The Society for Economic Studies
The University of Kitakyushu
Working Paper Series
No.2020-5(accepted in March 29, 2021)

On Keynes’ Business Cycle Theory

Jumpei Tanaka *The University of Kitakyusyu

Abstract
The purpose of this paper is to formulate Keynes’ own view regarding the business cycle
as simply and faithfully as possible by using a simple macro-dynamic model based on
the intertemporal optimization of the firm’s investment decision under demand
constraint. We demonstrate that under a specific assumption regarding the firm’s
expectation modification behavior Keynes’ view that economic fluctuations are caused by
optimistic soars in the marginal efficiency of capital and their inevitable collapses canbe faithfully formulated.
JEL Classification: E12, E22, E32
Keywords: business cycle, Keynesian economics, effective demand, intertemporal
optimization of investment decision, subjective expectation* Jumpei Tanaka

E-mail: j-tanaka@kitakyu-u.ac.jp

1. Introduction
The objective of this paper is to formulate Keynes’ own view regarding the businesscycle as simply and faithfully as possible. In Chapter 22 of General Theory (Keynes(1936)) he summarizes his view of business cycle as follows:(1) The most important factor that generates the business cycles is fluctuation in “themarginal efficiency of capital (MEC, hereafter)”. The move of investment caused by thefluctuation of MEC brings about the business cycles through the principle of effective
demand.
(2) A boom is mainly caused by excessive investment driven by an optimistic soar in the
MEC. Here, “excessive investment” is interpreted according to Keynes’ own words as a
type of investment that is “made in conditions which are unstable and can not endure
because it is prompted by expectations which are destined to disappointment”.
(3) The main reason why the economy recovers regularly from recessions is that capitalstock over-accumulated during a boom naturally depreciates and returns to its normal
level within several years. 1
This Keynes’ view of the business cycle seems to be very convincing, but this view has
not necessarily been formulated properly even in the context of post Keynesian businesscycle theories. 2 In the post Keynesian school, there are various types of business cycle
models. For example, Samuelson (1939), Hicks (1950) and Goodwin (1951) presented
business cycle models based on the acceleration principle, i.e., the assumption thatinvestment depends on the current changes in aggregate income. Kalecki (1935, 1937)

Keynes also pointed out the existence of inventory cost as a reason for regular
recoveries from recessions.
2 There seems to be little research that attempts to formulate faithfully this Keynes’view on business cycles in the mainstream new Keynesian (NK) framework. Recently,
Branch and McGough (2010), Branch and Evans (2011), De Grauwe (2011, 2012) and soon introduced heterogeneous boundedly rational expectation formations into the NK
model and formulated boom and bust cycles driven by ‘animal spirits’ (i.e., the waves of
optimism and pessimism). However, it is questionable whether these can be interpreted
as a model that faithfully reproduces Keynes's views described above. See also footnote
5 regarding this point.
2
1

constructed models built on the profit principle, i.e., the assumption that investment
depends on the current aggregate profits (or the rate of profit). A model by Kaldor (1940)
was built under the assumption that investment depends on the current total profit (orincome) and its functional form is the sigmoid shape. Finally, in Goodwin (1967)’s model,
business cycles are led by a certain income distribution structure between employees andcapitalists 3 . However, Keynes himself emphasized the MEC (or the expected rate of
return on capital) as the crucial determinant of investment, and these models are hardly
a faithful formulation of that point. 4
There are not many attempts to construct a Keynesian business cycle model in which
the expected return on capital is the primary determinant of investment. Benassy (1984)
and Murakami (2018, 2019, 2020) are two such attempts. Benassy (1984) constructed ashort-run non-Walrasian model in which the wage is adjusted in response to theunemployment level, and the expected demand as a determinant of investment isadjusted adaptively in response to its realized value. He then showed that a limit cycle
exists when the effect of destabilizing quantity dynamics exceeds the effects of stabilizing
price dynamics. Murakami (2018, 2019, 2020) presented a post Keynesian cyclical
growth model built on the assumption that investment depends on the expected rate of
return on capital and its functional form is the sigmoid shape like the Kaldor (1940)’s
model. He then showed that if investment is highly elastic to the expected rate of return,and if the expected rate of return is frequently revised in response to its realized value,then the unique limit cycle exists, and an economy starting from any state other thanthe long-run equilibrium will converge stably to the limit cycle.
However, these models are not adequate as a formulation of Keynes' own businesscycle theory in the following two respects. First, according to Keynes, the investment isdetermined so as to maximize the discounted present value of profits, given future
In this sense Goodwin (1967)’s model should be interpreted as a kind of “Marxian”
business cycle theory. But this model was subsequently extended to the Keynesianframework where the output level is determined by the principle of effective demand by
Wolfstetter (1982), Franke and Asada (1994), and so on.
4 This holds true for more sophisticated post Keynesian macroeconomic models such as
Flaschel et al. (1997) and Chiarella et al. (2005).
3
3

expectations (long-term expectations), but in these models the (dynamic) optimization
behavior of the firm is not made explicit. Secondly, Keynes thought that booms arecaused by the over-investment which is brought about by the optimistic expectations
that are destined to disappointment, and that the economy will enter a phase of recession
after the sudden collapse of such optimistic expectations. In these models, however, suchdiscontinuous features of economic fluctuations have not been modeled faithfully. 5
The purpose of this paper is to formulate Keynes’ business cycle theory as simply andfaithfully as possible. The model presented in this paper is post Keynesian in that it isbased on the principle of effective demand. However, it has a feature that is not often
seen in post Keynesian models in that we explicitly introduce the intertemporal
optimization of the firm’s investment decision. The reason why we introduce it is thatthe concept of MEC defined by Keynes can be well formulated using this framework as
explained by Nagatani (1981, ch6). In our model the firm derives the optimal investment
path under the subjective expectation regarding the aggregate demand, so the optimal
investment demand in each period depends on the expected aggregate demand. Once thecurrent investment demand is determined, the current equilibrium output is also
determined based on the principle of effective demand. However, there is no guaranteethat the firm’s expected aggregate demand will coincide with its realized value (i.e, theequilibrium output). In this paper we assume a simple expectation modification behavior
that the firm revises the expectation in response to its realized value only in the steadystate, and we show that under such an assumption Keynes’ business cycle theory can beformulated both simply and faithfully.
The remainder of this paper is organized as follows. In the next section we formulate

Recently, boom and bust cycles driven by ‘animal spirits’ (i.e., the waves of optimism
and pessimism) have been constructed by introducing heterogeneous boundedly
rational expectation formation in both new Keynesian (NK) and post Keynesian (PK)schools. In NK school the references listed in footnote 1 correspond to such attempts. In
PK school Franke (2012, 2014), Flaschel et.al (2018) and so on corresponds to suchattempts. These studies, however, are not faithful formulations of Keynes's own
business cycle theory in that they are not based on the firm’s optimal investment
decisions given long-term expectations.
45

the intertemporal optimization problem of the firm’s investment decision undersubjective expectations regarding the aggregate demand and the interest rate. In section
3 we investigate the macro-dynamics of the economy when subjective expectations are
fixed and not revised. In section 4 we consider the case where only the aggregate demand
expectation is revised in the steady state, and then reproduce the Keynes’ business cycle
theory as a deviation from the steady state where the expectation is self-fulfilling. In
section 5 we investigate the case where the expectations of both the aggregate demand
and the interest rate are revised simultaneously. Finally, we conclude the paper insection 6.

