Solow Model PDF

# Chapter 5: Solow's Model of Growth ## 5.1. Introduction The **neo-classical** critique of the Harrod-Domar growth theory starts from the basic proposition that the **instability** of the Harrod-Domar (H-D) growth path is not the essential property of a mature modern capitalistic economy which is...

# Chapter 5: Solow's Model of Growth ## 5.1. Introduction The **neo-classical** critique of the Harrod-Domar growth theory starts from the basic proposition that the **instability** of the Harrod-Domar (H-D) growth path is not the essential property of a mature modern capitalistic economy which is the framework for the Keynesian origin of the H-D growth theory, but is entirely **exclusive property** of these models because of their implicit technological assumption of **fixed coefficients of production** with no factor substitutability. In particular, Solow proceeds to show that the so-called **knife-edge problem** arising out of the **warranted rate of growth** diverging from the **natural rate of growth** is a consequence of the assumption of fixed proportions and that once this assumption is dropped and a **neo-classical production function** with factor substitution is assumed, a stable balanced growth path exists no matter whatever the natural rate of growth of labour force may be. In the H-D model the **capital-labour ratio** is a constant. The **isoquants** are L-shaped and there is no possibility of **substitution** between capital and labour. Solow argues that if this assumption is dropped and instead if it is assumed that capital and labour can be employed in **different proportions**, then the **warranted rate of growth** cannot diverge from the **natural rate of growth**. Whenever there is a divergence between the two the former will move towards the latter and in this way the **equality** between them will be maintained automatically. ## 5.2. Assumptions of the Model The Solow model is important because in the later growth models this model has been used as the basis. For this reason we shall consider this model in details. The model is based on the following assumptions: 1. The economy produces a single homogeneous commodity which can be used both as wage good and capital good. It is denoted by Y. When we **assume that there is only one good**, it implies that capital is "malleable"-- that is to say, capital stock can be adapted **without difficulty** to more or less capital-intensive technique of production. 2. The **saving ratio** in terms of this commodity is constant. In other words, a constant **marginal** (equal to average) propensity to save denoted by s is assumed. The saving function of the economy can be written as S = sY where S represents total saving and s is a constant, 0<s<1. 3. The **supply of labour** is growing at an exogenously given rate, n. The labour supply function can be written as $L_t = L_0e^{nt}$ where $L_t$ is the labour supply at any period t, and $L_0$ is the initial labour supply. Note that if $n = 0$, then $L_t = L_0$, which means that the labour supply is constant. 4. Output is produced with the help of two factors of production-- labour and capital. The production function can be written as Y=F(K,L) where K stands for capital stock. This production function is homogeneous of degree one, that is, subject to constant returns to scale. The production function is such that capital and labour can be substituted in the production process. The **marginal productivity** of each factor is positive but diminishing. This means $\frac{\partial Y}{\partial K} > 0$, $\frac{\partial Y}{\partial L} > 0$, $\frac{\partial^2 Y}{\partial K^2} < 0$ and $\frac{\partial^2 Y}{\partial L^2} < 0$. Production function is also assumed to be **"well behaved"**. Point-input flow-output technology is used. 5. In **equilibrium** the available labour supply is fully employed and capital and labour grow at the same rate. Equilibrium in the commodity market requires the equality of saving and investment. Investment is the net addition to the capital stock and is denoted by $\frac{dK}{dt}$ or K. 6. The economy is a **closed economy** and the real wage rate and the **rental on capital** are perfectly flexible. 