Women`s Liberation as a Financial Innovation

Women’s Liberation as a Financial Innovation∗
Moshe Hazan
David Weiss
Hosny Zoabi
Tel Aviv University and
CEPR
Tel Aviv University
The New Economic
School
April 2015
Preliminary and Incomplete
Abstract
Over the course of the second half of the 19th century, states in the US,
which were entirely dominated by men, gave married women property rights.
Before this ”women’s liberation”, married women were subject to the laws
of coverture. Coverture had detailed laws as to which spouse had ownership
and control over various aspects of property both before and after marriage.
This paper develops a general equilibrium model with endogenous determination of women’s rights in which these laws affect portfolio choices, leading
to inefficient allocations. We show how technological advancement eventually leads to men granting rights, and in turn how these rights affect development. We show how key implications of the model are consistent with
cross-state empirical evidence in the US. Specifically, the dynamics of nonagricultural employment both after and before rights are granted fit exactly
with the model’s prediction.
∗
We thank Yannay Shanan and Alexey Khazanov for excellent research assistance. Hazan:
Eitan Berglas School of Economics, Tel Aviv University, P.O. Box 39040, Tel Aviv 69978, Israel.
e-mail: moshehaz@post.tau.ac.il. Weiss: Eitan Berglas School of Economics, Tel Aviv University,
P.O. Box 39040, Tel Aviv 69978, Israel. e-mail: dacweiss@gmail.com. Zoabi: The New Economic
School, 100 Novaya Street, Skolkovo Village, The Urals Business Center, Moscow, Russian Federation. e-mail: hosny.zoabi@gmail.com.
1
Introduction
Over the course of the second half of the 19th century, states in the US, which
were entirely dominated by men, gave married women property rights, while
England granted similar rights in 1870. Before this ”women’s liberation”, married women were subject to the laws of coverture.1 Coverture had two aspects.
First, in the eyes of the law, husband and wife were the same person. Second,
there were detailed laws as to which spouse had ownership and control over
various aspects of property both before and after marriage. This paper focuses
on the second aspect of coverture, property laws, in developing a theory as to
why men gave women rights.
Property was divided into multiple types. Personal property, including money,
stocks, furniture and livestock, became the husbands property entirely. He could
sell or give the property away, and even bequeath it to others. There was a limitation on this freedom to paraphernalia, which was personal property such as
clothing and jewelry, which the husband could sell or give away, but not bequeath. Real assets, such as land and real estate, became under the husbands
partial control. He could manage the assets as he saw fit, including the income
generated by the assets, but he could not sell or bequeath the property without
his wife’s consent.2
We argue that these laws influenced the investment portfolio choices women
made, and also had the effect of distorting capital markets, and thus allocations.
Women investing predominantly in real assets, such as land, rather than moveable assets, such as capital, led to a misallocation between the associated sectors
1
Coverture was an inherent aspect of British common law, and as such applied both in England and her colonies, including those that formed the United States. The property aspects of
these laws were undone over the course of the 19th century in both England and America. Note
that Canada and Australia also saw women’s rights greatly expanded in this period, though we
do not focus on their experiences in this paper.
2
See Combs (2005) for a description of these rights. For an excellent description of the general
responsibilities husbands and wives had to one another under coverture, see Basch (1982) Tables
1 and 2.
1
of the economy.3 As the productivity of capital-intensive industries, such as railroads, grew the effects of this factor misallocation became worse.4 Eventually,
these distortions were significant enough for men to want to give women rights.5
We develop a model in order to study men’s incentive to give women property
rights in the context of financial market efficiency.6
In the model, men have utility defined over their own consumption and the
bequest they leave to their children. These in turn are determined by overall
household income and the man’s bargaining power in the household. Bargain3
The notion that women’s property laws affected portfolio choices is grounded in historical evidence. Combs (2005) finds that these laws induced women to hold their wealth strategically, and that portfolios changed after rights were granted. In her sample of British shopkeepers
wives, those who were subject to coverture and not subject to coverture had nearly the same total
amount of assets, but their portfolio composition was dramatically different. She exploits the fact
that when rights were granted in 1870 to married women, they were not granted retroactively.
Combs thus studies the difference in portfolios of women who died at the same time but had
married either before or after the reform. Women with rights had half as much money in real
assets having nearly twice the amount in personal property. Clearly, the effects of coverture on
portfolio allocations were dramatic.