2. Firm’s investment decision
Before constructing a macro-dynamic model, in this section we discuss the firm’s
investment decisions 6. Consider an economy where only one type of final product which
can be used for both consumption and investment is produced. We assume that there aremany homogeneous firms and normalize their number to one without loss of generality.
It is also assumed that, since the size of each firm is sufficiently small, each firm does
not imagine that its own investment demand is capable of affecting the aggregate
demand. At time 𝑠𝑠 the firm faces the following problem:
∞

𝑡𝑡 𝑒𝑒

max ∞ 𝑉𝑉𝑠𝑠 = � [𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) − 𝑤𝑤𝑡𝑡 𝐿𝐿𝑡𝑡 − {𝐼𝐼𝑡𝑡 + 𝑐𝑐(𝐼𝐼𝑡𝑡 )}]𝑒𝑒 − ∫𝑠𝑠 𝑟𝑟𝑢𝑢 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑∞

{𝐼𝐼𝑡𝑡 }𝑡𝑡=𝑠𝑠 ,{𝐾𝐾𝑡𝑡 }𝑡𝑡=𝑠𝑠𝑠𝑠s.t. 𝐾𝐾𝑡𝑡̇ = 𝐼𝐼𝑡𝑡 − 𝛿𝛿𝐾𝐾𝑡𝑡 ,𝑌𝑌𝑡𝑡𝑒𝑒 = 𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 )(1)(𝐾𝐾𝑠𝑠 : 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔)

where 𝐾𝐾𝑡𝑡 is the capital input, 𝐿𝐿𝑡𝑡 the labor input, 𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) the production function, 𝑤𝑤𝑡𝑡

the real wage (measured in terms of the final product), 𝐼𝐼𝑡𝑡 the investment expenditure,

𝑐𝑐(𝐼𝐼𝑡𝑡 ) the investment adjustment cost function, 𝛿𝛿 the depreciation rate, and 𝐾𝐾𝑡𝑡̇ the time

derivative of 𝐾𝐾𝑡𝑡 (i.e., 𝐾𝐾𝑡𝑡̇ ≡ 𝑑𝑑𝐾𝐾𝑡𝑡 ⁄𝑑𝑑𝑑𝑑 ). 𝑌𝑌𝑡𝑡𝑒𝑒 and 𝑟𝑟𝑡𝑡𝑒𝑒 are the subjective expectations about

the aggregate demand and the real interest rate, which will be explained in more detailbelow.
We mainly follow Nagatani (1981, ch6) regarding the argument in this section. See
also Grossman (1972) on the original formulation of the firm’s investment behavior
adopted here.
56

We assume that the production function 𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) satisfies the following usual

neoclassical properties:

𝐹𝐹𝐾𝐾 > 0, 𝐹𝐹𝐾𝐾𝐾𝐾 < 0, 𝐹𝐹𝐿𝐿 > 0, 𝐹𝐹𝐿𝐿𝐿𝐿 < 0, 𝐹𝐹𝐾𝐾𝐾𝐾 > 0,

lim 𝐹𝐹𝐾𝐾 = ∞,𝐾𝐾→0

where 𝐹𝐹𝐾𝐾 ≡ 𝜕𝜕𝜕𝜕 ⁄𝜕𝜕𝜕𝜕 and(2)

lim 𝐹𝐹𝐾𝐾 = 0, lim 𝐹𝐹𝐿𝐿 = ∞, lim 𝐹𝐹𝐿𝐿 = 0,

𝐾𝐾→∞

𝐿𝐿→0

𝐿𝐿→∞

𝐹𝐹𝐾𝐾𝐾𝐾 ≡ 𝜕𝜕𝐹𝐹𝐾𝐾 ⁄𝜕𝜕𝜕𝜕 . It is also assumed that the investment

adjustment cost function 𝑐𝑐(𝐼𝐼𝑡𝑡 ) is strictly convex:(3)

𝑐𝑐(0) = 0, 𝑐𝑐𝐼𝐼 (𝐼𝐼𝑡𝑡 ) > 0, 𝑐𝑐𝐼𝐼𝐼𝐼 (𝐼𝐼𝑡𝑡 ) > 0,

where 𝑐𝑐𝐼𝐼 (𝐼𝐼𝑡𝑡 ) ≡ 𝜕𝜕𝜕𝜕 ⁄𝜕𝜕𝐼𝐼𝑡𝑡 and 𝑐𝑐𝐼𝐼𝐼𝐼 (𝐼𝐼𝑡𝑡 ) ≡ 𝜕𝜕𝑐𝑐𝐼𝐼 ⁄𝜕𝜕𝐼𝐼𝑡𝑡 .

Problem (1) is an intertemporal optimization problem according to which the firm

maximizes the present discount value of the net cash flow stream, but it mainly differsfrom the standard setting in that the firm in our model has subjective expectations
regarding present and future aggregate demands {𝑌𝑌𝑡𝑡𝑒𝑒 }∞𝑡𝑡=𝑠𝑠 and aims to maximize theobjective function under the demand constraints: 𝑌𝑌𝑡𝑡𝑒𝑒 = 𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) for any 𝑡𝑡 7 . This
7F

constraint means that the firm produces a quantity of the final product which is equal
to the expected aggregate demand. The firm is also assumed to have the subjective
expectation about the interest rate in each point in time and to maximize the objectivefunction under it. Furthermore, the wage at each point in time is assumed to be
exogenous fixed in our model (namely, our model is a fixed wage model).
𝑒𝑒 ∞To simplify the analysis, we assume throughout the paper that {𝑌𝑌𝑡𝑡𝑒𝑒 }∞𝑡𝑡=𝑠𝑠 , {𝑟𝑟𝑡𝑡 }𝑡𝑡=𝑠𝑠 and{𝑤𝑤𝑡𝑡 }∞
𝑡𝑡=𝑠𝑠 are constant over time:(4)

𝑌𝑌𝑡𝑡𝑒𝑒 = 𝑌𝑌 𝑒𝑒 , 𝑟𝑟𝑡𝑡𝑒𝑒 = 𝑟𝑟 𝑒𝑒 , 𝑤𝑤𝑡𝑡 = 𝑤𝑤.

Under these assumptions, the firm’s optimization problem can be rewritten as:∞𝑒𝑒max ∞ 𝑉𝑉𝑠𝑠 = � [𝑌𝑌 𝑒𝑒 − 𝑤𝑤𝑤𝑤(𝐾𝐾𝑡𝑡 , 𝑌𝑌 𝑒𝑒 ) − {𝐼𝐼𝑡𝑡 + 𝑐𝑐(𝐼𝐼𝑡𝑡 )}]𝑒𝑒 −𝑟𝑟 (𝑡𝑡−𝑠𝑠) 𝑑𝑑𝑑𝑑∞

{𝐼𝐼𝑡𝑡 }𝑡𝑡=𝑠𝑠 ,{𝐾𝐾𝑡𝑡 }𝑡𝑡=𝑠𝑠

𝑠𝑠

(5)

s.t. 𝐾𝐾𝑡𝑡̇ = 𝐼𝐼𝑡𝑡 − 𝛿𝛿𝐾𝐾𝑡𝑡 (𝐾𝐾𝑠𝑠 : 𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔)

where the function 𝐿𝐿𝑡𝑡 = 𝐺𝐺(𝐾𝐾𝑡𝑡 , 𝑌𝑌 𝑒𝑒 ) can be derived from the demand constraint 𝑌𝑌 𝑒𝑒 =𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) and it satisfies the following properties:

Note that the firm’s expectations are subjective so there is no guarantee that they are
consistent with their realized values in macroeconomic equilibrium. Regarding theformulation of the firm’s expectation correction behavior, see section 4 and 5.
6

7

𝐺𝐺𝐾𝐾 < 0, lim 𝐺𝐺𝐾𝐾 = −∞,𝐾𝐾→0lim 𝐺𝐺𝐾𝐾 = 0, 𝐺𝐺𝐾𝐾𝐾𝐾 > 0, 𝐺𝐺𝑌𝑌 > 0, 𝐺𝐺𝑌𝑌𝑌𝑌 > 0, 𝐺𝐺𝐾𝐾𝐾𝐾 < 0.

