经济代写|宏观经济学代写Macroeconomics代考|Connecting Comparative Statics to Stability Analysis

Doug I. Jones

Doug I. Jones

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经济代写|宏观经济学代写Macroeconomics代考|Connecting Comparative Statics to Stability Analysis

Samuelson’s (1941) Econometrica article had a profound impact on the development of dynamics and stability analysis. ${ }^{16}$ On the one hand, the article clarified the Oxford discussion by providing the first stability analysis of a Keynesian system, which was called for by Frisch and Tinbergen. On the other hand, in connecting comparative statics to stability analysis through his “correspondence principle” (Samuelson, 1947: 5), ${ }^{17}$ Samuelson redirected the debate toward the problem of the stability of full employment equilibrium, pointing to a direction soon followed by Lange (see Chap.9).

Before we see, through the example of the “Keynesian system” presented by Samuelson, how the correspondence principle linked together comparative statics and stability analysis, we present his approach to stability, which is very similar to the econometricians’ and in contrast with Meade’s approach. Samuelson argued that the solution to the problem of stability, “presupposes a theory of dynamics,” namely a theory which determines the adjustment behavior of all variables outside of the equilibrium. He began with a very general approach to show what dynamic analysis could bring to comparative statics. The first sections were thus devoted to the examination of different adjustment mechanisms between supply and demand, from which Samuelson could derive different stability conditions which yielded “meaningful theorems” about the slope of the demand and supply curves. In doing so, he extended the arguments raised by Tinbergen and Frisch against Meade, who had argued that changing the dynamic hypotheses implicit behind his stability analysis could change the theorems obtained. Thus Samuelson underlined, like Tinbergen had done privately with Meade, that “[i]f alternative dynamic models are postulated, completely different conditions are deduced, which in turn lead to alternative theorems in comparative statics” (Samuelson, 1941: 103).

Samuelson argued that with $n$ time-dependent variables to explain $x_1(t), \ldots, x_n(t)$ and $n$ functional equations of the general form $f^i\left(x_1(t), \ldots, x_n(t)\right)$, then their behavior was determined once certain initial conditions are specified. This was made with explicit reference to Frisch’s methodology exposed in Frisch (1936), for instance when he argued that a set of equilibrium values $x_1^0, \ldots, x_n^0$ will satisfy the equations $f^i\left(x_1^0, \ldots, x_n^0\right)=0$ for all times $t$ (Samuelson, 1941: 100) .

For such equilibrium states, the system may be displaced, a displacement being equivalent to an arbitrary change in the initial conditions. Samuelson introduced four different types of stability, two “kinds” which could themselves hold “perfectly” or only “in the small.” Perfect stability of the first kind meant that once displaced, all the variables would approach their equilibrium values in the limit as time became infinite. When the system was only stable of the first kind in the small, equilibrium would be restored only for small displacements away from the equilibrium, highlighting the possibility that in the presence of multiple equilibria, large displacements may definitely destabilize the economy, an idea that Tinbergen had explored in several models (as it has been shown in Chap. 6). Stability in the second kind was related to the behavior of conservative systems showing undamped fluctuations, and was again divided into perfect stability of the second kind and stability of the second kind in the small. In the first case, the limit cycle (to use the modern term for those undamped cycles) was obtained from any initial conditions, while in the second case, the limit cycle was again enclosed in a stability “corridor,” outside of which it could not be attained.

经济代写|宏观经济学代写Macroeconomics代考|Meade’s Conditions Compared to Samuelson’s

Differentiating conditions (8.1) to (8.7), Meade found that his system was stable when the expectation elasticity $\pi$ was lower than the proportion of income going to profits $1-\lambda, \lambda$ being the profit share going to wages. In a second case, when the rate of interest is assumed to vary, the stability condition is less severe. This is because the rise in the interest rate resulting from the rise in production limits the rise in investment and the magnitude of the disequilibrium between the interest rate and the marginal efficiency of capital.

