经济代写|宏观经济学代写Macroeconomics代考|ECON1120

2022年12月29日

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经济代写|宏观经济学代写Macroeconomics代考|Stability and Cycles in Kalecki’s Model

Based on Kalecki’s conditions for cycles and an analysis of the stability conditions of the model, we can distinguish four regions corresponding to cycles or growth and to stability and instability in the plane given by the two parameters $m$ and $n$, and parameterized by $\theta{ }^{38}$ There are three different cases with two subcases for growth and one general case when we have cycles, each of these cases being mutually exclusive. The only restrictive assumption that we will keep here is that $\theta>0$, which amounts to saying that the production cannot be delivered before it is decided. These different cases allow us to find the equations separating the plane into the four regions shown in Fig. 5.1.

The graph in Fig. $5.1$ represents in red the stable regions and in gray the unstable regions, in dark the regions yielding cycles and in bright the regions yielding growth. Thus region A corresponds to damped cycles, region D to undamped cycles, while region B corresponds to solutions giving a stable growth path and region C gives an unstable growth path. This holds for $\theta=0.6$, and when this value is modified the frontiers between the regions are adjusted accordingly. ${ }^{39}$ We see again that $n>0$ is a necessary condition for stability, something that does not change as long as the lag remains positive (for the other stability conditions giving the equations of the separating lines in the above figure, see Appendix 1).

Kalecki (1936) did not exclude that $m$ could change, as evidenced by his reply to Frisch and his 1939 Essays, but he restricted it to a positive value, which meant that a positive change in profits could not cause a fall in investment decisions. In that case, we see that the region where the model is stable is rather small while the rest of the northeast quadrant gives rise to unstable solutions. Kalecki restricted the solution space even more by considering only the combinations on the line separating regions A and D, which correspond to self-sustained oscillations.

Kalecki rejected exponential solutions because he was only interested in the cyclical properties of the economy. In addition, he was interested in the solution with the longest period, following in this approach what Tinbergen had developed in 1931. From this point of view they both took a different path than Frisch.

By adding the initial condition that $f(0)=0$ and replacing in the equation for $I(t)$ Kalecki obtained the economic solution as a function of an imaginary root $\lambda=x+i y, i=\sqrt{-1}$
$$I(t)-U=e^{(m-x) \cdot \theta^{-1} \cdot t} c \cdot \sin \left(\frac{y}{\theta} t\right) .$$

经济代写|宏观经济学代写Macroeconomics代考|Putting the Ingredients Together: Kalecki’s

Kalecki modeled a simplified closed world with two classes of people: capitalists and workers. Workers are assumed to consume the entirety of their wages, while capitalists consume only the part of their profits that is not invested. In reference to his work presented above, this distinction means that the total income of capitalists is globally equal to their expenses and the more they spend as a class in consumption and investment, the more profits they earn. ${ }^{27}$

This paints a grim picture of an asymmetric world in which capitalists’ decisions rule the ups and downs of the business cycle, while workers have no other choice than to adapt to the resulting conditions. But this is only true for capitalists as a class, and not individually: at their level, they can only hope when they increase their expenses that other capitalists will do as well. The difficulty is increased by the fact that the current yield of investments depends on the stock of capital, which is itself a function of past investment decisions that are delivered only after a production period. Capitalists must then anticipate the future level of investment, and its impact on the future stock of capital which determines the yield of current investment projects (remembering also to incorporate a factor of depreciation and the evolving rate of interest). This is the main source of economic instability for Kalecki, and the driver of the ups and downs of the economy. ${ }^{28}$

To model this instability, Kalecki started from a situation with a constant stock of capital where profits are equal to the sum of the consumption of capitalists $C$ and the level of production $A: P=C+A$. Capitalists’ consumption consists of a constant amount $C_0$ and a part proportionate to profits; i.e., $C=C_0+\lambda P$ where $\lambda$ is the capitalists’ marginal propensity to consume. Substituting this consumption equation into the profit equation allows Kalecki to obtain the level of realized profits $P=$ $\frac{A+C_0}{1-\lambda}$ which corresponds to a state of “quasi-equilibrium,” that is, the equilibrium level of profit for any given level of investment spending and autonomous capitalist consumption. As will become clear, Kalecki was much more interested in the stability properties of his whole system, where this quasi-equilibrium remained obtained at each time, than in the adjustment process to this equilibrium. ${ }^{29}$ The dynamic process that he studied was the trajectory of this equilibrium point when the capital stock was allowed to vary.

宏观经济学代考

经济代写|宏观经济学代写Macroeconomics代考|Stability and Cycles in Kalecki’s Model

Kalecki (1936) 并没有排除 $m$ 可以改变，正如他对 Frisch 的回筫和他 1939 年的论文所证明的那样，但他将其限制为正值，这意味着利润 的正变化不会导致投资决策的下降。在那种情况下，我们看到模型稳 定的区域相当小，而东北象限的其余部分产生不稳定的解决方案。 Kalecki 通过仅考虑分隔区域 A 和 D 的直线上的组合进一步限制了解 空间，这对应于自持振荡。

Kalecki 拒绝指数解决方案，因为他只对经济的周期性特性感兴趣。此外，他对周期最长的解决方案很感兴趣，逻循丁伯根在 1931 年开发 的这种方法。从这个角度来看，他们都采取了与弗里苃不同的路径。

$$I(t)-U=e^{(m-x) \cdot \theta^{-1} \cdot t} c \cdot \sin \left(\frac{y}{\theta} t\right)$$

经济代写|宏观经济学代写Macroeconomics代考|Putting the Ingredients Together: Kalecki’s

Kalecki 模拟了一个简化的封闭世界，其中有两类人：资本家和工人。 假设工人消费了他们的全部工资，而资本家只消艴了他们末投资的那 部分利润。参考他上面介绍的工作，这种区别意味着资本家的总收入 在全球范围内等于他们的支出，他们作为一个阶层在消艴和投资上花 茀的越多，他们赚取的利润就越多。

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