SS 1.1 Basic Facts
The main conclusion of the Solow model is that accumulation of physical capital cannot account for either the vast growth over time in output per person or the vast geographic differences in output per person.
SS 1.2 Assumptions of the Solow Model
Model: $Y(t)= F\left( K\left(t\right), A\left(t\right) \cdot L\left(t\right) \right)$
where
Y= output
K= capital
A= knowledge (effectiveness of labor)
L= labor
t= time
- At any given time the economy has some level of $A$ and $L$.
- t does not enter the production function $F$ directly, it enters $F$ via $K$, $A$, and $L$. Hence, $Y$ changes over time only if $K$, $A$, or $L$ change over time.
- For given level of inputs, $Y$ increases (we have technological progress) only if $A$ increases. (check this)
- $A$ and $L$ enter multiplicatively as $AL$, which is labeled effective labor. Technological progress that enters $F$ as $AL$ is known as labor augmenting or Harrod-Neutral.
- Technological progress can enter $F$ in other ways
- $Y=F\left( AK,L \right)$, Capital Augmenting $Y=AF\left(K,L\right)$, Hicks-Neutral
- The assumption that $A$ enters $F$ as $AL$ will imply that the capital output ratio $K/Y$ settles down (behaves constant); in practice there is no upward or downward trend in $K/Y$. This assumption makes the analysis much easier.
####Assumptions Concerning F * Constant returns to scale: $F\left( cK, cAL \right)=cF\left( K, AL \right)$, for all $c\geq 0$. For example, doubling K and L (i.e. c=2), holding $A$ fixed, also doubles output. The assumption of constant returns to scale breaks down into two smaller assumptions: 1. The economy is large enough that gains from specialization have been exhausted. If the economy were small we would expect that further specialization would more than double output as we double $K$ and $L$. Hence, we expect that extra $K$ and $L$ will be used in the same way so that no extra gains are obtained from specialization. 2. Inputs other than capital, labor, and knowledge are not that important. In particular we are neglecting land and natural resources. If natural resources are important then doubling $K$ and $L$ can result in less than double output. In practice, natural resources do not appear to constraint growth. Hence, this assumption seems to be a reasonable approximation. Constant Returns assumption allows us to work with the production function in intensive form: Let $c=1/AL$ $\Rightarrow cF(K,AL)= F(cK,cAL)=F(K/AL, 1)$ That is, **EQ 1.3:** $F(K/AL, 1)= \dfrac{1}{AL} F(K,AL)$ * K/AL is capital per unit of effective labor. * Y/AL=F/AL is output per unit of effective labor. Let $k=K/AL$, $y=Y/AL$, and $f(k)=F(k,1)$. Then, $y=Y/AL=\dfrac{F(K,AL)}{AL}=F(K/AL,1)=F(k,1)=f(k)$. We can write equation 1.3 as output per unit of effective labor as a function of capital per unit of effective labor. **EQ 1.4:** $y=f(k)$. * It's easier to analyze the behavior of $k$ than to analyze the behavior of $K$ and $AL$. We study $k$ to learn about the behavior of $K$ and $AL$. **Note:** Equation 1.4 divides the economy into $AL$ small economies that depend on one unit of effective labor and $K/AL$ units of capital. Since $F$ has constant returns to scale, each of these economies produces $1/AL$ as much output as the large economy. Hence, the amount of output per unit of effective labor depends only on the quantity of capital per unit of effective labor and not on the over all size of the economy. ####Intensive Production Function Assumptions * $f(0)=0$ * $f'(k) > 0$ * $f''(k) < 0$ Using the chain rule and the fact that $F(K,AL)=AL f(K/AL)$, we obtain the following $\dfrac{\partial F(K,AL)}{\partial K}$ $=AL \dfrac{df(K/AL)}{dK} \cdot \dfrac{d(K/AL)}{dK}$ $=ALf'(K/AL) \cdot 1/AL \cdot \dfrac{dK}{dK}$ $=\dfrac{AL}{AL}f'(K/AL)$ $ = f'(K/AL)$ $=f'(k)$ Thus, the assumption that $f'(k)$ is positive and $f'(k)$ is negative imply that the marginal product of capital is positive, but that it declines as capital (per unit of effective labor) rises. * Inada Conditions: $\displaystyle \lim_{k \to 0} f'(k)= \infty$ $\displaystyle \lim_{k \to \infty} f'(k)= 0$ This assumption says that the marginal product of capital (MPK) is very large when the capital stock is sufficiently small (small amounts of capital go along way in this interval) and MPK becomes very small as capital stock becomes large (we might not be able to utilize such large amounts of capital). These assumptions ensure that the path of the economy does not diverge. ##Insert Figure 1.1 ####The Cob-Douglas example: $F=(K, AL)= K^{\alpha}(AL)^{1-\alpha}$, for $0 < \alpha <1 a="" actual="" alpha-1="" alpha="" approximation="" assumption:="" c-d:="" c-d="" c="" cal="" cf="" check="" ck="" constant="" decent="" dfrac="" f="" find="" form="" function="" functions.="" intensive="" is="" k="" left="" marginal="" of="" product:="" production="" returns="" right="" some="">0$ $f''(k)=\alpha (\alpha-1)k^{\alpha-2}= -\alpha(1-\alpha)k^{\alpha-2}