7. There is **no depreciation of capital**. 8. There is **no technical progress**. 9. All variables are **continuous functions of time (t)**. ## 5.3. Properties of Neo-classical Production Function In the one-commodity neo-classical model it is assumed that there is only one homogeneous commodity usable both for consumption and as a perfectly malleable capital good. Output and capital input are then alike, both being measured in real terms, i.e. in terms of the single homogeneous commodity. Returns to factors i.e. profits and wages are also measured in terms of this commodity. Only labour is left to be measured separately, in its natural units, e.g. man hours. On the assumption that the inputs K of capital and L of labour are continuously variable and continuously substitutable in production and that to each combination (K, L) there corresponds a unique output Y, we can write the production function $Y = F(K, L)$ defined on $K \geq 0$, $L \geq 0$....(1). The restrictions imposed on the production function are that it is continuous and twice differentiable. Denoting partial derivatives by $\frac{\partial Y}{\partial K} = F_K$ and $\frac{\partial Y}{\partial L} = F_L$ and similarly for those of second order, we assume : $F_K > 0$, $F_{KK} < 0$, $F_L > 0$ and $F_{LL} < 0$. The partial derivative $\frac{\partial Y}{\partial K}$ represents marginal productivity of capital and the partial derivative $\frac{\partial Y}{\partial L}$ represents marginal productivity of labour. $F_{KK} < 0$ implies that marginal productivity of capital ($F_K$) decreases as more units of capital are employed in the production process, labour remaining the same. Similar is the interpretation of $F_{LL} < 0$. Thus the production function is assumed to be subject to diminishing return to each factor. It is also assumed that the production function $Y = F (K, L)$ is linear homogeneous i.e. subject to constant returns to scale. This means $F(\lambda K, \lambda L) = \lambda F(K,L)$ for all $\lambda > 0$.....(2) It also means from Euler's theorem that $F_K.K+F_L.L=Y$ for all K and L (3) The interpretation of (3) is familiar : total product will be exhausted if each factor is paid according to its marginal productivity. From (2) we can get the production function in per capita terms. Suppose we put $\lambda = \frac{1}{L}$ in (2). Then we get $\frac{Y}{L} ( \frac{1}{L} ) = F ( \frac{K}{L}, 1 ) = \frac{1}{L} F(K,L)$ Suppose we put $y = \frac{Y}{L}$, $k = \frac{K}{L}$ and $f(k) = F(k, 1)$. Then the production function can be written as $y = f(k)$....(4) Equation (4) represents the production function in per capita terms. Here y is $\frac{Y}{L}$ or per capita output and $k = \frac{K}{L}$ represents per capita capital stock. Thus production function in per capita terms shows that per capita output is a function of per capita capital stock. The marginal productivities of capital and labour can be expressed in terms of this production function in per capita terms. Since $\frac{Y}{L} = f(k)$, $\frac{\partial Y}{\partial K} = L.f'(k) = f'(k)$ or, $\frac{\partial Y}{\partial x} = f'(k)$. Thus, f'(k) represents the marginal productivity of capital. Again $\frac{\partial Y}{\partial L} = f(k) + L.f'(k)$. = $f(k) + L.f'(k)(\frac{K}{L})$ = $f(k) - f(k)$ or, $\frac{\partial Y}{\partial L} = f(k)-k.f'(k)$. This gives the marginal productivity of labour. Under conditions of perfect competition the rate of profit (r) will be equal to marginal productivity of capital and the real wage rate (w) will be equal to the marginal productivity of labour. Hence $r=f'(k)$ and $w=f(k)-kf'(k)$. ## 5.4. Possibility of Steady State The equations of the model can be written as follows : The production function: $Y = F (K, L)$ ........