is therefore lE = 1 and equilibrium production is y E = 2 lE 1 2 1 = 2 12 = 2 12 Equilibrium pro...t is E = p2 p = p=p w w Equilibrium consumption is cE = w p 2 + p =1+ p =1+1=2 p Notice that y E = cE = 2 and therefore all the good produced is consumed. This is the Walras'law. We have found the real wage. But what about the nominal price p and the nominal wage w? We must consider the money market. According to the Cambridge approach the demand of money is a share of the nominal income: M d = hpy or according to the quantitative theory of money M d V = py where V is the velocity of circulation of money. The second formula can be rewritten as 1 M d = py V and so 1 = h: V The money supply is ...xed exogenously by government: Ms = M Therefore equilibrium in the money market requires M = hpy But we know M , h and y : therefore the price level is: p= M hy and since p=w we have M hy 1 Example. Suppose M = 1000 and h = 2 (V = 2). Since y = 2 we have w= p=w= 1000 = 1000 1 22 It follows a complete dichotomy between the real sector and the nominal sector. Labor market ...xes the real wage meanwhile the money market determines the nominal price and the nominal wage. 13 5 The basic Keynesian model We will consider in this chapter a closed economy, i.e. there are no exportations nor importations and capital ? ows between countries. 5.1 Elements of national accounting Let Y the real national income and let C , I , G be respectively aggregate consumption, aggregate investment and public spending. We have Y =C +I +G Ley now Yd = Y T be the available income and T the taxes. We have that the available income can be consumed or saved, i.e. Yd = C + S where S stands for saving. From the previous equations we have C +I +G=C +S+T which yields I = S + (T G) Notice that S is the private saving and T G is the public saving. Therefore investment is ...nanced by private and public saving. 5.2 Equilibrium production and employment Consumption is given by: C = C0 + c (Y T) where C0 is the autonomous consumption and c is the marginal propensity to consume. For the time being, we consider that investment is driven by animal spirits and is exogenous: I = I0 Eventually, government ...xes the level of public spending which is an exogenous variable too: G = G0 The aggregate demand AD is therefore given by AD = C0 + c (Y T ) + I0 + G0 = cY + C0 14 cT + I0 + G0 On the supply side we assume that ...rms are willing to satisfy whatever production level. This can be explained as it follows. Price p and nominal wage w are ...xed (we are in the short run). Firms solve the following maximization problem: maxY;l pY wl s:t: Y = f (l) where l is the number of employed, f (l) a deceasing returns to scale production 0 function with f 0 (l) > 0 and f 0 (l) < 0. We can write the unconstrained problem max pf (l) wl l whose ...rst order condition is f 0 (l) = w : p Therefore the notional labor demand is ld = f 0 1 w p which is a decreasing function of the real wage. It follows that the notional supply of good is w Y d = f ld = f f 0 1 p which is a decreasing function of the real wage too. We will suppose all though this chapter that aggregate demand fall down to Y d : AD < Y d : In such a case, ...rms will not produce the notional level of supply because they are constrained by the aggregate demand. Notice that for all Y < Y d pro...ts are increasing and therefore in such a region ...rms are willing to satisfy whatever level of the demand. Only when AD > Y d ...rm will stick to Y d . It follows that in the Y d ; Y plane the aggregate supply is perfectly elastic (horizontal). Equilibrium requires therefore Y = cY + C0 i.e. YE = C0 cT + I0 + G0 cT + I0 + G0 1c Once we have found Y E it is immediate to derive the unemployment rate. If the production function is Y = f (L) we have that LE = f 1 15 YE : Let L be the labor force. Unemployment is then LE U =L and unemployment rate is LE L One may wonder which is result of an increase in