𝐾𝐾→∞

(6)

The first order conditions of the problem (5) are
1 + 𝑐𝑐𝐼𝐼 (𝐼𝐼𝑡𝑡 ) = 𝑞𝑞𝑡𝑡

(7.a)

𝐾𝐾𝑡𝑡̇ = 𝐼𝐼𝑡𝑡 − 𝛿𝛿𝐾𝐾𝑡𝑡

(7.c)

𝑞𝑞𝑡𝑡̇ = (𝑟𝑟 𝑒𝑒 + 𝛿𝛿)𝑞𝑞𝑡𝑡 + 𝑤𝑤𝐺𝐺𝐾𝐾 (𝐾𝐾𝑡𝑡 , 𝑌𝑌 𝑒𝑒 )lim 𝑒𝑒𝑡𝑡→∞−𝑟𝑟 𝑒𝑒 (𝑡𝑡−𝑠𝑠)

𝑞𝑞𝑡𝑡 𝐾𝐾𝑡𝑡 = 0

(7.b)

(7.d)

where 𝑞𝑞𝑡𝑡 is the shadow price of capital. By eliminating 𝑞𝑞𝑡𝑡 in (7.a) and (7.b), thedynamic system of the optimal investment plan chosen by the firm can be derived as𝐼𝐼𝑡𝑡̇ =

(𝑟𝑟 𝑒𝑒 + 𝛿𝛿)[1 + 𝑐𝑐𝐼𝐼 (𝐼𝐼𝑡𝑡 )] + 𝑤𝑤𝐺𝐺𝐾𝐾 (𝐾𝐾𝑡𝑡 , 𝑌𝑌 𝑒𝑒 ),𝑐𝑐𝐼𝐼𝐼𝐼 (𝐼𝐼𝑡𝑡 )𝑒𝑒𝐾𝐾𝑡𝑡̇ = 𝐼𝐼𝑡𝑡 − 𝛿𝛿𝐾𝐾𝑡𝑡 ,lim 𝑒𝑒 −𝑟𝑟 (𝑡𝑡−𝑠𝑠) [1 + 𝑐𝑐𝐼𝐼 (𝐼𝐼𝑡𝑡 )]𝐾𝐾𝑡𝑡 = 0.

𝑡𝑡→∞

(8.a)
(8.b)
(8.c)

Figure 1 shows the phase diagram of this system. The figure clearly shows that thissystem has a unique optimal investment path and that this path satisfies saddle-point
stability.

(Figure 1 around here)

How can the “marginal efficiency of capital” (hereafter MEC), defined by Keynes (1936)
as the special discount rate which equates the present discount value of the net cash flow
stream from an additional investment with its investment cost, be formulated within
this framework? It is well known that the marginal benefit from an addtional investment
is represented by the shadow price of capital 𝑞𝑞𝑠𝑠 , because by integrating (7.b) we have
∞

𝑒𝑒

∞

𝑞𝑞𝑠𝑠 = � (−𝑤𝑤𝐺𝐺𝐾𝐾 )𝑒𝑒 −(𝑟𝑟 +𝛿𝛿)(𝑡𝑡−𝑠𝑠) 𝑑𝑑𝑑𝑑 = �𝑠𝑠𝑠𝑠𝜕𝜕𝑛𝑛𝑛𝑛𝑡𝑡 −(𝑟𝑟 𝑒𝑒+𝛿𝛿)(𝑡𝑡−𝑠𝑠)𝑒𝑒𝑑𝑑𝑑𝑑,𝜕𝜕𝐾𝐾𝑡𝑡(9)where 𝑛𝑛𝑛𝑛𝑡𝑡 ≡ 𝑌𝑌 𝑒𝑒 − 𝑤𝑤𝑤𝑤(𝐾𝐾𝑡𝑡 , 𝑌𝑌 𝑒𝑒 ) − {𝐼𝐼𝑡𝑡 + 𝑐𝑐(𝐼𝐼𝑡𝑡 )} is the net cash flow at time 𝑡𝑡 . This 𝑞𝑞𝑠𝑠 isknown as Tobin’s marginal 𝑞𝑞, which can be interpreted roughly as a firm’s stock price 8.

Strictly speaking, the shadow price 𝑞𝑞𝑠𝑠 is not equal to the firm’s stock price in ourmodel. As Hayashi (1982) demonstrated, they become equal if and only if (i) the firm
behaves competitively, and (ii) the investment adjustment cost function is linearly78

It can be easily verified that 𝑞𝑞𝑠𝑠 decreases with the capital stock and the expectedinterest rate, and increases with the expected aggregate demand and the real wage:
𝑞𝑞𝑠𝑠 = 𝑞𝑞(𝐾𝐾𝑠𝑠 , 𝑟𝑟 𝑒𝑒 , 𝑌𝑌 𝑒𝑒 , 𝑤𝑤).

𝜕𝜕𝑞𝑞𝑠𝑠𝜕𝜕𝑞𝑞𝑠𝑠𝜕𝜕𝑞𝑞𝑠𝑠𝜕𝜕𝑞𝑞𝑠𝑠
�
< 0,
< 0,
> 0,
> 0, �𝑒𝑒𝑒𝑒
𝜕𝜕𝐾𝐾𝑠𝑠
𝜕𝜕𝑟𝑟
𝜕𝜕𝑌𝑌
𝜕𝜕𝜕𝜕

(10)

On the other hand, since the investment cost (or more precisely, the marginal cost of thefirst unit investment) is given by 1 + 𝑐𝑐𝐼𝐼 (0), the MEC at time 𝑠𝑠 can be derived as the

special discount rate 𝑅𝑅𝑠𝑠 which satisfies the following equation:
1 + 𝑐𝑐𝐼𝐼 (0) = 𝑞𝑞(𝐾𝐾𝑠𝑠 , 𝑅𝑅𝑠𝑠 , 𝑌𝑌 𝑒𝑒 , 𝑤𝑤).

(11)

It can easily be confirmed from (10) that the MEC ( 𝑅𝑅𝑠𝑠 ) thus derived satisfies thefollowing:𝜕𝜕𝑅𝑅𝑠𝑠𝜕𝜕𝑅𝑅𝑠𝑠𝜕𝜕𝑅𝑅𝑠𝑠
> 0,
> 0,
<0
𝑒𝑒
𝜕𝜕𝑌𝑌
𝜕𝜕𝜕𝜕
𝜕𝜕𝐾𝐾𝑠𝑠(12)

Therefore, in our model the MES changes when exogenous variables such as 𝑌𝑌 𝑒𝑒 or 𝑤𝑤
change 9.

Lastly, the effects of changes in exogenous variables on the firm’s optimal investment
plan are summarized as follows. An increase in the expected aggregate demand 𝑌𝑌 𝑒𝑒 atthe initial time 𝑠𝑠 shifts the 𝐼𝐼𝑡𝑡̇ = 0 line in Figure 1 upward, causing instantaneous
upward jumps in the MEC (𝑅𝑅𝑠𝑠 ), the stock price (𝑞𝑞𝑠𝑠 ) and the investment level (𝐼𝐼𝑠𝑠 ) at time𝑠𝑠, and then raises the capital stock in the steady state. An increase in the real wage 𝑤𝑤

also shifts the 𝐼𝐼𝑡𝑡̇ = 0 line upward and yields the same changes as an increase in 𝑌𝑌 𝑒𝑒 . The

reason why an increase in 𝑤𝑤 stimulates investment is that such a change lowers therelative price of capital and encourages factor substitution from labor to capital. Finally,an increase in the expected interest rate 𝑟𝑟 𝑒𝑒 shifts the 𝐼𝐼𝑡𝑡̇ = 0 line downward, leading toan instantaneous drop in the stock price and the investment level at time 𝑠𝑠 ,subsequently reducing the capital stock in the steady state. This is because an increasehomogenous in 𝐼𝐼 and 𝐾𝐾. In our framework, neither of these conditions hold.
9 Nagatani (1981, ch 6) further clarified the difference between the schedule of the
MEC (i.e., the relationship between 𝑅𝑅 and 𝐾𝐾 which satisfies (11)) and the schedule ofthe marginal efficiency of investment (i.e., the relationship between 𝑟𝑟 and 𝐼𝐼 which
satisfies (7.a)), and stressed that the schedule which determines the current rate ofinvestment is not the former but the latter.
8

in the expected interest rate shrinks the present discount value of the net cash flow
stream and discourages the firm’s investment activity.

3. “Short-run” macro-dynamics
In this section we construct a “Keynesian” macro-dynamic model and explore theproperties of its “short-run” behavior. Here, the term “Keynesian” means that theequilibrium output at any time is determined by the principle of effective demand. The
term “short-run” will be defined later.
Let us investigate macroeconomic equilibrium. Regarding the behavior of the firm, we
suppose the followings. First, the firm finances the total investment cost (i.e., 𝐼𝐼𝑠𝑠 + 𝑐𝑐(𝐼𝐼𝑠𝑠 ))from the household savings through the external financial market. 10 Second, the firm isassumed to distribute all sales to the household sector in the form of either the wage
income (𝑤𝑤𝐿𝐿𝑠𝑠 ) or the profit (𝜋𝜋𝑠𝑠 ≡ 𝑌𝑌𝑠𝑠 − 𝑤𝑤𝐿𝐿𝑠𝑠 ).

