Solow Growth Model

*IntroductionProduction FunctionGeneral FormHomogeneous of Degree OneIntensive FormSlope of Production FunctionConsumption and Investment FunctionsEvolution of the InputsInputsChange in the Stock of CapitalThe Dynamics of the ModelThe dynamics of kThe Balanced Growth PathOutline

*The Impact of changes in Saving RateOn OutputOn Capital StockOn Balanced Growth PathThe Impact of changes in Population GrowthOn OutputOn Capital StockOn Balanced Growth PathThe Impact of changes in Technological ProgressOn OutputOn Capital StockOn Balanced Growth PathConclusion

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*Solow (1956) presented this model.The idea was how output responds to the changes in SavingsPopulation GrowthTechnology in the form of Knowledge

According to SolowYt = f (Kt , AtLt)Whereas: Yt = Level of Output in time tKt = Units of Capital in time t, Lt = Units of Labour in time tAt = Technical Progress in the form of Knowledge in time tAt Lt = Labour Augmented Technical Progress in time t Introduction

*Solow (1956) assumed that his production function is homogeneous with degree one.Slope of his production function is positive. CRS are possible for large nations where land and other natural resources are ignored.LettingYt = J x Yt , Kt = J x Kt , and Lt = J x LtWhere,J = Any Constant Number,Therefore, JYt = f (JKt , At JLt)Let JYt = Y*t soY*t = f (JKt , At JLt)Y*t = J x f (Kt , At Lt)Y*t = J1 x YtContinued...

*This shows that multiplying production function of Solow with any constant number say J and in the end, that constant number has completely factors out with degree, hence it indicates that Production function of Solow is Homogeneous with degree one.

Intensive Form of Solows Production FunctionAs Yt = f (Kt, AtLt)Dividing both sides with AtLt(Yt / AtLt)= f (Kt / AtLt , AtLt / AtLt)or(Yt / AtLt)= f (Kt / AtLt , 1)Ignoring 1(Yt / AtLt)= f (Kt / AtLt )Continued...

*Or Yt = AtLt x f(Kt / AtLt )

This is required Intensive form of Solows Model

Now for obtaining the production function we consider the equation(Yt / AtLt) = f(Kt / AtLt )Let Yt / AtLt = ytand Kt / AtLt = kt

Therefore, yt = f(kt)

This is the required Production FunctionContinued...MPKDiminishing Marginal Product

*This production function states that output per effective worker depends on capital per effective worker.Slope of the Production FunctionAs we know that Slope of the Production Function = d/dK (Yt) = MPK > 0Or d/dK (Yt) = d/dK [AtLt x f(Kt / AtLt)]= AtLt x [d/dK {f(Kt / AtLt)}]= AtLt x [f/(Kt / AtLt ) x d/dK (Kt / AtLt)]= AtLt x [f/(kt) x {AtLt x d/dK (Kt) - Kt x d/dK (AtLt)} / (AtLt)2] = AtLt x [f/(kt) x {AtLt - 0} / (AtLt)2] = AtLt x [f/(kt) x {AtLt / (AtLt)2]= f/(kt) x [AtLt x AtLt / (AtLt)2]Slope of the Production Function = d/dK (Yt) = MPK = f/(kt) > 0

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*As we know that AD = Yt = Ct + It . . (a)andAS = Yt = Ct + St . . (b)Solving equation (b) for C:Ct = Yt Stwhereas; In the long run:St = sYtso, Ct = Yt sYt = Yt (1 s)now substituting C in eq. (a)Yt = Yt (1 s) + It or Yt Yt (1 s) = ItYt (1 1 + s) = ItorYt (s) = ItDividing with AtLt on both sides (sYt) / (AtLt) = (It) / (AtLt)or Yt / AtLt = yand It / AtLt = iso,sy = iory = f (k) so,s . f (k) = irequired Investment FunctionDemand for Goods and Consumption Functions

*Now rewriting equation (a) as:Yt / AtLt = Ct / AtLt + It / AtLt ory = c + ic = y iandor c = f (k) s . f (k)required Consumption Function

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*Amount of Depreciation:

The amount of depreciation increases as stock of capital per efficient worker increases. Therefore, it is upward slopping.

Amount of Depreciation = (k)Whereas: = rate of depreciationk = stock of capital per efficient workerContinued...