(1) The saving function: $S = sY$ .........(2) The equilibrium condition in the market for commodities: I = S or, $K = s Y$ .(3) The labour supply function: $L = L_o e^{nt}$ .......(4) Equations (1) - (4) can be transformed into a single equation $K = sF(K, L_o e^{nt})$.............(5) This equation determines the time path of capital accumulation that must be followed if all available labour is to be fully employed. In other words, equation (5) is a differential equation in the single variable K(t). Its solution gives the time path of capital stock which will fully employ the available labour. Let us see how economic growth proceeds from one period to another in this model. At any moment of time the available labour supply is given by equation (4) and the available stock of capital is a datum. Since the flexibility of wages and rentals will ensure full employment of labour and capital we can use the production function, equation (1), to find the current rate of output. Then the propensity to save determines the amount of net output saved and invested. Hence we know the net accumulation of capital during the current period. Added to the already accumulated stock this gives the capital available for the next period, and the whole process can be repeated. Solow now proves that there is always a capital accumulation path consistent with any rate of growth of labour force. In other words, a balanced capital accumulation path exists on which capital and labour grow at the same rate. To see how this happens let us introduce a new variable k = $\frac{K(t)}{L(t)}$ which represents the capital-labour ratio or a measure of capital intensity. Then $K(t) = kL(t) = k. L_o e^{nt}$ Differentiating both sides with respect to t $\frac{dK(t)}{dt} = kL_o e^{nt} + k.n. L_o e^{nt}$ or, $ K = L_o e^{nt} ( k + nk )$ But by (5) $K = sF(K, L_o e^{nt})$ .:. $sF(K, L_o e^{nt}) = L_o e^{nt} ( k + nk )$ Since the production function is homogeneous of degree one, we can divide each variable in F by L provided we multiply the function by L. Then $sf(k) = F ( \frac{K}{L}, 1 ) = L_o e^{nt} [\frac{K}{L} + n \frac{K}{L}]$ or, $sf(k) = (k + nk)$ where $k = \frac{K}{L}$ or, $k = sf(k) -nk$ ......(6) This is a differential equation in the capital-labour ratio (k) alone. This equation will give us the time path of the capital-labour ratio. The differential equation (6) can also be obtained in an alternative way. K = $k$ Since L .. $log k = log K - log L$ Differentiating with respect to t, $\frac{1}{k} \frac{dk}{dt} = \frac{1}{K} \frac{dK}{dt} - \frac{1}{L} \frac{dL}{dt}$ or, $\frac{dk}{dt} = \frac{k}{K} \frac{dK}{dt} - \frac{k}{L} \frac{dL}{dt}$ But $ \frac{dK}{dt} = \frac{K}{K} sf(k) = sf (k)$ Hence $\frac{dk}{dt} = sf (k) -nk$ or, $k= sf(k)- nk$ Now divide L out of F(K, L) as before to get $k = \frac{1}{L} . K -nk$ or, $k = sf(k)-nk$ which is the same equation as obtained before. The function f(k) is easy to interpret. From the production function $Y = F(K, L)$ we get $\frac {Y}{L} = F ( \frac{K}{L}, 1) = f(k)$. Thus f(k) shows $\frac{Y}{L}$ or output per worker as a function of capital per worker. It is also the total product curve as varying amount k of capital are employed with one unit of labour. Equation (6) states that the rate of change of **capital-labour ratio** is the difference of two terms, one representing the **increment of capital** and one the **increment of labour**. In order that we may have balanced growth, it is necessary to fulfill the condition that capital stock increases at the same rate as the rate of increase of labour force. On the balanced growth path $k = \frac{K}{L} $. This means that $\frac {dK}{dt} = \frac {K}{L}= 0$, that is, $k = 0$ which implies $k = 0$. This means that the capital-labour ratio remains constant on the balanced growth path. If $k = 0$, equation (6) can be written as $sf(k) = nk$....