4
Maltby, Rutterford, Green, Ainscough, and van Mourik (2011) note two interesting facts
about railroads in England, which was clearly a capital-intensive industry. First, between 1853
and 1914, railroad stocks rose dramatically to represent roughly 40% of dividend and interest paying assets traded in London, representing the national portfolio (pg 161-162). Furthermore, there
was a great democratization of the stock market over this time period, as “In the years between
the 1840s and 1914, there was a transformation of the composition of both investments and the
investing public. No longer were investors confined to a wealthy elite largely located in London,
for they were increasingly found throughout the country and among the middle classes.” (pg
156). In particular, it is estimated that between 150,000 and 300,000 people held stock in British
railways by 1886 (pg 163). It is hard to imagine that the railroad industry would have been as
successful without the overall deepening of financial markets over this time period.
5
We argue that it was no coincidence that property rights were given in England in the middle
of a period of massive capital market development. For instance, Maltby, Rutterford, Green,
Ainscough, and van Mourik (2011) argues that there was “an enormous expansion in the volume
and variety of securities available to the investing public, especially from the 1860s onwards.
Between 1870 and 1913, new issues on the London capital market, for example, totaled 5.7 billion
pounds and among them were an increasing number of shares from the likes of British industrial
and commercial companies and foreign mines and plantations.” (pg 161).
6
One possible criticism of the idea that women’s property rights were important for aggregate
outcomes is the notion that perhaps women didn’t have much in the way of assets. Married
women’s labor force participation was low, even after rights were granted, so where would they
have the money to invest? We note that bequests were a major source of wealth in this period, as
in DeLong (2003). As such, all we need to assume is that parents bequeathed assets appropriate to
their children in order for the distortions to exist. Specifically, as long as parents internalize that
their bequest to their daughter will be taken from her unless it’s in the form of land, the claims of
this paper stand.
2
ing power depend on the relative income of the spouses both from the labor market and from assets.7 Before marrying, individuals make their portfolio choice
with women potentially underinvesting in capital when they do not have rights.8
Thus, when deciding whether to grant women rights, men face a tradeoff. On one
hand, granting rights may increase overall output, and thus household income,
while on the other hand, granting rights reduces men’s bargaining power within
the household, reducing their share of household income.9
We model two different sectors; agriculture, which uses land, and manufacturing, which uses capital. As technology in manufacturing increases, the demand for capital grows and the effect on factor misallocation becomes worse.
One prediction of the model is that higher levels of industrialization are associated with giving women property rights. The model is a general equilibrium
model with endogenous determination of women’s rights.
After solving the model, we present a numerical example in order to illustrate
how the model works. This exercise clearly shows the tradeoff men face when
considering granting rights. On one hand, if they grant rights, total household
income goes up.10 On the other hand, granting women rights reduces men’s bar7
Combs (2006) argues that, after property rights were given, women had higher fraction of
household wealth, invested more in ’moveable property’ despite returns decreasing in that area,
and perhaps received transfer from husband due to bargaining power.
8
This is equivalent to men and women bargaining before marriage and portfolio choices, subject to the constraint of no commitment on the men’s side to implement any promises.
9
Although couples bargain in a cooperative way, the model leads to Pareto inefficient decision on the women’s part, and a corresponding undersupply of capital. This noncooperative
ingredient is reminiscent of Basu (2006), who finds that when the threat points depend in part
on endogenous decisions, multiple equilibria may exist. The nature of our model, however, is
different from that of Basu (2006). In that paper, all decisions are made in the same period. In
contrast, our models assumes two periods, and the inefficiency is dynamic in the sense that it is
due to the initial underinvestment in capital in the first period caused by expectations of behavior
in the second period, rather than any inefficiency in the second period. In this respect, our model
is close to Konrad and Lommerud (1995) who assume a two-period model where individuals
invest first in education, then marry.
