# Monopoly, long-run capital accumulation and bubbles

Bertrand Crettez

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Bertrand Crettez. Monopoly, long-run capital accumulation and bubbles. 1996. halshs-05090944

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CAHIERS ECO & MATHS

96.26

MONOPOLY, LONG-RUN CAPITAL ACCUMULATION
AND BUBBLES

Bertrand CRETTEZ

CEME
Monopoly, Long-run Capital accumulation and
Bubbles

Bertrand CRETTEZ

C.E.R.D.O, University of Paris IX-Dauphine
and C.E.M.E, University of Paris I, U.R.A. C.N.R.S. D0924,
12 Place du Panthéon, 75005 Paris, France

January 1996
Résumé

Cet article analyse les effets de portefeuilles macroéconomiques dus aux
rentes de monopoles. Nous montrons que les monopoles peuvent être à
l'origine d'une éviction du capital physique dans les portefeuilles des agents.
De ce fait, ils peuvent être à l'origine d'une amélioration du bien être dans
une économie connaissant une suraccumulation du capital. Néammoins, dans
une telle situation une politique de la concurrence est toujours désirable.
Cette politique a pour conséquence une diminution des rentes de monopoles
puisqu'elle favorise l'entrée sur les marchés peu concurrentiels. Cependant,
la baisse des rentes n'impliquent pas nécessairement un accroissement de
l'accumulation du capital. En effet, nous montrons que la valeur des rentes
capitalisées des rentes de monopole se transforme en bulles. Ceci suffit pour
freiner l'accumulation de capital et dans le même temps, les distorsions
causées par les monopoles tendent à disparaître. Nous montrons aussi que si
l'équilibre concurrentiel est inefficace, alors l'équilibre avec monopoles se rap-
proche de l'allocation de la règle d'or lorsque le nombre de firmes présentes
sur les marchés tend vers l'infini.

Numéro de classification du JEL : D42, D43, D91, D92, E64, K21.

Mots clés : Modèles à générations imbriquées d'agents, monopole, Bulle.

Abstract

This paper investigates the macroeconomic portfolio implications of monopoly rents. It is shown that monopolies may be an important way of diverting capital from portfolios and then welfare improving by reducing capital over-accumulation. However, an anti-trust policy is still desirable. Such policy implies a decrease in rents by increasing entry but does not result in an in-crease in capital accumulation. Indeed, it is shown that the capitalized value of monopoly rents become a pure bubble. This is sufficient to refrain the over-accumulation of capital while in the same time distortions caused by monopolies tend to vanish. We also show that, if the competitive equilibrium is inefficient, then in an equilibrium with monopolies, the steady-state capi-tal increases towards its golden rule level when the number or firms tends to infinity.

JEL Classification Number : D42, D43, D91, D92, E64, K21.

Keywords : Overlapping Generations Models, Monopoly, Bubble.
# 1 Introduction

The macroeconomic portfolio implications of monopoly rents seem to have received little attention in the literature. Two important exceptions are Laitner [1982] analysis of the effect of monopoly on the capital stock and Eaton [1989] study of the elimination of rent-creating policies on monopoly wealth and international debt.

Laitner shows that capitalized monopoly rents reduce welfare in steady state by displacing capital in individual portfolios.

Eaton shows that shifts in policy that affect the capitalized value of monopoly rents in a small open economy generate correlated movements in the current account and the real exchange rate.

Both analyses make use of an overlapping generations model (Laitner constructs a two sectors version of the Diamond model whereas Eaton studies an imperfect competition version of the Blanchard [1985] model).

In his seminal analysis, Laitner studies the case of an economy whose steady-state competitive equilibrium is under the golden rule of capital accumulation. In this context, an equilibrium with monopolies is Pareto-dominated by a competitive equilibrium for two reasons. Indeed the welfare loss due to a a lower steady-state capital stock<fnref n="1" /> adds to the welfare losses caused by monopolies. Using numerical simulations, Laitner shows that the first kind of welfare loss is far more important than the second. Indeed, the first kind of loss amounts to 10% of gross national product whereas the second kind of loss is barely higher than 1% of gross national product. This result is a strong argument in favor of antitrust policies.

In this paper we aim at discussing this later conclusion when in a competitive equilibrium the economy is over-accumulating (its steady-state capital stock is greater than the golden rule).