(7). This is the condition of **steady state equilibrium** in Solow's model of growth. Let us now interpret this condition with the help of a diagram (figure 5.2). In the figure we plot capital per man (k) on the horizontal axis and output per man on the vertical axis. We then get f(k) curve. Let us suppose that this curve is concave to the origin and passes through the origin. Since s is a constant fraction, we can find sf(k) curve from the f(k) curve. This curve will lie below the f(k) curve. Let us also plot nk against k. It will be a straight line through the origin. Now condition (7) is satisfied at point P where the sf(k) curve intersects the nk line and $k = k^*$. Thus at k = k* steady state growth exists. Let us define a function $(k)$ such that $\phi (k) = sf(k) -nk$. In fact $\phi (k) = k$ so that $\phi(k) = 0$ is the condition of steady state equilibrium. By plotting the vertical distances between sf(k) and nk we can get the $(k)$ curve. This curve is also shown in figure 5.2. The point where $(k)$ curve intersects the horizontal axis is the point where $(k)=0$. At this point $k=k^*$. This is the steady state equilibrium value of capital-labour ratio, k. At this equilibrium point P, all the macro variables grow at the same rate. Since $sf(k) = nk$, $\frac{Y}{L} = \frac{S}{L} = \frac{s Y}{L} = \frac{sf(k)}{k} = n$ or, $\frac{\dot{Y}}{Y} = n$ or, $\frac{\dot{K}}{K} = n$ or, $\frac{I}{K} = n$ or, $\frac{\dot{K}}{L} =n$. Further, by assumption $\frac{\dot{L}}{L} = n$. Since the production function is homogeneous of degree one, output will also grow at the same rate, n. This can be proved as follows : From the production function $Y = F (K, L)$ we get $dY=F_K dK + F_L.dL$ $\frac{dY}{dt} = F_K \frac{dK}{dt} + F_L . \frac{dL}{dt}$ or, $\dot{Y} = F_K. K + F_L.L$ $ \frac {\dot{Y}}{Y} = \frac{F_K.K}{Y}+ \frac{F_L.L}{Y} = \frac {F_K.K}{Y} \frac{K}{K} + \frac {F_L.L}{Y}\frac{L}{L} = \frac {F_K.K}{Y.K} + \frac {F_L.L}{Y.L} = \frac{(F_K.K+F_L.L)}{Y} = \frac{Y}{Y}=n$ .:. We get $\frac{\dot{Y}}{Y}=n$. Again $S = \frac{SY}{Y}$. Therefore $s = \frac{\dot{S}}{Y}$ and $\frac{\dot{S}}{S} = \frac{SY}{S} \frac{\dot{Y}}{Y} = \frac{SY}{SY} \frac{\dot{Y}}{Y}$. Thus we see that at point P, $\frac{\dot{K}}{K} = \frac{\dot{S}}{S} = \frac{\dot{Y}}{Y}$. This shows that all the macro variables grow at the same rate, n. Such a situation is known as the golden age growth path or steady state growth path. ## 5.5. Stability of Steady State Equilibrium For stability, consider the values $k_1$ and $k_2$ other than the equilibrium value $k^*$. At $k = k_1$, $sf(k) > nk$. By (6) this means that $\dot{k} > 0$. It implies that k increases towards k*. Similarly at $k = k_2$, $sf(k) < nk$ which implies that $\dot{k} <0$. Therefore k decreases towards k*. Thus the equilibrium value k* is stable. Even if we start from a value of k which is different from k*, we shall ultimately move towards k*. However, the strong stability in figure 5.2 is a consequence of the way in which the sf(k) curve has been drawn. Stability is ensured if the production function f(k), is "well behaved". According to Inada the production function f(k) is well behaved if (i) f(0) = 0 (ii) f(x) = $\infty$, (iii) f'(k)>0, (iv) f''(k)<0, (v) f'(0) = $\infty$ and (vi) f'(x)=0. This means that f'(k) is continuously downward sloping and asymptotic to both the axes. This condition may not always be fulfilled. We can therefore say that