10
Our mechanism depends on women investing more of their assets in capital when they have
rights, which reduces spreads in the returns of assets, and thus increases efficiency. Acemoglu
and Zilibotti (1997) argue that financial market development allowed for greater diversification of
risk and higher productivity. Granting women property would allow for the same mechanism:
greater capital investment allows for more diversification and higher productivity. Incorporat-
3
gaining power within the household. Furthermore, we are able to study how
coverture affects the dynamics of the economy, with a focus on the interaction
between capital accumulation, the rate of return of capital, and the level of financial distortion in the economy. We discuss how men’s incentive to give women
property rights evolves over the course of economic development, and how these
rights in turn affect development. Thus, this paper is connected to a growing
literature on both how development affects women’s empowerment and how
women’s empowerment affects development.11
We estimate the dynamic effects of the change in laws on non-agricultural
employment in order to confirm three of the model predictions with empirical
evidence. Using U.S. data and cross-state variation in the granting of married
women’s property rights, we show that: (i) When rights were granted, there
was a discrete increase in the fraction of workers in the non-agricultural sector;
(ii) The effects on the non-agricultural sector grew at a diminishing rate after
rights were granted; (iii) Prior to rights being granted, the fraction of workers in
non-agriculture grew at a concave rate. These facts jointly suggest that granting
women rights allowed for greater growth of the non-agricultural sector.12
People were aware of the financial market implications of women’s rights
when these rights were granted. Chused (1985) argues that “It is now generally
agreed that the first wave of married women’s acts were adopted in part because
of the dislocations caused by the Panic of 1837.”, implying that the financial maring their model into our own would allow for yet another mechanism through which women’s
property rights affects growth.
11
For more on this topic, see Doepke and Tertilt (2014).
12
Geddes and Lueck (2002) show that states with a greater fraction of the population in cities,
higher wealth, and more educated women were more likely to enact married women’s property
rights laws. States that were more urbanized, and thus likely to be more industrialized, with
more wealth likely experienced greater distortions due to misallocation of assets under coverture,
which also goes along with our hypothesis. Khan (1996) shows that granting women property
rights led to increased involvement of women in commercial activity, as measured by patent
records. While we argue that property rights increased efficiency in the financial markets, the
idea that rights also increased research and development is clearly complementary to the story
we present in this paper.
4
ket implications of women’s rights were indeed a cause of reform.13 Lawmakers
at the time were also aware of the implications. Thomas Herrtell, of the New
York Legislature, argued that women’s property rights “would open appropriate segments of the economy to women, reduce pauperism, and thereby save the
public considerable expense.” Basch (1982) pg 115. Furthermore, Combs (2013)
argues that trusts established for women during coverture allowed for women to
protect their husbands assets during bankruptcy, effectively committing sophisticated fraud, and shows that people were mindful of these realities during the
debate over granting property rights.14 The historical evidence fits well with our
argument that men were aware that giving women rights would improve capital
markets and general allocations.
There is a growing literature on why men gave women rights in the 19th century. Doepke and Tertilt (2009) argue that men faced a tradeoff between wanting
their own wives to have no power, and other men’s wives to have power, and
thus increase investment in human capital.15 Fern´andez (2014) argues that men
faced a tradeoff between not wanting their own wives to have any rights and
wanting to be able to leave a bequest for their daughters. This paper adds to this
literature in three ways. First, we propose a novel complementary mechanism
through which men choose to give women rights, which does not depend on the
desire to help their daughters. While men are altruistic towards their children,
they are selfish in the sense that they permit women’s rights in order to maximize
their own consumption. Second, the story we propose is based on the details of
13
This notion is further reflected in Basch (1982) “It is worth noting that the two major statutes
of 1848 and 1860 followed the depressions of 1839-43 and 1857” (page 122).
14
Chused (1985) argues the same occurred in Oregon, showing that the same phenomenon was
present in the United States. Notice that this story of property rights reducing fraud is a complementary mechanism to our own as to how granting married women property rights would be a
financial innovation that improves capital markets.
15
Doepke and Tertilt (2009) use the growing importance of human capital as their trigger
through which men eventually give women rights. Galor and Moav (2006) study the interaction between physical and human capital complementaries and development. Our paper shows
how women’s rights affect physical capital accumulation, which in turn affect the returns to human capital, as in Galor and Moav (2006), and thus feedback into the story presented in Doepke
and Tertilt (2009).
5
the property rights given and how the legal regime that existed prior to these
rights distorted capital markets. Finally, our story is consistent with several facts
in the data, including the dramatic change in portfolio choices and the dynamics
of industrialization as discussed above.
We proceed as follows. Section 2 develops the model. In Section 3 we solve
the model, define equilibrium, and outline the intuition for various stages of development. Section 4 outlines the solution methodology for numerically solving
the model and discusses the results of the numerical exercise. Section 5 presents
the cross-state empirical evidence. We conclude in Section 6.