Our strategy for deriving a geometrical representation of the “core” of the model is to construct two schedules in the $\left(p_i, y_i\right)$ space, with $p_i$ the supply ( $\left.p_i^s\right)$ or demand $\left(p_i^d\right)$ price of investment goods and $y_i$ the output of investment goods. The first of these schedules is given by equations (8.1) and (8.2) and a production function that we can write as $y_i=A n_i^\lambda$ :
p_i^s=\frac{1}{\lambda} A^{\frac{-1}{\lambda}} W y_i^{\frac{1-\lambda}{\lambda}}
Equation 8.23 represents the short-run aggregate supply curve of investment goods, with parametric wages, or what Keynes called the investment supply price, that is, the price for which producers of investment goods are ready to start to produce.

The $p_i^d$ line results from the expression of profits in terms of the output of investment goods. If we assume that $P=(1-\lambda) Y$, in line with Meade’s hypotheses, we have $P=(1-\lambda) \frac{p_i{ }^d y_i}{s}$. By inserting it in the condition of equilibrium between the marginal efficiency of capital and the interest rate $r=\frac{E(P)}{p_i{ }^4}$ we obtain:
p_i^d=\frac{1}{r}\left[(1-\lambda) \frac{p_i{ }^d y_i}{s}\right]^\pi=r^{\frac{-1}{1-\pi}}\left(\frac{(1-\lambda)}{s}\right)^{\frac{\pi}{1-\pi}} y_i^{\frac{\pi}{1-\pi}},
where we have assumed $E(P)=P^\pi$, with $\pi$ is the elasticity of expectations, that is, the expectation of future profits depends on current profits. This schedule can be interpreted as follows: given this price and the level of output $y_i$, the ordinate of the $p_i^d$ curve shows the “investment demand price,” that is the price that investors are ready to pay to purchase capital equipment. Equilibrium is determined at the intersection of both curves. ${ }^{30}$


经济代写|宏观经济学代写Macroeconomics代考|Connecting Comparative Statics to Stability Analysis

萨缪尔森 (Samuelson) (1941) 的计量经济学文章对动力学和稳定性分析的发展产生了深远的影响。16一方面,这篇文章通过提供凯恩斯系统的第一个稳定性分析澄清了牛津的讨论,这是 Frisch 和 Tinbergen 所呼吁的。另一方面,在通过他的“对应原则”(Samuelson,1947:5)将比较静态与稳定性分析联系起来时,17 Samuelson 将争论转向了充分就业均衡的稳定性问题,指出Lange 紧随其后(见第 9 章)。

在我们看到之前,通过萨缪尔森提出的“凯恩斯系统”的例子,对应原理如何将比较静态和稳定性分析联系在一起,我们介绍了他的稳定性方法,这与计量经济学家的方法非常相似,与米德的方法形成对比. 萨缪尔森认为,稳定性问题的解决方案“以动力学理论为前提”,即决定平衡之外所有变量的调整行为的理论。他从一种非常普遍的方法开始,展示了动态分析可以为比较静态学带来什么。因此,第一部分专门研究供需之间的不同调整机制,从中萨缪尔森可以推导出不同的稳定性条件,这些条件产生了关于需求和供给曲线斜率的“有意义的定理”。在这样做的过程中,他扩展了丁伯根和弗里施提出的反对米德的论点,米德认为改变他的稳定性分析背后隐含的动态假设可能会改变所获得的定理。因此,萨缪尔森强调,就像丁伯根私下与米德所做的那样,“[i]如果假设了替代动态模型,则推导出完全不同的条件,这反过来导致比较静态中的替代定理”(萨缪尔森,1941:103)。