stability depends on the nature of the production function. Inada conditions can also be represented as follows : $Lt_{k\to 0} \frac{f(k)}{k} = \infty $ and $Lt_{k \to \infty} \frac{f(k)}{k} = 0$ Now the condition of steady state can be written as $\phi (k) = sf(k) - nk = 0$ or, $sf(k) = nk$ or, $s \frac{f(k)}{k} = n$. This shows that $\phi (k)$ has a positive slope at the origin. Further when $k \to \infty$, $f'(k) \to 0$ from the Inada condition. This shows that the slope of $\phi(k)$ will be negative for large enough values of k. It can also be seen that $\phi(k)$ is itself positive for small values of k but as k becomes large enough $\phi(k)$ will become negative. Now $\phi(k)$ can become negative if $f(k) < \frac{n}{s}$. But from L' Hospital's rule¹ we get $Lt_{k \to \infty} \frac{f(k)}{k} = Lt_{k \to \infty} f'(k)$. But $Lt_{k \to \infty} f'(k) = 0$ from the Inada condition. Hence $Lt_{k \to \infty} \frac{f(k)}{k} = 0$ which is clearly less than $\frac{n}{s}$. Thus we see that for some large value of k, $f(k) < \frac{n}{s}$ and therefore $\phi(k)$ is negative. Thus $\phi(k)$ is positive for some small values of k while $\phi(k)$ is negative for some large values of k. Hence if $\phi(k)$ is continuous, there must exist at least one value of k for which $\phi(k)$ is zero. That proves the result that there exists at least one value of k (say k*) for which $\phi(k)$ is zero. In our figure 5.2 we have drawn the $\phi(k)$ function also. This function is obtained by plotting the vertical differences between the sf(k) curve and the nk line. The $\phi(k)$ curve is first upward rising and then downward sloping. At k*, the $\phi(k)$ curve intersects the horizontal axis and therefore $\phi(k) = 0$ at this point. Alternatively, steady state equilibrium requires $\frac{f(k)}{k} = \frac{n}{s}$. Now $\frac{f(k)}{k} \to \infty $ and $\frac{f(k)}{k} \to 0$ as $k \to \infty$. Thus as k varies from $0$ to $\infty$, $\frac{f(k)}{k}$ varies from $\infty$ to 0. 'L' Hospital's rule states that if $Lt_{x \to a} \frac{g(x)}{h(x)} = \frac{0}{0}$ or $\frac{\infty}{\infty}$, then $Lt_{x \to a} \frac{g(x)}{h(x)} = Lt_{x \to a} \frac{g'(x)}{h'(x)}$ if $g'(x)$ and $h'(x)$ exist and are continuous. The equation determines the time path of capital accumulation that must be followed if all available labour is to be fully employed. In other words, equation (5) is a differential equation in the single variable K(t). Its solution gives the time path of capital stock which will fully employ the available labour. Let us see how economic growth proceeds from one period to another in this model. At any moment of time the available labour supply is given by equation (4) and the available stock of capital is a datum. Since the flexibility of wages and rentals will ensure full employment of labour and capital we can use the production function, equation (1), to find the current rate of output. Then the propensity to save determines the amount of net output saved and invested. Hence we know the net accumulation of capital during the current period. Added to the already accumulated stock this gives the capital available for the next period, and the whole process can be repeated. Solow now proves that there is always a capital accumulation path consistent with any rate of growth of labour force. In other words, a balanced capital accumulation path exists on which capital and labour grow at the same rate. To see how this happens let us introduce a new variable k = $\frac{K(t)}{L(t)}$ which represents the capital-labour ratio or a measure of capital intensity. Then $K(t) = kL(t) = k. L_o e^{nt}$ Differentiating both sides with respect to t $\frac{dK(t)}{dt} = kL_o e^{nt} + k.n. L_o e^{nt}$ or, $ K = L_o e^{nt} ( k + nk )$ But by (5) $K = sF(K, L_o e^{nt})$ .:. $sF(K, L_o e^{nt}) = L_o e^{nt} ( k + nk )$ Since the production function is homogeneous of degree one, we can divide each variable in F by L provided we multiply the function by L. Then $sf(k) = F ( \frac{K}{L}, 1 ) = L_o e^{nt} [\frac{K}{L} + n \frac{K}{L}]$ or, $sf(k) = (k + nk)$ where $k = \frac{K}{L}$ or, $k = sf(k) -nk$ ......