2
Model
The economy consists of overlapping generations of men and women who live
for two periods. In every period the economy produces a single homogeneous
final good that can be used for consumption and investment. There are two different assets: Land, T , and capital, K.16 The final good is produced by two intermediate goods: agriculture, A, and manufactured goods, M . While agriculture
uses labor and land, manufacture utilizes labor and capital as factors of production. We assume the both assets fully depreciate within a period.17
16
Land corresponds to the ’real’ assets over which married women always had partial rights
to, while capital represents the ’moveable’ assets that immediately and forever became the husband’s property upon marriage. We maintain the interpretation that women are investing in
land, rather than all real estate such as commercial buildings, by assuming that the financing of
commercial buildings required complex financial instruments such as stocks. These stocks would
have been taken over by the husband upon marriage. So while technically, commercial real estate
should appear in women’s portfolio choice, practically we believe this assumption to be valid.
17
As will be clear below, the assumption of full depreciation is not necessary for our analytic
results. Rather, it simplifies the solution by allowing us to abstract from relative changes of asset
prices over time and the corresponding implication for portfolio choice of households.
6
2.1
Production
Production takes place in three different sectors: the final good sector, the agricultural sector, and the manufacturing sector.
2.1.1
The Final Good
The output of the final good in the economy in period t, Yt , is given by aggregating the agricultural intermediate good, YtA , and the manufacturing intermediate
good, YtM , according to the following neoclassical constant elasticity of substitution (CES) production technology:
(1/ρ)
Yt = (YtA )ρ + (YtM )ρ
,
(1)
where ρ ∈ (0, 1).18
2.1.2
The Agricultural Intermediate Good
Production of the agricultural intermediate good occurs within a period according to a neoclassical, CRS, Cobb-Douglas production technology, using labor and
land. The output produced at time t, YtA , is
α
A (1−α)
YtA = AA
,
t (Tt ) (Lt )
(2)
A
where AA
t is the level of technology in the agricultural sector, Tt and Lt are the
land and the number of workers, respectively, employed by the agricultural sector in period t, and α ∈ (0, 1) is the weight on land. Notice that the amount of
land in the economy is not fixed. This is to capture the fact that real estate includes
buildings, including commercial buildings, which are not fixed in supply.
18
The implicit assumption here is that manufacturing goods and agriculture goods are gross
substitutes, rather than complements. This allows for structural transformation away from the
agricultural sector over the process of development, in line with historical data. For further discussion on modeling structural transformation, see Herrendorf, Rogerson, and Valentinyi (2014).
7
2.1.3
The Manufacturing Intermediate Good
Production of the manufacturing intermediate good occurs within a period according to a neoclassical CRS Cobb-Douglas production technology using labor
and moveable asset. The output produced at time t, YtM , is
α
M (1−α)
YtM = AM
,
t (Kt ) (Lt )
(3)
M
where AM
T is the level of technology in the manufacturing sector, and Kt and Lt
are the capital stock and the number of workers, respectively, employed by the
manufacturing sector in period t.19
2.2
Individuals
In every period a generation, consisting of a unit measure of men and of women,
is born. Individuals live for two periods, childhood and adulthood. Children
make no decisions. Adults receive bequest from their parents, bt−1 , and then the
men decide whether to grant women property rights. Single men and women
then invest their bequest in land and capital. After the investment decision, they
form households and decide on consumption for each spouse, along with a bequest for the next generation. We assume that the man supplies his one unit of
time inelastically while the woman does not work.20 Since there is no heterogeneity within genders, we analyze the representative agent problem of married
households along with the investment decisions of single men and women.
Preferences of individual i ∈ {m, f }, for male and female, who is born in
period t are defined over second-period consumption, cit+1 , and a transfer to both
offsprings, 2bt+1 . They are represented by a log-linear utility function
U (cit , bt ) = log(cit ) + γ log(2bt ),
19
(4)
Notice that we use the same elasticity of production with respect to labor in the manufacturing and agriculture sector. This assumption is helpful in solving the model analytically.
20
None of our results hangs on this assumption as labor is assumed to be exogenous.
8
where γ is the weight put on children.
Denote Kti and Tti as the capital and land, respectively, that member i ∈ {m, f }
has, the single’s budget constraint is given by
Kti + Tti = bt−1 .
(5)
Each household has a son and daughter. Using income from their assets and
the man’s wage, each husband and wife cooperatively allocate their resources
f
between the husband’s consumption, cm
t , the wife’s consumption, ct , and equal
bequest to each of their progeny, bt .
The budget constraint that a couple faces in the second period of life is thus
f
cm
t + ct + 2bt = It ,
(6)
where It is household’s income, which is given then by
It = rtK Kt + rtT Tt + wt ,
(7)
where rtK and rtT are the returns of capital and land, respectively, and wt is the
wage earned by the husband. The household budget constraint includes both the
man and woman’s assets. That is, Kt = Ktm + Ktf and Tt = Ttm + Ttf .