It is well known that land (Allais [1947], Samuelson [1979], Homburg [1992]), public debt (Diamond [1965]), social security (Samuelson [1975]) or bubbles (Tirole [1985]) can prevent an economy from over-accumulating. In that respect, monopolies may be another important way of diverting capital from portfolios and then welfare improving by reducing capital over-accumulation. However, in the latter case, one must balance the efficiency

<footnote n="1">¹Capitalized monopoly rents displace capital in individual portfolios.</footnote>
gains due to the decrease in the stock of capital with the welfare losses caused by monopolies.

Suppose then that an antitrust policy enhancing free-entry is suddenly implemented. As this implies a decrease in rents, this policy should result in an increase in capital accumulation and may therefore lead to a decrease in the steady-state welfare.

The purpose of this paper is to discuss this conjecture. Using a very simple version of a two sectors overlapping generations model first proposed by Allais [1947], we offer a counter-example to the above conjecture : an antitrust policy directed towards free entry can result in an increase in welfare despite the fact that monopoly-rents are decreasing.

We show indeed, that the capitalized value of monopoly rents become a pure bubble. This is sufficient to refrain the over-accumulation of capital while in the same time distortions caused by monopolies tend to vanish.

The paper is organized as follows.

Section two sets out the model. Section three is devoted to the study of the golden rule. Section four studies the equilibrium of the economy with imperfect (Cournot) competition. In section five we present a proof of our assertion that, in an overaccumulating economy, an antitrust policy does not increase the capital stock and further is welfare improving.

# 2 The Model

We make use of an overlapping generations model à la Allais [1947]. The economy consists of agents and firms.

## 2.1 Agents

In each date $t$ $L_t$ identical non altruistic agents are born. Each individual lives for two periods, supplies inelastically one unit of labor and consumes in both periods.

An agent born in $t$ has a lifetime utility function of the form:

$$U_t = (1-s) \ln c_t + s \ln d_{t+1} \quad (1)$$

where : $0 < s < 1$.
## 2.2 Firms

There are two productions sectors. Firms act competitively in the first sector; Cournot competition prevails in the second sector. It is assumed that there is one representative firm in the first sector and N firms in the second one.

The representative firm in the first sector produces a capital good according to a linear production function :

$$Y_t^k = aL_t^1 \quad (2)$$

where $a > 0$, and $L_t^1$ is the amount of labor used in the capital good sector.

The N firms in the second sector produce a consumption good according to a Cobb-Douglas production function :

$$Y_{i,t}^c = (K_{i,t}^2)^\alpha (L_{i,t}^2)^{1-\alpha} \quad (3)$$

where $0 < \alpha < 1$, $K_{i,t}^2$ and $L_{i,t}^2$ stand respectively for the capital and the labor used by firm i. We shall assume that capital fully depreciates within a period.

There are three aspects on which the above model is simpler than the Laitner's one.

First, there is only one consumption good, whereas in the Laitner's model there are two goods (the good produced by the first production sector may either be consumed or invested).

Second, instead of using a CRRA lifetime utility function with Cobb-Douglas sub-utilities as Laitner does, we only use a Cobb-Douglas lifetime utility function.

Third, we use a linear production function in the capital good sector whereas Laitner uses the same Cobb-Douglas production function in both sector<fnref n="2" />.

<footnote n="2">²One should also add that we depart slightly from the original Allais [1947] model since we exclude land as an input in the production function of the consumption good. Also, the number of agents is not constant. With this respect, our model can be thought of as being a complete information version of the Zeira [1987] model. There are two differences between our model and Zeira's one. First, whereas in our model the first production sector produces a capital good, in the Zeira model, the first sector produces a good which can be either used as a capital or as a consumption good. Second, an agent consumes both consumption goods but the (Cobb-Douglas) preferences are stochastically determined.</footnote>
These different assumptions have been introduced in order to simplify the study of the dynamics of the system and to allow us to concentrate on the steady-state of the economy.

The main purpose of the Laitner's paper is to give some plausible order of magnitude of the welfare losses due to the under-accumulation of capital caused by monopolies rents. Our purpose is somewhat different, this is why we choose a different and simpler modelling strategy.

## 3 The Golden Rule

In this section, we compute a version of the Allais [1947]-Diamond [1965]- Samuelson [1958] social optimum.