*Solow assumed that inputs like Labour and Technology (Lt , At) grows at constant rate say n and g respectively. So,L*t = n (Lt) or L*t / Lt = n and A*t = g (At) or A*t / At = gThe log form of the above information may be stated asln L*t = ln [n (Lt)]ln A*t = ln [n (At)]ln L*t = ln (n) + ln (Lt)ln A*t = ln (g) + ln (At)ln L*t = n (t) + ln (Lt)ln A*t = g (t) + ln (At)Let t = 0so,ln L*0 = n (0) + ln (L0)ln A*0 = g (0) + ln (A0)ln L*0 = 0 + ln (L0) ln A*0 = 0 + ln (A0)ln L*0 = ln (L0) ln A*0 = ln (A0)so,ln L*t = n (t) + ln (L0)ln A*t = g (t) + ln (A0)Taking this information in the power of exponentioale [ln (L*t)] = e [n (t) + ln (L0)] e [ln (A*t)] = e [g (t) + ln (A0)]The Evolution of the Inputs

*e [ln (L*t)] = e [n (t) + ln (L0)] e [ln (A*t)] = e [g (t) + ln (A0)]L*t = e [n (t)] x e [ln (L0)] A*t = e [g (t)] x e [ln (A0)]L*t = e [n (t)] x L0 A*t = e [g (t)] x A0L*t = L0 e [n (t)] A*t = A0 e [g (t)]

This shows that both inputs grow at exponential rate over time.Now the changes in the stock of capital could be expressed as:

Change in stock capital = investment amount of depreciationk = s f (k) - (k)Continued...

*The Dynamics of kAs Lt and At are exogenously determined and it is difficult to handle Kt. So, we are proceeding with the dynamics of adjusted k.So, k = (Kt / AtLt)Taking change of capital with respect to time.dk / dt = k = (dKt / dt) x (AtLt) (d AtLt / dt) x (Kt) / (AtLt)2= (K/t) x (AtLt) / (AtLt)2 [Ltx(dAt / dt) + Atx(dLt / dt)] x (Kt) / (AtLt)2= (K/t) / (AtLt) (Kt) x [Ltx (A/t)] / (AtLt)2 + [Atx (L/t)] / (AtLt)2= (K/t) / (AtLt)[{(Kt) / (AtLt) x (A/t) / (At)} + {(Kt) / (AtLt) x (L/t) / (Lt)}]= (K/t) / (AtLt) [(Kt) / (AtLt) x (g + n)] andK/t = It - Kt = sYt - KtThe Dynamics of the Model

*Therefore,= (sYt Kt) / (AtLt) [(Kt) / (AtLt) x (g + n)]= (sYt / (AtLt) (Kt) / (AtLt) [(Kt) / (AtLt) x (g + n)] or= s (Yt / AtLt) (Kt) / (AtLt) [(Kt) / (AtLt) x (g + n)] or= s (Yt / AtLt) (Kt) / (AtLt) [ + (g + n)] ork = sy (k) x [ + g + n] ork = i (n + g + ) x (k)

If i > (n + g + ) x (k)then k > o so, k will increaseIf i < (n + g + ) x (k)then k < o so, k will decreaseIf i = (n + g + ) x (k)then k = o so, k will remain same

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*Actual and Break-even Investment kk*sf(k)[n+g+d ]kactual saving break-even investmentInvestment per efficiency unit of labourCapital per efficiency unit of labour

*Phase Diagram For Solow Modelkk*k*0dk/(t)/dtsf(k(t))- [n+g+d ]k balanced growth path k = k*k = k* when dk(t)/dt = 0; so sf(k*) = [n+g+d]k* which determines k*

*The Balanced Growth Path (BGP): How does the model behave once k has converged to k* ?Once k has converged to k* :

K(t) / A(t)L(t) = k*, where k* is constant. Or K(t) = A(t)L(t)k*Now gL(t) = n and gA(t) = g, so K(t) grows at the rate (n+g).

Since K(t) and A(t)L(t) both grow at rate (n+g), and the production function exhibits Constant Returns to Scale, Y(t) grows at rate (n+g).Since C(t) = (1-s)Y(t) consumption also grows at rate (n+g).K(t)/Y(t) is constant.K(t)/L(t) and Y(t)/L(t) grow at rate g. Thus g determines the growth rate of income per capita.