Regarding the household sector, we suppose that there are many homogenous

households and their number is normalized to one without any loss of generality. Eachhousehold is endowed with constant units of labor 𝐿𝐿� at each point in time, but thehousehold supplies labor at a level which is equal to the firm’s labor demand 𝐿𝐿𝑠𝑠 due tothe existence of the demand constraint. Since the household is also an owner of the firm,
it receives not only the wage income 𝑤𝑤𝐿𝐿𝑠𝑠 as an employee, but also the firm’s profit 𝜋𝜋𝑠𝑠 (≡𝑌𝑌𝑠𝑠 − 𝑤𝑤𝐿𝐿𝑠𝑠 ) as a stockholder at time 𝑠𝑠. Under this setting the household receives the total

income equal to the firm’s output level 𝑌𝑌𝑠𝑠 . For decisions regarding the household’s

consumption, we assume the following most simple Keynesian consumption function:
𝐶𝐶𝑠𝑠 = 𝑐𝑐𝑌𝑌𝑠𝑠(0 < 𝑐𝑐 < 1)(13)

where 𝑐𝑐 is the constant marginal (average) propensity to consume. Since thegovernment and foreign sectors are neglected in our model, the actual aggregate demand

If we assume that the investment cost is internally financed (i.e., financed by thefirm’s retained earnings), the equilibrium output cannot be determined as usual
through the principle of effective demand. Since the choice between internal andexternal finance does not have any impact on the investment plan itself under perfectfinancial market, this assumption is justifiable.
910

of the final product is
𝐴𝐴𝐴𝐴𝑠𝑠 = 𝐶𝐶𝑠𝑠 + 𝐼𝐼𝑠𝑠 + 𝑐𝑐(𝐼𝐼𝑠𝑠 ) = 𝑐𝑐𝑌𝑌𝑠𝑠 + 𝐼𝐼𝑠𝑠 + 𝑐𝑐(𝐼𝐼𝑠𝑠 ),(14)

where the investment demand 𝐼𝐼𝑠𝑠 is derived from the firm’s investment problem (5) inthe previous section. Here, note that the adjustment cost of investment (𝑐𝑐(𝐼𝐼𝑠𝑠 )) is assumed

to take the form of expenditure of the final product. The equilibrium condition of the
final product market is given by
𝑌𝑌𝑠𝑠 = 𝑐𝑐𝑌𝑌𝑠𝑠 + 𝐼𝐼𝑠𝑠 + 𝑐𝑐(𝐼𝐼𝑠𝑠 ).

Therefore, the equilibrium output 𝑌𝑌𝑠𝑠∗ at time 𝑠𝑠 can be derived as follows:
𝑌𝑌𝑠𝑠∗ = 𝑠𝑠 −1 [𝐼𝐼𝑠𝑠 + 𝑐𝑐(𝐼𝐼𝑠𝑠 )](𝑠𝑠 ≡ 1 − 𝑐𝑐)

(15)

(16)

where 𝑠𝑠(≡ 1 − 𝑐𝑐) is the marginal (average) propensity to save. From (16) we can confirm

that 𝑌𝑌𝑠𝑠∗ is a monotonically increasing function of 𝐼𝐼𝑠𝑠 .

Note that under the “short-run” macroeconomic equilibrium derived in (16), the firm’s

subjective expectations (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) is not self-fulfilling (i.e., those expectations are notconsistent with their realization values (𝑌𝑌𝑠𝑠∗ , 𝑟𝑟𝑠𝑠∗ )). In this paper we use the term “short-run” as the period during which such inconsistency remains. In other words, the term“short-run” means the situation where even if the firm’s subjective expectations (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )deviate from their realization values (𝑌𝑌𝑠𝑠∗ , 𝑟𝑟𝑠𝑠∗ ) , the firm does not correct the
expectations. 11

Since in the short-run the firm’s expectations remain fixed, the optimal investment
plan derived in (8.a), (8.b) and (8.c) becomes equal to the actual investment path. Oncethe investment path is determined, the output path {𝑌𝑌𝑡𝑡∗ }∞𝑡𝑡=𝑠𝑠 is also determined by (16).

Therefore, the short-run macro-dynamics is essentially the same as one shown in Figure

1 in the previous section, and the effects of changes in exogenous variables on the shortrun macro-dynamics are summarized as follows. First, an increase in the expected
aggregate demand 𝑌𝑌 𝑒𝑒 or the fixed wage 𝑤𝑤 stimulates MEC, stock price, investment

and output at that time, and raises output and capital stock in the steady state. Second,an increase in the expected interest rate 𝑟𝑟 𝑒𝑒 discourages MEC, stock price, investment
Note that the term “short-run” used in this paper is different from the usual
meaning of the term. The same is true for the concepts of “long-run” defined later.
10

11and output at that time, and lowers output and capital stock in the steady state. Finally,an increase in the saving rate 𝑠𝑠 does not have any direct impact on the firm’s investment

decision (i.e., it has no impact on MEC, stock price and investment at that time andcapital stock in the steady-state), while such a change lowers the equilibrium output at
any point in time through a decline in the aggregate demand.

4. Business cycle as a deviation from the “long-run” steady state
In this section we focus on the “long-run” behavior of the economy. In this paper the“long-run” is defined as the situation in which the firm’s expectations (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) coincidewith their realization values (𝑌𝑌𝑠𝑠∗ , 𝑟𝑟𝑠𝑠∗ ) (i.e., the expectations are self-fulfilling). From (8.a),(8.b) and (16), the short-run steady state is characterized as follows.
(𝑟𝑟 𝑒𝑒 + 𝛿𝛿)[1 + 𝑐𝑐𝐼𝐼 (𝐼𝐼 ∗ )] + 𝑤𝑤𝐺𝐺𝐾𝐾 (𝛿𝛿 −1 𝐼𝐼 ∗ , 𝑌𝑌 𝑒𝑒 ) = 0
𝑌𝑌 ∗ = 𝑠𝑠 −1 [𝐼𝐼 ∗ + 𝑐𝑐(𝐼𝐼 ∗ )](17.a)(17.b)

When the firm’s expectations differ from their realization values in the short-run steadystate shown by (17.a) and (17.b), the expectations will continue to be betrayedpermanently. In such a situation any firm which is not irrational will correct its mistaken
expectation. In this paper we adopt the following assumption regarding the firms’
expectation revision behavior:

Assumption: When the firm’s expectations deviates from their realization values in the

short-run steady state, the firm modifies the expectations toward their realization value.

In other words, this assumption states that when the economy stays on the transition

process to the short-run steady state, the firm does not modify the expectations even if
they deviate from their realization value. This assumption is necessary in formulating
Keynes's business cycle theory in its simplest form, though it is somewhat ad hoc.
Following justifications may be possible for this assumption. First, the firm’s capital

11

investments such as building a factory, for example, take time 12, and it is not realistic tocontinuously revise expectations and change investment plans during the construction
phase. In this sense, the above assumption is not so unrealistic. Second, this assumption
may not be valid when the economy stays on the transition process for a fairly long period.
In this paper, however, we focus on the case where the economy mainly stays in the longrun steady state, sometimes deviating from it for a while due to changing expectations.
In such a case this assumption is not so problematic. It will be a future task to improve
our model by adopting a more convincing expectation formation hypothesis. (Seeconcluding remarks on this point.)
In this section, we further assume that only the expected aggregate demand 𝑌𝑌 𝑒𝑒 isadjusted in the short-run steady state (i.e., the expected interest rate 𝑟𝑟 𝑒𝑒 is not adjusted

and remains fixed) to simplify the discussion. The case where both 𝑌𝑌 𝑒𝑒 and 𝑟𝑟 𝑒𝑒 areadjusted simultaneously will be discussed in the next section.