The physical constraints faced by the social planner are<fnref n="3" />:

$$Y_t^c = L_t c_t + L_{t-1} d_t \quad (4)$$

$$Y_t^k = K_{t+1} \quad (5)$$

$$L_t = L_t^1 + L_t^2 \quad (6)$$

$$K_t = K_t^2 \quad (7)$$

It is useful to restate the model with *per capita* variables and to concentrate on the steady-state.

Define:

$$\varphi = \frac{L_t^2}{L_t}$$

$$k_t = \frac{K_t}{L_t} = \frac{a(1 - \varphi)}{1 + n}$$

$$q_t = \frac{Y_t^c}{L_t} = k^\alpha \varphi^{1-\alpha} = \left(\frac{a(1-\varphi)}{1+n}\right)^\alpha \varphi^{1-\alpha} \quad (8)$$

The problem of the social planner is then:

<footnote n="3">³In this section we suppose without loss of generality that there exists only one firm in the second sector.</footnote>
$$ \max_{c, d, \varphi} (1-s) \ln c + s \ln d $$

such that :

$$ q = \left(\frac{a}{1+n}\right)^{\alpha} (1-\varphi)^{\alpha} \varphi^{(1-\alpha)} = c + \frac{d}{1+n} $$

The optimal choice of $\varphi$ will help us to determine the steady-state golden rule capital.

It is immediate to check that the optimal $\varphi$ satisfies:

$$ \varphi = 1 - \alpha \qquad (9) $$

Hence, the golden rule capital $k^g$ is:

$$ k^g = \frac{a\alpha}{1+n} \qquad (10) $$

# 4 Equilibrium

This section is devoted to the study of the equilibrium of our economy.

In the sequel we shall assume that agents buy the capital good produced during their youth and will rent it to firms during the next period. As will be shown, firms in the consumption good sector have a strictly positive value, namely the discounted value of current and anticipated future monopolies rents. Hence, wealth of agents consists of capital and claims on firms.

Without loss of generality, we will take the capital good as the *numéraire*.

## 4.1 Agents

Let $s_t^k$ and $s_t^m$ be respectively the part of an agent savings invested in capital and in the total claims on firms. $p_t$, $M_t$ and $\Pi_t$ will denote respectively the price of the consumption good in term of the capital good, the total value of the claims on firms, and the total monopoly rents. $w_t$ is the wage rate and $r_t$ is the rate of interest between period $t-1$ and $t$. Both are expressed in term of the capital good.

The problem of an agent born at time $t$ is:
$$ \max_{c_t, d_{t+1}, s_t^k, s_t^m} (1-s) \ln c_t + s \ln d_{t+1} $$

such that :

$$ \begin{cases} p_t c_t + s_t^k + M_t s_t^m = w_t \\ p_{t+1} d_{t+1} = (1+r_{t+1}) s_t^k + (M_{t+1} + \Pi_{t+1}) s_t^m \end{cases} \quad (11) $$

Individual can invest their savings in claims on firms or in capital so that the total return on claims equalizes that on capital :

$$ 1 + r_{t+1} = \frac{M_{t+1} + \Pi_{t+1}}{M_t} \quad (12) $$

Under this assumption, the optimal intertemporal choices of the individual are given by :

$$ c_t = \frac{(1-s)w_t}{p_t} \quad (13) $$

$$ d_{t+1} = s_t \frac{(1+r_{t+1})w_t}{p_{t+1}} \quad (14) $$

$$ s_t = sw_t = s_t^k + M_t s_t^m \quad (15) $$

where $0 < s_t^m < 1$.

## 4.2 Firms

First, let us consider the capital good sector. The profit of the representative firm in that sector is given by :

$$ B_t^k = Y_t^k - w_t L_t^1 = (a - w_t) L_t^1 \quad (16) $$

Clearly, in equilibrium, $w_t = a$ and the firm is ready to produce any level of capital good.

Now, consider the consumption good sector.