public spending, in investment and in taxes. We have the following results: L u= YE = YE = YE = 1 1 c 1 1 c c 1 c G0 > 0 I0 > 0 T > 0: The term 1 1 c is called the Keynesian multiplier: he tells us in which proportion a unitary increase of public spending or investment translate into an increase of equilibrium production. From the previous considerations, we have that if the economy is in a recession (because e.g. private investment stagnates), government must increase public spending and/or reduce taxes. In both cases, of course, there will be a public de...cit, D = G T : In such cases government ...nances public de...cit by borrowing, i.e. selling bonds. Now we want to study the impact on production of a balanced public spending, i.e. G0 = T: In such a case it is easy to verify that c T + G0 = 1c YE = c G0 + G0 = 1c G0 Therefore a balanced public spending has a positive impact on equilibrium production, although less important than in the case public spending is ...nanced by the emissions of bonds. 6 6.1 The IS LM model The curb IS We now suppose that ...rms' investment decision depends negatively upon the interest rate r; i.e. good market is at equilibrium when Y = C0 + cY T + I0 r + G: Why investment is negatively correlated to the interest rate? The explication of Keynes is the following. Suppose there are n investment projects. Each of 16 them entails a cash exit of Ai , i = 1; ::; n: Then in the following h periods it gets an amount of Bi;j , j = 1; ::; j i.e. V1 V2 Vn B11 B12 B 1h + 2 + ::::: + h 1 + r (1 + r) (1 + r) B 2h B21 B22 + ::::: + = A2 + + h 1 + r (1 + r)2 (1 + r) :::::::::::::::::::::::::::::::::::::::::: B n1 B n2 Bnh = An + + + ::::: + h 1 + r (1 + r)2 (1 + r) = A1 + Notice that all the present values of the cash ? ows Vi are a decreasing function of the interest rate. Let r be the market interest rate and let ri the interest rates solving V1 V2 :::::: Vi :::: Vn = = = = = = 0 0 0 0 0 0 Then the ...rm will retain only those investment projects such that ri > r In fact this means that is more pro...table to carry out the investment project rather than investing in the ...nancial market, or equivalently, that the present value of the cash ? is positive when evaluated at the market interest rate. ow Therefore the lower r the larger the number of the projects to a¤ord. Equilibrium in the good market requires therefore Y= 6.2 C0 T + I0 1c 1r +G = C0 T + I0 + G 1c 1 1 c r The curb LM Consider now the money market. The supply of nominal balances is exogenously ...xed by public sector, i.e. Ms = M d Agents decide how many real balances M to hold. The demand of real balances p depends positively upon the income Y . In fact, the higher the income the higher the need of money for transactional and precautionary scopes. Conversely, the demand of real balances is negatively related to the interest rate: the latter indeed represents the opportunity cost of money holding. Namely, by holding 17 money, agents renounce to invest wealth in rentable bonds yielding the interest rate. Therefore the demand of money can be resumed by Md = p 1Y 2r where 1 and 2 are positive parameters. Equilibrium in the money market implies M = M d i.e. Md = 1Y 2 r: p Solving for production, we have M + 1p Y= 6.3 2 r 1 Equilibrium Combining the two equations we can solve for the interest rate r : C0 T + I0 + G 1c 1 1 c r= M + 1p 2 r 1 which yields 2 + 1 1 1 c i.e. r= C0 r= C0 T +I0 +G 1c 2 1 M 1p T + I0 + G 1c + M 1p 1 1c and aggregate demand is therefore Y= C0 T + I0 + G 1c C0 T +I0 +G 1c 1 1 c 2 1 + M 1p 1 1c which can be rewritten Y= C0 T + I0 + G 1c C0 T +I0 +G 1c 1 1 c 2 1 + + 1 1c M 1p 1 1 c 2 1 + 1 1c i.e. Y = (C0 0 T + I0 + G) @ 1 1 1 1 c 1 c 2 1 18 + 1 1c 1 A+ M 1p 1 1 c 2 1 + 1 1c i.e. Y = (C0 T + I0 + G) 1 1 c 0 1 @1 1 1 1c 1 2 1 + M 1p 1 1 c 2 1 + 1 1c >0 1c 1 1c 1 + 2 1 which can be rewritten 1 1c 1 1c 1> + 2 11 i.e. 2 1 1c A+ 1 In fact is must be 1> + 2 Notice that 1 1c 1 1 + 11 1 1 c 1...