When the firm revises the expected aggregate demand 𝑌𝑌 𝑒𝑒 according to the above

assumption, does an arbitrarily chosen expectation stably converge to a self-fulfillingone? From (17.a), the firm’s investment in the short-run steady state ( 𝐼𝐼 ∗ ) can be

expressed as

𝐼𝐼 ∗ = 𝐼𝐼(𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ),𝐼𝐼𝑌𝑌∗ > 0, 𝐼𝐼𝑟𝑟∗ < 0,(18)

where 𝐼𝐼𝑌𝑌∗ ≡ 𝜕𝜕𝐼𝐼 ∗ ⁄𝜕𝜕 𝑌𝑌 𝑒𝑒 and 𝐼𝐼𝑟𝑟∗ ≡ 𝜕𝜕𝐼𝐼 ∗ ⁄𝜕𝜕 𝑟𝑟 𝑒𝑒 . Accordingly, from (17.b) and (18), the relation

between the expected aggregate demand 𝑌𝑌 𝑒𝑒 and its realization value in the short-run

steady state 𝑌𝑌 ∗ is𝑌𝑌 ∗ = 𝑠𝑠 −1 �𝐼𝐼(𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) + 𝑐𝑐�𝐼𝐼(𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )��,where 𝑌𝑌𝑌𝑌∗ ≡ 𝜕𝜕𝑌𝑌 ∗ ⁄𝜕𝜕 𝑌𝑌 𝑒𝑒 and 𝑌𝑌𝑟𝑟∗ ≡ 𝜕𝜕𝑌𝑌 ∗ ⁄𝜕𝜕 𝑟𝑟 𝑒𝑒 .

𝑌𝑌𝑌𝑌∗ > 0, 𝑌𝑌𝑟𝑟∗ < 0,(19)

In this paper, we assume that the function 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 ) is a strictly concave function as

depicted in the Figure 2. To do so, the following three assumptions are necessary.
(𝑟𝑟 𝑒𝑒 + 𝛿𝛿)[1 + 𝑐𝑐𝐼𝐼 (0)] + 𝑤𝑤𝐺𝐺𝐾𝐾 (0,0) = 0(20.a)

According to empirical evidences, it is reasonable to estimate that the average
length from start to completion of a new investment project is about 2 years. See Mayer(1960), Taylor (1982), Montogomery (1995), Peeters (1996) and Koeva (2000) for thispoint.
1212

𝑌𝑌𝑌𝑌∗ |𝑌𝑌 𝑒𝑒=0 �≡

𝜕𝜕𝑌𝑌 ∗�� = 𝑠𝑠 −1 �1 + 𝑐𝑐𝐼𝐼 �𝐼𝐼(0, 𝑟𝑟 𝑒𝑒 )��𝐼𝐼𝑌𝑌∗ (0, 𝑟𝑟 𝑒𝑒 ) > 1
𝜕𝜕𝑌𝑌 𝑒𝑒 𝑌𝑌 𝑒𝑒=0∗
𝑌𝑌𝑌𝑌𝑌𝑌�≡𝜕𝜕𝑌𝑌𝑌𝑌∗∗ (1� = 𝑠𝑠 −1 [𝐼𝐼𝑌𝑌𝑌𝑌+ 𝑐𝑐𝐼𝐼 ) + (𝐼𝐼𝑌𝑌∗ )2 𝑐𝑐𝐼𝐼𝐼𝐼 ] < 0
𝜕𝜕𝑌𝑌 𝑒𝑒(20.b)(20.c)(20.a) means that the function 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 ) is a curve passing through the origin. (20.b)

indicates that the slope of the function 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 ) at 𝑌𝑌 𝑒𝑒 = 0 is greater than 1. (20.c) showsthat the second-order derivative of 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 ) is negative.

(Figure 2 around here)

In order to consider these conditions more concretely, let us specify the production
function and the investment adjustment cost function as follows.
𝐹𝐹(𝐾𝐾𝑡𝑡 , 𝐿𝐿𝑡𝑡 ) = 𝐾𝐾𝑡𝑡𝛼𝛼 𝐿𝐿1−𝛼𝛼,𝑡𝑡

𝑐𝑐(𝐼𝐼𝑡𝑡 ) = 𝜓𝜓𝐼𝐼𝑡𝑡2 /2(21)

In this case the equation (17.a) can be rewritten as
𝑌𝑌 𝑒𝑒 = 𝐴𝐴(1 + 𝜓𝜓𝐼𝐼 ∗ )1−𝛼𝛼 𝐼𝐼 ∗ .

�𝐴𝐴 ≡ 𝛿𝛿 −1 �

(1 − 𝛼𝛼)(𝑟𝑟 𝑒𝑒 + 𝛿𝛿) 1−𝛼𝛼
� �
𝛼𝛼𝛼𝛼

(22)

From (22) and (19) we can see that (20.a) holds under this specification. From (20.b), (21)and (22) we can derive
𝑌𝑌𝑌𝑌∗ |𝑌𝑌 𝑒𝑒=0 �≡

𝜕𝜕𝑌𝑌 ∗
𝛿𝛿 (1 − 𝛼𝛼)(𝑟𝑟 𝑒𝑒 + 𝛿𝛿) −(1−𝛼𝛼)
�
�= �
�
.
𝜕𝜕𝑌𝑌 𝑒𝑒 𝑌𝑌 𝑒𝑒=0
𝑠𝑠
𝛼𝛼𝛼𝛼

(23)

Therefore, under the specification (21), the assumption (20.b) is equivalent to𝛿𝛿 (1 − 𝛼𝛼)(𝑟𝑟 𝑒𝑒 + 𝛿𝛿) −(1−𝛼𝛼)
�
�
> 1.
𝑠𝑠
𝛼𝛼𝛼𝛼

(24)

Concerning (20.c), from (22) we can show∗
𝐼𝐼𝑌𝑌∗ > 0, 𝐼𝐼𝑌𝑌𝑌𝑌
< 0.

(25)

∗
in (20.c) is still ambiguous.
However, even taking this result into account, the sign of 𝑌𝑌𝑌𝑌𝑌𝑌

Therefore, even under the specification (21), (20.c) needs to be assumed.

If the function 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 ) is as shown in Figure 2, then the expectation modification

process under the Assumption above is globally stable, so any short-run steady state

converges to the unique long-run steady state 𝑌𝑌 𝑒𝑒∗ even if the initial value of 𝑌𝑌 𝑒𝑒 ischosen arbitrarily.

13Based on the expectation revision process described so far, let us examine themechanism of business cycles driven by changes in the expectation. Consider thesituation in which the economy initially has been staying in a long-run steady state(𝐸𝐸0 (𝑌𝑌0∗ , 𝐾𝐾0∗ ) in Figure 4 below) where the expectation is self-fulfilling (i.e., 𝑌𝑌 𝑒𝑒 = 𝑌𝑌0∗ ). Whensome change in the firm’s expectation has occurred, how does the economy respond tosuch an exogenous expectation shock?
Figure 3.a and Figure 3.b illustrate a series of changes in the economy when the firm’s
expected aggregate demand suddenly rise to a level (𝑌𝑌1𝑒𝑒 ) which is higher than the selffulfilling one (𝑌𝑌 𝑒𝑒 = 𝑌𝑌0∗ ). Figure 3.a depicts the change in the optimal investment pathwhen expectations change. Figure 3.b depicts the relationship between the expectationand its realization value (i.e., equilibrium output in the short-run steady state).

(Figure 3.a and Figure 3.b around here)

First of all, when the firm adopts the new expectation 𝑌𝑌1𝑒𝑒 , the 𝐼𝐼𝑡𝑡̇ = 0 line shifts upward(i.e., the change ↑(a) in Figure 3.a and 3.b), and the economy moves from the initial long-run steady state 𝐸𝐸0 to the new short-run steady state 𝐸𝐸1 (𝑌𝑌1∗ , 𝐾𝐾1∗ ). As is obvious from theargument in the previous section, during this process the economy experiences a “boom”
because MEC, stock price, investment and equilibrium output instantaneously jump,and equilibrium output and capital stock in the new short-run steady state (𝐸𝐸1 ) also rise.
However, the economy cannot continue to stay in 𝐸𝐸1 , because it can be confirmed from

Figure 3.b that the firm’s expectation 𝑌𝑌1𝑒𝑒 exceeds its realization value 𝑌𝑌1∗ in 𝐸𝐸1 .