Recall that there are $N$ similar firms. In what follows, we will compute the Cournot-Nash equilibrium of that sector given the factor prices.
A firm *i* has a profit which writes :

$$B_{i,t}^{c} = p_{t}Y_{i,t}^{c} - w_{t}L_{i,t}^{2} - (1+r_{t})K_{i,t}^{2} \quad (17)$$

The cost function of such a firm is :

$$C(Y_{i,t}^c) = w_t^{1-\alpha}(1+r_t)^\alpha \alpha^{-\alpha} (1-\alpha)^{-(1-\alpha)} Y_{i,t}^c = \Omega_t Y_{i,t}^c \quad (18)$$

where $\Omega_t \equiv w_t^{1-\alpha}(1+r_t)^\alpha \alpha^{-\alpha}(1-\alpha)^{-(1-\alpha)}$.

Let $D_t$ denotes the total value of the demand to the second sector the amount of which being perfectly anticipated by firms. In equilibrium, one has :

$$D_t = L_t(1-s)w_t + L_{t-1}s_{t-1}(1+r_t)w_{t-1} \quad (19)$$

The equilibrium of the market of the second sector is:

$$Y_t^c = \sum_{i=1}^{N} Y_{i,t}^c = \frac{D_t}{p_t} \quad (20)$$

The profit of a firm *i* in the second sector is then:

$$B_{i,t}^{c} = \frac{D_t}{Y_t^c} Y_{i,t}^{c} - \Omega_t Y_{i,t}^{c} \quad (21)$$

In equilibrium, abstracting from Ford effects, firms choose their quantities such that :

$$p_t = \frac{N}{N-1} \Omega_t \quad (22)$$

The total monopoly rents, of the sector are then :

$$\Pi_t = \frac{Y_t^c}{N-1} \Omega_t \quad (23)$$

## 4.3 The equilibrium

We shall be interested in the dynamics of the model. Let us first write the equations of the equilibrium :
$$Y_t^k = K_{t+1} = L_t s_t^k \qquad (24)$$

$$Y_t^c = L_t c_t + L_{t-1} d_t \qquad (25)$$

$$K_t = \sum_{i=1}^{N} K_{i,t}^2 \qquad (26)$$

$$L_t = L_t^1 + \sum_{i=1}^{N} L_{i,t}^2 \qquad (27)$$

$$1 = L_t s_t^m \qquad (28)$$

We already know that in equilibrium $w_t$ equals $a$. Let us look for the equilibrium value of $r_t$. After some tedious algebra<fnref n="4" />, one arrives at :

$$1 + r_t = \frac{a(1 - s)}{k_t \frac{N}{\alpha(N-1)} - \frac{as}{1+n}} \qquad (29)$$

where $k_t = \frac{K_t}{L_t}$.

Let us add equations (24) and (25) to get :

$$K_{t+1} + M_t = L_t s_t = L_t a s \qquad (30)$$

In *per capita* variables this writes :

$$(1 + n)k_{t+1} + m_t = a s \qquad (31)$$

Now after some algebra one obtains<fnref n="5" /> :

$$m_t = (1 + n)(m_{t+1}(k_{t+1}\frac{N}{\alpha(N-1)} - \frac{as}{1+n})\frac{1}{a(1-s)} + \frac{k_{t+1}}{\alpha(N-1)}) \qquad (32)$$

Combining these two equations, one finally has:

$$m_{t+1} = \left( m_t \left( 1 + \frac{1}{\alpha(N-1)} \right) - \frac{as}{\alpha(N-1)} \right) \frac{a(1-s)}{as\left( \frac{N}{\alpha(N-1)} - 1 \right) - m_t \frac{N}{\alpha(N-1)}} \qquad (33)$$

<footnote n="4">⁴Combining (22) and (23), one gets : $P_t Y_t^c = N \Pi_t$. Now one can easily show that : $\Pi_t = \frac{(1+r_t)K_t}{\alpha(N-1)}$. Using these results and equations (13)-(15) in order to rearrange equation (25), one gets (29).</footnote>

<footnote n="5">⁵To get this result use (12), (29) and the fact that : $\frac{\Pi_{t+1}}{L_{t+1}(1+r_{t+1})} = \frac{k_{t+1}}{\alpha(N-1)}$.</footnote>
It is worth noting that there are no initial conditions for $m_t$. This is a pure forward-looking equation (see Sargent and Wallace [1973] and e.g. Farmer [1993]).

Indeed, the value of claims today depends on the value on claims tomorrow an so on.