(6) This is a differential equation in the capital-labour ratio (k) alone. This equation will give us the time path of the capital-labour ratio. The differential equation (6) can also be obtained in an alternative way. K = $k$ Since L .. $log k = log K - log L$ Differentiating with respect to t, $\frac{1}{k} \frac{dk}{dt} = \frac{1}{K} \frac{dK}{dt} - \frac{1}{L} \frac{dL}{dt}$ or, $\frac{dk}{dt} = \frac{k}{K} \frac{dK}{dt} - \frac{k}{L} \frac{dL}{dt}$ But $ \frac{dK}{dt} = \frac{K}{K} sf(k) = sf (k)$ Hence $\frac{dk}{dt} = sf (k) -nk$ or, $k= sf(k)- nk$ Now divide L out of F(K, L) as before to get $k = \frac{1}{L} . K -nk$ or, $k = sf(k)-nk$ which is the same equation as obtained before. The function f(k) is easy to interpret. From the production function $Y = F(K, L)$ we get $\frac {Y}{L} = F ( \frac{K}{L}, 1) = f(k)$. Thus f(k) shows $\frac{Y}{L}$ or output per worker as a function of capital per worker. It is also the total product curve as varying amount k of capital are employed with one unit of labour. Equation (6) states that the rate of change of **capital-labour ratio** is the difference of two terms, one representing the **increment of capital** and one the **increment of labour**. In order that we may have balanced growth, it is necessary to fulfill the condition that capital stock increases at the same rate as the rate of increase of labour force. On the balanced growth path $k = \frac{K}{L} $. This means that $\frac {dK}{dt} = \frac {K}{L}= 0$, that is, $k = 0$ which implies $k = 0$. This means that the capital-labour ratio remains constant on the balanced growth path. If $k = 0$, equation (6) can be written as $sf(k) = nk$....(7). This is the condition of **steady state equilibrium** in Solow's model of growth. Let us now interpret this condition with the help of a diagram (figure 5.2). In the figure we plot capital per man (k) on the horizontal axis and output per man on the vertical axis. We then get f(k) curve. Let us suppose that this curve is concave to the origin and passes through the origin. Since s is a constant fraction, we can find sf(k) curve from the f(k) curve. This curve will lie below the f(k) curve. Let us also plot nk against k. It will be a straight line through the origin. Now condition (7) is satisfied at point P where the sf(k) curve intersects the nk line and $k = k^*$. Thus at k = k* steady state growth exists. Let us define a function $(k)$ such that $\phi (k) = sf(k) -nk$. In fact $\phi (k) = k$ so that $\phi(k) = 0$ is the condition of steady state equilibrium. By plotting the vertical distances between sf(k) and nk we can get the $(k)$ curve. This curve is also shown in figure 5.2. The point where $(k)$ curve intersects the horizontal axis is the point where $(k)=0$. At this point $k=k^*$. This is the steady state equilibrium value of capital-labour ratio, k. At this equilibrium point P, all the macro variables grow at the same rate. Since $sf(k) = nk$, $\frac{Y}{L} = \frac{S}{L} = \frac{s Y}{L} = \frac{sf(k)}{k} = n$ or, $\frac{\dot{Y}}{Y} = n$ or, $\frac{\dot{K}}{K} = n$ or, $\frac{I}{K} = n$ or, $\frac{\dot{K}}{L} =n$. Further, by assumption $\frac{\dot{L}}{L} = n$. Since the production function is homogeneous of degree one, output will also grow at the same rate, n. This can be proved as follows : From the production function $Y = F (K, L)$ we get $dY=F_K dK + F_L.dL$ $\frac{dY}{dt} = F_K \frac{dK}{dt} + F_L . \frac{dL}{dt}$ or, $\dot{Y} = F_K. K + F_L.L$ $ \frac {\dot{Y}}{Y} = \frac{F_K.K}{Y}+ \frac{F_L.L}{Y} = \frac {F_K.K}{Y} \frac{K}{K} + \frac {F_L.L}{Y}\frac{L}{L} = \frac {F_K.K}{Y.K} + \frac {F_L.L}{Y.L} = \frac{(F_K.K+F_L.L)}{Y} = \frac{Y}{Y}=n$ .:. We get $\frac{\dot{Y}}{Y}=n$. Again $S = \frac{SY}{Y}$. Therefore $s = \frac{\dot{S}}{Y}$ and $\frac{\dot{S}}{S} = \frac{SY}{S} \frac{\dot{Y}}{Y} = \frac{SY}{SY} \frac{\dot{Y}}{Y}$. Thus we see that at point P, $\frac{\dot{K}}{K} = \frac{\dot{S}}{S} = \frac{\dot{Y}}{Y}$. This shows that all the macro variables grow at the same rate, n. Such a situation is known as the golden age growth path or steady state growth path. ## 5.5. Stability of Steady State Equilibrium For stability, consider the values $k_1$ and $k_2$ other than the equilibrium value $k^*$. At $k = k_1$, $sf(k) > nk$. By (6) this means that $\dot{k} > 0$. It implies that k increases towards k*. Similarly at $k = k_2$, $sf(k) < nk$ which implies that $\dot{k} <0$. Therefore k decreases towards k*. Thus the equilibrium value k* is stable. Even if we start from a value of k which is different from k*, we shall ultimately move towards k*. However, the strong stability in figure 5.2 is a consequence of the way in which the sf(k) curve has been drawn. Stability is ensured if the production function f(k), is "well behaved". According to Inada the production function f(k) is well behaved if (i) f(0) = 0 (ii) f(x) = $\infty$, (iii) f'(k)>0, (iv) f''(k)<0, (v) f'(0) = $\infty$ and (vi) f'(x)=0. This means that f'(k) is continuously downward sloping and asymptotic to both the axes. This condition may not always be fulfilled. We can therefore say that stability depends on the nature of the production function. Inada conditions can also be represented as follows : $Lt_{k\to 0} \frac{f(k)}{k} = \infty $ and $Lt_{k \to \infty} \frac{f(k)}{k} = 0$ Now the condition of steady state can be written as $\phi (k) = sf(k) - nk = 0$ or, $sf(k) = nk$ or, $s \frac{f(k)}{k} = n$. This shows that $\phi (k)$ has a positive slope at the origin. Further when $k \to \infty$, $f'(k) \to 0$ from the Inada condition. This shows that the slope of $\phi(k)$ will be negative for large enough values of k. It can also be seen that $\phi(k)$ is itself positive for small values of k but as k becomes large enough $\phi(k)$ will become negative. Now $\phi(k)$ can become negative if $f(k) < \frac{n}{s}$. But from L' Hospital's rule¹ we get $Lt_{k \to \infty} \frac{f(k)}{k} = Lt_{k \to \infty} f'(k)$. But $Lt_{k \to \infty} f'(k) = 0$ from the Inada condition. Hence $Lt_{k \to \infty} \frac{f(k)}{k} = 0$ which is clearly less than $\frac{n}{s}$. Thus we see that for some large value of k, $f(k) < \frac{n}{s}$ and therefore $\phi(k)$ is negative. Thus $\phi(k)$ is positive for some small values of k while $\phi(k)$ is negative for some large values of k. Hence if $\phi(k)$ is continuous, there must exist at least one value of k for which $\phi(k)$ is zero. That proves the result that there exists at least one value of k (say k*) for which $\phi(k)$ is zero. In our figure 5.2 we have drawn the $\phi(k)$ function also. This function is obtained by plotting the vertical differences between the sf(k) curve and the nk line. The $\phi(k)$ curve is first upward rising and then downward sloping. At k*, the $\phi(k)$ curve intersects the horizontal axis and therefore $\phi(k) = 0$ at this point. Alternatively, steady state equilibrium requires $\frac{f(k)}{k} = \frac{n}{s}$. Now $\frac{f(k)}{k} \to \infty $ and $\frac{f(k)}{k} \to 0$ as $k \to \infty$. Thus as k varies from $0$ to $\infty$, $\frac{f(k)}{k}$ varies from $\infty$ to 0. 'L' Hospital's rule states that if $Lt_{x \to a} \frac{g(x)}{h(x)} = \frac{0}{0}$ or $\frac{\infty}{\infty}$, then $Lt_{x \to a} \frac{g(x)}{h(x)} = Lt_{x \to a} \frac{g'(x)}{h'(x)}$ if $g'(x)$ and $h'(x)$ exist and are continuous.