Accordingly, the firm corrects the expectation downward and adopts a new expectation,for example, 𝑌𝑌2𝑒𝑒 which is smaller than 𝑌𝑌1𝑒𝑒 . This change shifts the 𝐼𝐼𝑡𝑡̇ = 0 line downward(i.e., the change ↓(b) in Figure 3.a and 3.b), and the economy moves from 𝐸𝐸1 to the next

short-run steady state 𝐸𝐸2 (𝑌𝑌2∗ , 𝐾𝐾2∗ ) . During this transition process the economy
experiences a “slump” when MEC, stock price, investment and equilibrium output

instantaneously fall, and equilibrium output and capital stock in the new steady state( 𝐸𝐸2 ) also decline relative to those in 𝐸𝐸1 . During this process the level of capital
14

depreciation (𝛿𝛿𝐾𝐾𝑠𝑠 ) exceeds the gross investment level (𝐼𝐼𝑠𝑠 ) and the capital stock declines

continuously. Therefore, when the capital depreciation rate becomes larger, the over-accumulated capital induced by the firm’s optimistic demand expectation depreciates
more rapidly, and as a result the period of slump that the economy experiences becomesshorter.
In the case where 𝑌𝑌2∗ > 𝑌𝑌0∗ holds, the firm must still revise its expectation downward

because it is still higher than its realization value in the steady state, and the economy
will eventually converge to the initial long-run steady state 𝐸𝐸0 . On the other hand, if thefirm modifies the expectation to an excessively low level such that 𝑌𝑌2𝑒𝑒 < 𝑌𝑌0∗ , the extent of

the downward shift of the 𝐼𝐼𝑡𝑡̇ = 0 line becomes fairly large (i.e., the change ↓(c) in Figure
3.a and 3.b), and the economy experiences a more severe slump (namely, a “depression”)

during the transition from 𝐸𝐸1 to 𝐸𝐸3 . Such a deep slump is often called a “negative

bubble”, because in such a situation the firm’s revised expectation is too pessimistic inthat it is lower even than the self-fulfilling expectation in the long-run steady state 𝐸𝐸0 ,causing an excessive drop in the stock price. However, the economy recovers from thissevere slump sooner or later, returning to the initial long-run steady state 𝐸𝐸0 , because

the firm’s expectation is lower than its realization value in 𝐸𝐸3 and the firm accordingly

revises the expectation upward.

It is clear that a business cycle model formulated in this manner corresponds exactly
to Keynes’ view of business cycles already stated at the beginning of the introduction. It
is remarkable that this view of Keynes regarding the business cycle cannot be formulated
if we assume the “perfect foresight” of the firm. In fact, if such an assumption is imposed,
the economy always stays in the long-run steady state and as a consequence the businesscycle cannot be described as being governed by optimistic and pessimistic surges in
expectations. In this sense our model is different from, for example, Farmer (2016) whichattempts to formulate Keynes’ “animal spirit” using the concept of indeterminacy andsunspot shock under the mainstream macroeconomic theory.

5. Analysis of the case where both are 𝒀𝒀𝒆𝒆 and 𝒓𝒓𝒆𝒆 adjusted
15

In the previous section we examined the case where only the expected aggregate
demand 𝑌𝑌 𝑒𝑒 is adjusted and the expected interest rate 𝑟𝑟 𝑒𝑒 remains fixed. In this section
we investigate the case where both 𝑌𝑌 𝑒𝑒 and 𝑟𝑟 𝑒𝑒 are adjusted.

The actual interest rate at time 𝑠𝑠 can be defined as 𝑟𝑟𝑠𝑠 = (𝑌𝑌𝑠𝑠 − 𝑤𝑤𝐿𝐿𝑠𝑠 )⁄𝐾𝐾𝑠𝑠 , so from (17.b)

and 𝐼𝐼 ∗ = 𝛿𝛿𝐾𝐾 ∗ , the actual interest rate in the short-run steady state can be written as
𝑟𝑟 ∗ =𝑌𝑌 ∗ − 𝑤𝑤𝑤𝑤(𝐾𝐾 ∗ , 𝑌𝑌 𝑒𝑒 ) 𝛿𝛿
𝑐𝑐(𝐼𝐼 ∗ )𝐺𝐺(𝛿𝛿 −1 𝐼𝐼 ∗ , 𝑌𝑌 𝑒𝑒 )
=
�1
+
�
−
𝛿𝛿𝛿𝛿
.
𝐾𝐾 ∗
𝑠𝑠
𝐼𝐼 ∗
𝐼𝐼 ∗

(26)

In the following analysis, we specify the production function and the investment
adjustment cost function as in (21). In this case we have𝑟𝑟 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) =𝛿𝛿
11 − 𝛼𝛼 𝑒𝑒(𝑟𝑟 + 𝛿𝛿)[1 + 𝜓𝜓𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )].
�1 + 𝜓𝜓𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )� −
𝑠𝑠
2
𝛼𝛼

(27)

Here, we make the following assumption.

𝛿𝛿 1 − 𝛼𝛼 𝑒𝑒(𝑟𝑟 + 𝛿𝛿) > 0
−
2𝑠𝑠
𝛼𝛼

(28)

Under this assumption we can easily confirm
𝑟𝑟𝑌𝑌∗ > 0,(29)

𝑟𝑟𝑟𝑟∗ < 0,where 𝑟𝑟𝑌𝑌∗ ≡ 𝜕𝜕𝑟𝑟 ∗ ⁄𝜕𝜕 𝑌𝑌 𝑒𝑒 and 𝑟𝑟𝑟𝑟∗ ≡ 𝜕𝜕𝑟𝑟 ∗ ⁄𝜕𝜕 𝑟𝑟 𝑒𝑒 . 𝑟𝑟𝑌𝑌∗ > 0 means that a rise in the expected

aggregate demand 𝑌𝑌 𝑒𝑒 , which increases output (aggregate income) in the short-run

equilibrium (see (19)), raises the capital income (i.e., the profit of the firm). Similarly,𝑟𝑟𝑟𝑟∗ < 0 means that a rise in the expected interest rate 𝑟𝑟 𝑒𝑒 , which reduces output(aggregate income) in the short-run equilibrium (see (19)), lowers the capital income.

As in the previous section, we assume that if the firm’s expectations (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) deviate

from their realized values in the short-run steady state, the firm adjusts the expectations

in response to their realized values. The expectation revision process therefore can beformulated as follows.
𝑌𝑌𝑒𝑒̇ = 𝛽𝛽1 [𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) − 𝑌𝑌 𝑒𝑒 ]𝑟𝑟𝑒𝑒̇ = 𝛽𝛽2 [𝑟𝑟 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) − 𝑟𝑟 𝑒𝑒 ](𝛽𝛽1 > 0)(𝛽𝛽2 > 0)(30.a)(30.b)

As is proved in the Appendix 1, there exists the unique steady state (𝑌𝑌 𝑒𝑒∗ , 𝑟𝑟 𝑒𝑒∗ ) in thisdynamic system. Figure 4 depicts the phase diagram of this system. As can be seen fromthis figure, the steady state is stable. (See Appendix 2 for a mathematical proof of local
16

stability). Therefore, even if the initial values of (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) are chosen arbitrarily, theeconomy eventually converges to this long-run steady state where both expectations are
self-fulfilling. In the following, we explain how the economy evolves when the expected
aggregate demand 𝑌𝑌 𝑒𝑒 suddenly rises in the long-run steady state.
(Figure 4 around here)

Suppose that the economy was initially located at 𝐸𝐸0 in Figure 4. We denote the firm’s

expectations at 𝐸𝐸0 as (𝑌𝑌0𝑒𝑒 , 𝑟𝑟0𝑒𝑒 ). Since 𝐸𝐸0 is the long-run steady state, 𝑌𝑌0𝑒𝑒 = 𝑌𝑌 𝑒𝑒∗ and

𝑟𝑟0𝑒𝑒 = 𝑟𝑟 𝑒𝑒∗ holds. Suppose that 𝑌𝑌0𝑒𝑒 suddenly rises while 𝑟𝑟0𝑒𝑒 remains unchanged (i.e., 𝑌𝑌1𝑒𝑒 >𝑌𝑌0𝑒𝑒 and 𝑟𝑟1𝑒𝑒 = 𝑟𝑟0𝑒𝑒 ). Then, the economy jumps from 𝐸𝐸0 to 𝐸𝐸1 . We denote the realized output
and interest rate in the short-run steady state under these new expectations (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) as
(𝑌𝑌1∗ , 𝑟𝑟1∗ ). Recall that the followings hold (see (19) and (29)).
𝑌𝑌𝑌𝑌∗ > 0, 𝑌𝑌𝑟𝑟∗ < 0,

𝑟𝑟𝑌𝑌∗ > 0,

𝑟𝑟𝑟𝑟∗ < 0.