萨缪尔森认为,用 $n$ 时间相关变量来解释 $x_1(t), \ldots, x_n(t)$ 和 $n$ 一般形式的函数方程 $f^i\left(x_1(t), \ldots, x_n(t)\right)$ ,那么一旦指定了某些初始条 件,它们的行为就确定了。这是明确参考 Frisch (1936) 中公开的 Frisch 方法论,例如当他论证一组平衡值 $x_1^0, \ldots, x_n^0$ 将满足方程 $f^i\left(x_1^0, \ldots, x_n^0\right)=0$ 对于所有 时间 $t$ (Samuelson, 1941: 100)。
对于这样的平衡状态,系统可能会发生位移,位移相当 于初始条件的任意变化。萨缪尔森介绍了四种不同类型 的稳定性,两种“种类”本身可以“完美地”保持或只能“在很 小的范围内”保持。第一类完全稳定性是指一旦发生位 移,随着时间变得无限大,所有变量都将在极限内趋近 于它们的平衡值。当系统只有小的第一类稳定时,只有 小的偏离平衡的位移才会恢复均衡,突出了在多重均衡 存在的情况下,大的位移肯定会破坏经济稳定的可能 性,这是丁伯根的想法已经在几个模型中进行了探索 (如第 6 章所示) 。第二类稳定性与保守系统表现出无 阻尼波动的行为有关,又分为完全第二类稳定性和小第 二类稳定性。在第一种情况下,极限环 (使用现代术语 来表示那些无阻尼循环)是从任何初始条件获得的,而 在第二种情况下,极限环再次包含在稳定性“走廊”中,在 该走廊之外它可以达不到。

经济代写|宏观经济学代写Macroeconomics代考|Meade’s Conditions Compared to Samuelson’s

对条件 (8.1) 到 (8.7) 进行微分,米德发现当预期弹性低 于收入占利润的比例是利润占工资的比例。在第二种情 况下,当假设利率发生变化时,稳定性条件不那么严 格。这是因为生产增加引起的利率上升限制了投资的增 加以及利率与资本边际效率之间不均衡的程度。 $\pi$ $1-\lambda, \lambda$
我们推导模型“核心”的几何表示的策略是在空间中构建两 个时间表,其中是供应 (或需求投资品的价格和投资品的 产量。生产函数给出: $\left.\left(p_i, y_i\right) p_i p_i^s\right)\left(p_i^d\right) y_i$ $y_i=A n_i^\lambda$
p_i^s=\frac{1}{\lambda} A^{\frac{-1}{\lambda}} W y_i^{\frac{1-\lambda}{\lambda}}
方程式 8.23 表示投资品的短期总供给曲线,其中包含参 数工资,或者屾恩斯所说的投资供给价格,即投资品生 产商准备开始生产的价格。
p_i线是根据投资货物的产出来表示利润的结果。如果我 们假设与米德的假设一致,我们有。将其代入资本边际 效率与利率均衡的条件下,我们得到: 假设,其中 $p_i^d$
& P=(1-\lambda) Y P=(1-\lambda) \frac{p_p{ }^d y_i}{s} r=\frac{E(P)}{p_i{ }^4} \
& p_i^d=\frac{1}{r}\left[(1-\lambda) \frac{p_i{ }^d y_i}{s}\right]^\pi=r^{\frac{-1}{1-\pi}}\left(\frac{(1-\lambda)}{s}\right)^{\frac{\pi}{1-\pi}} y_i^{\frac{\pi}{1-\pi}},
$$p_i线是根据投资货物的产出来表示利润的结果。如果我 们假设与米德的假设一致,我们有。将其代入资本边际 效率与利率均衡的条件下,我们得到: 假设,其中 $p_i^d$
& P=(1-\lambda) Y P=(1-\lambda) \frac{p_i{ }^d y_i}{s} r=\frac{E(P)}{p_i{ }^4} \
& p_i^d=\frac{1}{r}\left[(1-\lambda) \frac{p_i{ }^d y_i}{s}\right]^\pi=r^{\frac{-1}{1-\pi}}\left(\frac{(1-\lambda)}{s}\right)^{\frac{\pi}{1-\pi}} y_i^{\frac{\pi}{1-\pi}},
$E(P)=P^\pi \pi$ 是期望的弹性,即对末来利润的期望取
决于当前利润。该表可以解释如下: 给定该价格和产出
买资本设备的价格。平衡在两条曲线的交点处确定。 $y_i$ $p_i^{d 30}$

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术语 广义线性模型(GLM)通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归,以及方差分析和方差分析(仅含固定效应)。



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