The above equation can be restated as: $m_{t+1} = \phi(m_t)$. In the references quoted above, it is shown that the fixed point $m$ of $\phi$ - if it exists -, i.e. the steady-state value of $m$, is the only bounded equilibrium value of $m_t$ if an only if $\phi'(m) > 1$. The steady-state value of $m_t$, $m$ is :

$$m = \left[ \sqrt{\Delta} - \left( 1 + \frac{1}{\alpha(N-1)} + \frac{s}{1-s}\left(1 - \frac{N}{\alpha(N-1)}\right) \right) \right] \frac{\alpha a(N-1)(1-s)}{2N} \quad (34)$$

where :

$$\Delta = \left( 1 + \frac{s}{1-s} + \frac{1}{\alpha(N-1)} \left( 1 - \frac{Ns}{1-s} \right) \right)^2 + \frac{4Ns}{\alpha(1-s)(N-1)^2} \quad (35)$$

This condition is clearly checked on the following picture on which we graph the function $\phi$.

Insert figure one

The preceding argument guarantees that the only dynamic equilibrium of our model is the steady-state equilibrium. Hence we have :

**Proposition 1** *The only dynamic equilibrium is the steady-state equilibrium.*

This implies that the economy jumps to the steady-state equilibrium in one period.

The price to pay for this result is the simplifications introduced in the model, the most important one being the linearity of the production function in the capital good sector.

However, our framework is simple enough to perform comparative static exercises.

Before we do this, a few comments on the competitive equilibrium are in order. Suppose indeed that competition prevails in both sectors. Then there are no more profits in the consumption good sector. The remainder part of
the competitive equilibrium is quite alike the one we have just studied. In particular, the equilibrium equation for the market for capital goods is now : $K_{t+1} = N_t s W_t$; this reduce to $k_{t+1} = k = \frac{sa}{1+n}$. From this, one can draw the following conclusion: if if $s < \alpha$, there is under-accumulation with respect to the golden rule (equation (10); whereas if $s > \alpha$ there is over-accumulation.

# 5 Antitrust policy, Monopoly and Welfare

In this section, we first study the equilibrium of our economy when the number of firms in the sector of the consumption good is increasing. Actually, we will restrain ourself to the study of the equilibrium when the number of firms tends to infinity.

Such a situation could arise if an antitrust policy were implemented and permitted free-entry in the sector of the consumption good. What should happen seems pretty clear at a first glance. Indeed, as the number of firms increases, there is more competition. Then one would think that monopoly rents shrink and the value of claims on firms vanish.

In fact, if the competitive equilibrium is dynamically inefficient (i.e. its steady-state capital stock is over the golden rule), then, in an equilibrium with monopolies the monopoly rents indeed tend to zero when the number of firms in the second sector increases to infinity. Nevertheless the value of the firms remains strictly positive: there is a bubble on the value of firms in the second sector.

However, if the competitive equilibrium is efficient (i.e. its steady-state capital is under the golden rule), then both the monopoly rents and the value of claims on firms vanish when the number of firms in the second sector tend to infinity. This latter case is the one studied by Laitner [1982].

One has then:

**Proposition 2** • Suppose that, $s > \alpha$, so that in the absence of monopoly, the equilibrium is dynamically inefficient; then :

$$ \lim_{N \to \infty} m(N) = a(s - \alpha) $$

• Suppose that, $s < 0$, so that in the absence of monopoly, the equilibrium is efficient; then :
$$ \lim_{N \to \infty} m(N) = 0 $$

**Proof.** The proof follows from a close inspection of the dynamic equation of $m_t$ (equation (33)) or the equation of the steady-state value $m$ (equation (34)). ■

The graph below illustrates both cases.

Insert figure 2 and 3

The following result, which is a corollary of the preceding proposition, is of particular interest. Is says that, if the competitive equilibrium is inefficient ($s > \alpha$), then in an equilibrium with monopolies the steady-state per capita capital increases towards its golden rule level when the number or firms tends to infinity.