(31)

Considering (31), we can confirm 𝑌𝑌1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) > 𝑌𝑌0∗ (𝑌𝑌0𝑒𝑒 , 𝑟𝑟0𝑒𝑒 ) and 𝑟𝑟1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) > 𝑟𝑟0∗ (𝑌𝑌0𝑒𝑒 , 𝑟𝑟0𝑒𝑒 ) ,where 𝑌𝑌1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) and 𝑟𝑟1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) are the realized output and interest rate in 𝐸𝐸1 . Namely,when the economy jumps from 𝐸𝐸0 to 𝐸𝐸1 , the realized output and interest rate in the newsteady state become higher. Since the realized output is an increasing function of theinvestment level 𝐼𝐼𝑡𝑡 (see (17.b)) and 𝐼𝐼𝑡𝑡 is an increasing function of the “stock price” 𝑞𝑞𝑡𝑡

(see (7.a)), the investment level and the stock price in the short-run steady state of 𝐸𝐸1

are also higher than those of 𝐸𝐸0 . On the other hand, how the MEC (the marginal

efficiency of capital defined in (11)) changes when the economy jumps from 𝐸𝐸0 to 𝐸𝐸1 isambiguous. This is because the MEC increases as 𝑌𝑌 𝑒𝑒 increases and decreases as 𝐾𝐾𝑡𝑡

increases (see (12)), and both 𝑌𝑌 𝑒𝑒 and 𝐾𝐾𝑡𝑡 in the short-run steady state of 𝐸𝐸1 are higher
than those of 𝐸𝐸0 .

As can be seen from the directions of arrows in Figure 4, at 𝐸𝐸1 the expected aggregate

demand 𝑌𝑌1𝑒𝑒 is revised downward while the expected interest rate 𝑟𝑟1𝑒𝑒 is revised upward,

which implies that 𝑌𝑌1∗ < 𝑌𝑌1𝑒𝑒 and 𝑟𝑟1∗ > 𝑟𝑟1𝑒𝑒 hold. As a result, the economy will eventuallyreach 𝐸𝐸2 . Taking (31) into account, we can easily confirm 𝑌𝑌2∗ (𝑌𝑌2𝑒𝑒 , 𝑟𝑟2𝑒𝑒 ) < 𝑌𝑌1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ) and17

𝑟𝑟2∗ (𝑌𝑌2𝑒𝑒 , 𝑟𝑟2𝑒𝑒 ) < 𝑟𝑟1∗ (𝑌𝑌1𝑒𝑒 , 𝑟𝑟1𝑒𝑒 ). Namely, when the economy moves from 𝐸𝐸1 to 𝐸𝐸2 , the realized

output and interest rate in the short-run steady state become lower. Furthermore, since
the realized output declines during this process, the investment level and the stock pricealso fall.
From the directions of arrows in Figure 4, both 𝑌𝑌2𝑒𝑒 and 𝑟𝑟2𝑒𝑒 are revised downward at

𝐸𝐸2 , which implies that 𝑌𝑌2∗ < 𝑌𝑌2𝑒𝑒 and 𝑟𝑟2∗ < 𝑟𝑟2𝑒𝑒 hold. As a result, the economy will

eventually reach 𝐸𝐸3 . It is ambiguous how the realized output and interest rate willchange when the economy moves from 𝐸𝐸2 to 𝐸𝐸3 . Namely, it is not clear which is larger,

𝑌𝑌3∗ (𝑌𝑌3𝑒𝑒 , 𝑟𝑟3𝑒𝑒 ) or 𝑌𝑌2∗ (𝑌𝑌2𝑒𝑒 , 𝑟𝑟2𝑒𝑒 ) , (and 𝑟𝑟3∗ (𝑌𝑌3𝑒𝑒 , 𝑟𝑟3𝑒𝑒 ) or 𝑟𝑟2∗ (𝑌𝑌2𝑒𝑒 , 𝑟𝑟2𝑒𝑒 ) ). This is because a downward

revision of 𝑌𝑌 𝑒𝑒 has negative impacts on both the realized output and interest rate, while

a downward revision of 𝑟𝑟 𝑒𝑒 has positive impacts on them. Since the change in therealized output is ambiguous, the changes in the investment level and the stock pricefrom 𝐸𝐸2 to 𝐸𝐸3 are also ambiguous.

Subsequent changes in the economy can be discussed in the same way. The results are

summarized in Table 1. Changes in the investment level and the stock price are omitted
because they move in the same direction as the realized output. Changes in the MEC are
also omitted because they are always ambiguous.

(Table 1 around here)

6. Concluding remarks
In this paper we have constructed a simple model that closely reflects Keynes' ownview of business cycle which is stated in Chapter 22 of General Theory. Our model istraditional in that the equilibrium output is determined through the principle of effective
demand, but the main difference from the standard post Keynesian models is to explicitly
introduce the intertemporal optimization of the firm’s investment decision undersubjective expectations. Using this framework, we demonstrated that Keynes’ view ofbusiness cycles can be reproduced as a deviation from the long-run steady state where
the expectations are self-fulfilling.
18

In this paper we made a very simplifying assumption regarding the expectation
formation that the firm adaptively adjusts the expectations in response to their realized
value only in the short-run steady state. While this assumption is necessary informulating Keynes's view in its simplest form, this assumption is rather ad hoc. Keynes
himself thought that booms are caused by the over-investment which is brought about
by the optimistic expectations that are destined to disappointment, and that the economy
will enter a phase of recession after the sudden collapse of such optimistic expectations.
As pointed out in footnotes 1 and 5, attempts to formulate boom and bust cycles driven
by waves of optimism and pessimism have been vigorously developed by Branch and
McGough (2010), Branch and Evans (2011), De Grauwe (2011, 2012), Franke (2012,2014), Flaschel et.al (2018) and so on. In these studies there are two types of agents (i.e.,
one who forms forward-looking expectation and the other backward-looking expectation)and the ratio of these two types changes endogenously. While it seems difficult to regard
these models as a faithful formulation of Keynes's business cycle theory, these models
are superior to our model in that they formulate the waves of optimism and pessimism
based on the findings of behavioral economics and construct models of endogenous
business cycles. It will be a future task to elaborate our model by referring to suchformulations of expectation formation.

Appendix 1: Proof of uniqueness of the long-run steady state in expectation dynamics
In this appendix we show that there exists the unique long-run steady state for
expectation dynamics given by (30.a) and (30.b). In the long-run steady state we havethe followings.
�𝑌𝑌𝑒𝑒̇ = 0�(𝑟𝑟𝑒𝑒̇ = 0)First, we can easily show

�𝑌𝑌𝑒𝑒̇ = 0�(𝑟𝑟𝑒𝑒̇ = 0)

𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) = 𝑌𝑌 𝑒𝑒
𝑟𝑟 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) = 𝑟𝑟 𝑒𝑒

𝑑𝑑𝑟𝑟 𝑒𝑒 1 − 𝑌𝑌𝑌𝑌∗=< 0,
𝑑𝑑𝑌𝑌 𝑒𝑒𝑌𝑌𝑟𝑟∗

𝑑𝑑𝑟𝑟 𝑒𝑒
𝑟𝑟𝑌𝑌∗
=
> 0,
𝑑𝑑𝑌𝑌 𝑒𝑒 1 − 𝑟𝑟𝑟𝑟∗
19(A.1.a)(A.1.b)(A.2.a)(A.2.b)where we use the fact that under the assumptions (20.b) (or (24)) and (20.c) the value of
𝑌𝑌𝑌𝑌∗ evaluated with (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) that satisfies 𝑌𝑌𝑒𝑒̇ = 0 satisfies 𝑌𝑌𝑌𝑌∗ < 1 (or 1 − 𝑌𝑌𝑌𝑌∗ > 0). (A.2.a)(resp. (A.2.b)) means that the 𝑌𝑌𝑒𝑒̇ = 0 (resp. 𝑟𝑟𝑒𝑒̇ = 0) line is an downward (resp. upward)
sloping curve in the (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) plane.