*Investmentand depreciationCapital per worker, ki1* = k1*k1*k2*Depreciation, kInvestment, s1 f(k)Investment, s2f(k)The Solow Model shows that if the saving rate is high, the economy will have a large capital stock and high level of output. If the saving rate is low, the economy will have a small capital stock and a low level of output.i2* = k2*An increase in the saving rate causesthe capital stock to grow to a newsteady state.New Balanced Growth PathOld Balanced Growth PathChange in Saving rate

*Change in Saving ratesOLDssNEWk*NEWk*OLDdk/dttt tt0t0t0k

*Change in Saving RateThus a change in the saving rate affects only the level of activity in steady state (not its growth rate).Nonetheless, the change in level may be quantitatively substantial.Consider the effect on steady state y* = f(k*)

Now dy* /ds = f (k*) dk*/ ds k* is defined by the condition that dk/ dt =0 and satisfies

s f (k*) = [n + + g ] k*,and k* = k*(s, n, , g).

Differentiating w.r.t s: f(k*) + sf (k*) dk*/ds = [n + + g] dk*/ds.Now, gather terms in dk*/ds and re-arrange:dk*/ds = f (k*) / [[n + + g] - sf(k*)] (>0 see phase diag).Sody*/ds = f (k*) f(k*) / [[n + + g ] - s f (k*)]>0.

*Change in Savings RateRewrite as an elasticity (unit less, more convenient):

ey*,s = (s/y) (dy/ds) = ak(k*) / [1 - ak(k*) ]

where ak(k*) = k*f (k*)/f(k*) = elasticity of y w.r.t. k at k*= income share of k at k* Suppose ak(k*) = 1/3, then ey*,s = 1/2.

Then a 10% rise in s (from 0.2 to 0.22) raises y by 5%.So the effect of s on the level of output (per efficiency unit of labour) is relatively modest.In the long-run, changes in the saving rate (& other parameters) only have level effects on k* and y*, they do not affect the growth rate.

*Population GrowthThe basic Solow model shows that capital accumulation, alone, cannot explain sustained economic growth. High rates of saving lead to high growth temporarily, but the economy eventually approaches a steady state in which capital and output are constant.

To explain the sustained economic growth, we must expand the Solow model to incorporate the other two sources of economic growth.

So, lets add population growth to the model. Well assume that the population and labor force grow at a constant rate n.

*The Impact of Population GrowthLike depreciation, population growth is one reason why the capital stock per worker shrinks. If n is the rate of population growth and is the rate of depreciation, then ( + n)k is break-even investment, which is the amount necessary to keep constant the capital stock per worker k.

An increase in the rate of population growth shifts the line representing population growth and depreciation upward.

The new steady state has a lower level of capital per worker than the initial steady state. Thus, the Solow model predicts that economies with higher rates of population growth will have lower levels of capital per worker and therefore lower incomes.

*The Impact of Population GrowthAn increase in the rate of population growth from n1 to n2 reduces the steady-state capital stock from k*1 to k*2.Old Balanced Growth PathNew Balanced Growth Pathin reality

*The Impact of Technological ProgressLike depreciation and population growth, technological progress is another reason for decline in the capital stock per worker.If g is the rate of technological progress and ( + n) is the rate of depreciation after population growth, then ( + n + g)k is break-even investment, which is the amount necessary to keep constant the capital stock per worker k. An increase in the rate of technological progress shifts the line representing technological progress; population growth and depreciation upward.The new steady state has a lower level of capital per worker than the initial steady state. Thus, the Solow model predicts that economies with higher level of technological progress will have lower levels of capital per worker and therefore lower incomes.

*The Impact of Technological ProgressAn increase in the technological progress from g1 to g2 reduces the steady-state capital stock from k*1 to k*2.Old Balanced Growth PathNew Balanced Growth Pathin reality(d + n + g1) k

*Convergence: Theory and EvidenceA key implication of the Solow model is that economies converge towards their balanced growth paths

Solow model predicts rate of return (on capital) lower when K/L high. This would cause funds to flow to poorer countries, so we should expect poor countries to grow more quickly.

Solow model predicts that countries converge to BGPs; so if Y/L differences arise from differences relative to respective BGPs, we should expect poor countries to grow more quickly.

However in reality neither empirical results support the Solows proposition of Convergence nor the speed of adjustment.

*ConclusionIn the long run, an economys saving determines the size of k and thus y.

The higher the rate of saving, the higher the stock of capital and the higher the level of y.

An increase in the rate of saving causes a period of rapid growth, but eventually that growth slows as the new steady state is reached.

The higher the rate of population growth and technological progress, the higher the rate of depreciation, the lower the stock of capital per efficient worker and hence the lower the level of output per efficient worker respectively.

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