On the contrary, if the competitive equilibrium is efficient ($s < \alpha$), then in an equilibrium with monopolies the steady-state per capita capital increases towards its competitive steady-state level when the number or firms tends to infinity. One has:

**Proposition 3** • Suppose that, $s > \alpha$, then,

$$ \lim_{N \to \infty} k(N) = \frac{a\alpha}{1+n} $$

• Suppose that, $s < \alpha$, then,

$$ \lim_{N \to \infty} k(N) = \frac{as}{1+n} = k^g $$

**Proof.** The proof is easily obtained after combining equation (31) and the preceding proposition. ■

According to this proposition, an antitrust policy is much more efficient when there is over-accumulation of capital (in a competitive equilibrium) than when there is under-accumulation. In our view, this reinforces the case for an antitrust policy.
There is one issue which we have neglected. Indeed, we would like to address the following question: although it is desirable to implement a policy aiming at increasing entry, such a policy is not always possible. In this respect it would be interesting to know whether or not welfare is higher in an equilibrium with monopolies than in a competitive equilibrium. In fact, it is actually difficult to prove any statement by a direct comparison of steady-state utility levels. This is due to the cumbersome equilibrium expression of $m$. However, if one does not want to resort to a numerical exercise, one can rely on a local argument. In a neighborhood of infinity (i.e. when $\frac{1}{N} \to 0$ as $N \to \infty$), one can always find a number of firms which is such that the corresponding equilibrium Pareto-dominates the competitive equilibrium. Hence one has:

**Proposition 4** If the competitive equilibrium is dynamically inefficient, there exists a number of firms $\overline{N}$ such that, for $N \ge \overline{N}$, the equilibrium with $N$ firms Pareto-dominates the competitive equilibrium.

**Proof.** One has only to resort to an argument of continuity. We already know from the preceding proposition that if $s > \alpha$, i.e., if the competitive equilibrium is dynamically inefficient, then the limit equilibrium with monopolies is the golden rule of accumulation when the number of firms in the second sector tends to infinity. By definition, the golden rule delivers the highest level of utility in a steady-state. By continuity, one can always find a number $\overline{N}$ of firms such that the corresponding economy is as close as we want to the golden rule. Clearly, any economy with a number of firms greater or equal than $\overline{N}$ will deliver a welfare strictly greater than the one obtained in the inefficient competitive equilibrium. ■

How one is to interpret this result? It is well-known that a competitive equilibrium of an economy with overlapping generations may yield a Pareto-inefficient allocation of resources. Clearly an equilibrium with monopolies is not Pareto-efficient. But if the number of firms tend to infinity, i.e. if one approaches free competition from a situation of imperfect competition, the limit equilibrium may coincide with the golden rule and *not* the inefficient competitive equilibrium. This is because the value of firms is in fact a pure bubble.
# 6 Conclusion

This paper has investigated the macroeconomic portfolio implications of monopoly rents.

It has been shown that monopolies may be another important way of diverting capital from portfolios and then be welfare improving by reducing capital over-accumulation. However, an antitrust policy is still desirable.

Such policies imply a decrease in rents by increasing entry but does not result in an increase in capital accumulation when the competitive equilibrium of a given economy is dynamically inefficient. Indeed, we show that the capitalized value of monopoly rents become a pure bubble. This is sufficient to refrain the over-accumulation of capital while in the same time distortions caused by monopolies tend to vanish.

As a corollary, we have shown that, if the competitive equilibrium is inefficient, then in an equilibrium with monopolies the steady-state *per capita* capital increases towards its golden rule level when the number or firms tends to infinity.

Let us close on two issues which characterize the present analysis and the solution of which is on the agenda of future research.

First, one should test the robustness of our results by weakening the assumptions regarding preferences and technology made in the analysis.

Second, one should also test the robustness of our results by introducing different kind of imperfect competitions in the non-competitive sector.
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Eaton J (1989), Monopoly Wealth and International Debt, International Economic Review, Vol. 30, 1, 33-48.

Homburg S (1992), *Efficient Economic Growth*, Springer Verlag, Economic Studies.

Laitner J (1982), Monopoly and Long-run Capital Accumulation, Bell Journal of Economics, 13, 143-158.

Samuelson PA (1975), Optimum Social Security in a Life Cycle Growth Model, International Economic Review, 16, 3, 539-544.

Samuelson PA (1979), Land and the Rate of Interest, in *Theory for Economic Efficiency: Essays in Honor of Abba P. Lerner*, H.I. Greenfield, A.M. Levenson, W. Hamovitch, E. Rotwein editors, The M.I.T. Press, Cambridge, Massachusetts.

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<figure><img src="image_2.png" /><figcaption>Figure 1:</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2:</figcaption></figure>

case $s < \alpha$
<figure><img src="image_4.png" /><figcaption>Figure 3:</figcaption></figure>

case $s > \alpha$