Next, let us examine the values of 𝑌𝑌 𝑒𝑒 which satisfy 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , ∞) = 𝑌𝑌 𝑒𝑒 and 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 0) =𝑌𝑌 𝑒𝑒 in turn, where 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , ∞) ≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟 𝑒𝑒 →∞ 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) and 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 0) ≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟 𝑒𝑒 →0 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ).

From (22), 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , ∞)�≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟 𝑒𝑒 →∞ 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )� = 0 holds. From (19) and (21), 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) is

given by 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) = 𝑠𝑠 −1 [𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ) + 2−1 𝜓𝜓{𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )}2 ]. Hence, 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , ∞) = 0 also holds.

In sum, when 𝑟𝑟 𝑒𝑒 approaches to infinity, the value of 𝑌𝑌 𝑒𝑒 which satisfies 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , ∞) = 𝑌𝑌 𝑒𝑒
approaches to zero.

On the other hand, when 𝑟𝑟 𝑒𝑒 approaches to zero, the coefficient 𝐴𝐴 in (22) approaches

to some finite value 𝐴𝐴̅�= 𝛿𝛿 −𝛼𝛼 [(1 − 𝛼𝛼)/𝛼𝛼]1−𝛼𝛼 𝑤𝑤 −(1−𝛼𝛼) � , so 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 0)�≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟 𝑒𝑒 →0 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )�

can be depicted as follows.

(Figure 5 around here)

Accordingly, under the assumptions (20.b) (or (24)) and (20.c), 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 0) can also be

depicted by Figure 2. Therefore, the value of 𝑌𝑌 𝑒𝑒 which satisfies 𝑌𝑌 ∗ (𝑌𝑌 𝑒𝑒 , 0) = 𝑌𝑌 𝑒𝑒 will besome finite value.

Finally, let us examine the values of 𝑟𝑟 𝑒𝑒 which satisfies 𝑟𝑟 ∗ (0, 𝑟𝑟 𝑒𝑒 ) = 𝑟𝑟 𝑒𝑒 , where𝑟𝑟 ∗ (0, 𝑟𝑟 𝑒𝑒 ) ≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑌𝑌 𝑒𝑒→0 𝑟𝑟 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 ). From (22), 𝐼𝐼 ∗ (0, 𝑟𝑟 𝑒𝑒 )�≡ 𝑙𝑙𝑙𝑙𝑙𝑙𝑌𝑌 𝑒𝑒→0 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 𝑟𝑟 𝑒𝑒 )� = 0 holds. From(27), 𝑟𝑟 ∗ (0, 𝑟𝑟 𝑒𝑒 ) can be calculated as𝑟𝑟 ∗ (0, 𝑟𝑟 𝑒𝑒 ) =𝛿𝛿 1 − 𝛼𝛼 𝑒𝑒(𝑟𝑟 + 𝛿𝛿).
−𝑠𝑠
𝛼𝛼Hence the value of 𝑟𝑟 𝑒𝑒 which satisfies 𝑟𝑟 ∗ (0, 𝑟𝑟 𝑒𝑒 ) = 𝑟𝑟 𝑒𝑒 approaches to
1 1 − 𝛼𝛼𝑟𝑟 𝑒𝑒 = 𝛼𝛼𝛼𝛼 � −
�.
𝑠𝑠
𝛼𝛼

(A.3)

(A.4)

We can confirm that the sign of 𝑟𝑟 𝑒𝑒 in (A.4) is positive under the assumption (28).

From the above discussion, both the 𝑌𝑌𝑒𝑒̇ = 0 line and the 𝑟𝑟𝑒𝑒̇ = 0 line can be depicted asshown in Figure 4, and there exists the unique long-run steady state for the expectation20

dynamics.

Appendix 2: Proof of local stability of expectation dynamics
In this appendix we show the local stability of expectation dynamics given by (30.a)and (30.b). By linearly approximating (30.a) and (30.b) in the neighborhood of the longrun steady state (𝑌𝑌 𝑒𝑒∗ , 𝑟𝑟 𝑒𝑒∗ ), we have𝑒𝑒̇𝑌𝑌 𝑒𝑒 − 𝑌𝑌 𝑒𝑒∗
�𝑌𝑌𝑒𝑒 � = 𝐵𝐵 � 𝑒𝑒
�.
𝑟𝑟 − 𝑟𝑟 𝑒𝑒∗
𝑟𝑟 ̇𝛽𝛽 (𝑌𝑌 ∗ − 1)�𝐵𝐵 ≡ � 1 𝑌𝑌 ∗𝛽𝛽2 𝑟𝑟𝑌𝑌

𝛽𝛽1 𝑌𝑌𝑟𝑟∗��
𝛽𝛽2 (𝑟𝑟𝑟𝑟∗ − 1)

The signs of trace and determinant of the coefficient matrix 𝐵𝐵 are as follows.
𝑡𝑡𝑡𝑡(𝐵𝐵) = 𝛽𝛽1 (𝑌𝑌𝑌𝑌∗ − 1) + 𝛽𝛽2 (𝑟𝑟𝑟𝑟∗ − 1) < 0det(𝐵𝐵) = 𝛽𝛽1 𝛽𝛽2 [(𝑌𝑌𝑌𝑌∗ − 1)(𝑟𝑟𝑟𝑟∗ − 1) − 𝑌𝑌𝑟𝑟∗ 𝑟𝑟𝑌𝑌∗ ] > 0(B.1)(B.2.a)(B.2.b)Hence, two eigenvalues of the coefficient matrix 𝐵𝐵 are both negative, which means thatthe expectation dynamics is locally stable.

21

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24

𝐼𝐼𝑡𝑡

𝐾𝐾𝑡𝑡̇ = 0

saddle path𝐼𝐼𝑡𝑡̇ = 0
(Figure 1: The optimal investment path)

𝑌𝑌 ∗

𝐾𝐾𝑡𝑡

45 degree line25

𝑌𝑌 𝑒𝑒∗(Figure 2: Figure of function (19))

𝑌𝑌 𝑒𝑒

𝐼𝐼𝑡𝑡

𝐾𝐾𝑡𝑡̇ = 0

𝐸𝐸2

saddle path
𝐸𝐸3

𝐸𝐸1

(b)

𝐸𝐸0

(c)
(a)
𝐼𝐼𝑡𝑡̇ = 0

𝐾𝐾𝑡𝑡

(Figure 3.a : Business cycle as a deviation from the long-run steady state)

26

𝑌𝑌 ∗

45 degree line

𝐸𝐸0

𝐸𝐸3

𝐸𝐸2

𝐸𝐸1

(a)
(b)
(c)
𝑌𝑌3𝑒𝑒

𝑌𝑌0∗

𝑌𝑌2𝑒𝑒

𝑌𝑌1𝑒𝑒

𝑌𝑌 𝑒𝑒

(Figure 3.b : Business cycle as a deviation from the long-run steady state)

𝑟𝑟 𝑒𝑒

𝐸𝐸3
𝐸𝐸4

𝐸𝐸2
𝐸𝐸0

𝐸𝐸5

𝑟𝑟 𝑒𝑒̇ = 0

𝐸𝐸1

𝑌𝑌 𝑒𝑒̇ = 0

𝑌𝑌 𝑒𝑒(Figure 4: Phase diagram of (30.a) and (30.b))

𝑌𝑌 𝑒𝑒
27

𝑌𝑌 𝑒𝑒 = 𝐴𝐴(1 + 𝜓𝜓𝐼𝐼 ∗ )1−𝛼𝛼 𝐼𝐼 ∗𝑌𝑌 𝑒𝑒 = 𝐴𝐴̅(1 + 𝜓𝜓𝐼𝐼 ∗ )1−𝛼𝛼 𝐼𝐼 ∗(Figure 5: Figure of function 𝐼𝐼 ∗ (𝑌𝑌 𝑒𝑒 , 0))

𝑌𝑌 ∗
𝑟𝑟

∗

𝐸𝐸0 → 𝐸𝐸1＋＋

𝐸𝐸1 → 𝐸𝐸2

𝐸𝐸2 → 𝐸𝐸3

－

？

－

？

𝐼𝐼 ∗

𝐸𝐸3 → 𝐸𝐸3＋＋

𝐸𝐸4 → 𝐸𝐸5？
？

(Table 1: The effects of a sudden rise in 𝑌𝑌 𝑒𝑒 in the dynamic system of (30.a) and (30.b))28