A husband and wife decide cooperatively how to allocate their resources between the three goods: husband’s consumption, wife’s consumption, and bequest. Thus, economic choices are therefore determined by maximizing the following weighted average utility:
f
m
{cft , cm
t , bt } = argmax{θt log(ct ) + (1 − θt ) log(ct ) + γ log(2bt )},
(8)
where θt is the wife’s weight in household’s decision and (1−θt ) is the husband’s.
This maximization is subject to the constraint (6).
θ in turn depends both on the relative income of the spouses as well as the
9
political regime chosen by the men, as discussed below. Thus people take the
political regime into account when deciding upon their investments when single
and allocations when married.
2.3
The No Rights Regime (NR)
In the NR regime the husband owns and controls all the capital the household
has and manages all its land. Recall that, while wives own their land, their management is under husbands’ control. To capture this reality in a parsimonious
manner, we assume that the husband extracts λ ∈ (0, 1) of the returns on land
that the wife brings to the household.21 Therefore, wife’s weight in the household’s decision is given by the share 1 − λ of the returns on wife’s land out of the
total household’s resources.
That is, the wife’s share of household’s choice would be given by
θt =
2.4
(1 − λ)rtT Ttf
.
I
(9)
The Rights Regime (R)
In the R regime each member owns, manages and controls her (his) assets. Thus,
the wife controls all the returns of all the assets she brings to the household. In
this case, the wife’s welfare weight would be given by
rtT Ttf + rtK Ktf
θt =
,
I
(10)
21
The legal reality was that the men controlled the rental income from the wife’s land, but
could not sell or bequeath the land without the wife’s permission, and the land would return to
her upon dissolution of the marriage. We thus think of λ as capturing the rental flow of the land
over the course of the marriage.
10
2.5
Determination of the Political Regime
The political regime is determined by a vote among the male population before
marriages take place. Individuals’ portfolio depends upon the outcome of the
men’s decision, as described above. Under the assumption that men will vote for
N R when both regimes yield the same utility, granting rights will occur if and
only if:
(Utm )R > (Utm )N R .
(11)
Two economic forces dictate whether inequality (11) holds. The first is that,
under the NR regime, husbands have control over the women’s capital and a
fraction λ of their land, leading to greater power within the household and thus
a male preference for the NR regime. However, the NR regime distorts the
women’s perception of the different assets, which may lead to inefficiency in resource allocation within the economy. In what follows, we examine these tradeoffs in more detail, and derive conditions under which men prefer to share power
with their wives.
3
Model Solution
We begin by solving for the production side of the economy, taking as given the
investment choices made by households. Then we solve for individual choices of
individuals in the model by backwards induction. Given a rights regime and the
corresponding portfolio of each spouse, we calculate the consumption allocation
and bequest for the children. Foreseeing the solution to the household problem,
singles make their portfolio choice. Notice that in order to solve for the portfolio
choice of men and women, we need to know the returns to both land and capital,
as solved for in the production side of the economy. As noted, production in turn
depends on the portfolio allocations, yielding a general equilibrium problem.
As a last step, men take into account how their choice of granting women’s
11
rights affects both investment choices of singles before marriage along with household allocations of couples after marriage.
3.1
Production and Factor Prices
The final good producers take as given prices of intermediate goods and maximize their profits:
{YtA , YtM } = argmax
n
(YtA )ρ + (YtM )ρ
ρ1
− PtM YtM − PtA YtA
o
(12)
In turn, profit maximization by the final good producer, using the first order
conditions of (12), give the following inverse demand functions for the intermediate goods:
1 −1
PtM = (YtM )ρ−1 (YtA )ρ + (YtM )ρ ρ
1 −1
PtT = (YtA )ρ−1 (YtA )ρ + (YtM )ρ ρ .
(13)
The intermediate agricultural good producers maximize the following profit
T A
α
A 1−α
{Tt , LA
− rtT Tt − wt LA
,
t } = argmax Pt At (Tt ) (Lt )
t
(14)
and for the intermediate manufacturing good producers it is given by
M M
α
M 1−α
− rtK Kt − wt LM
.
{Kt , LM
t
t } = argmax Pt At (Kt ) (Lt )
(15)
These maximization problems give the following first order conditions
rtT
rtK
=
αPtT AA
t
=
αPtM AM
t
12
LA
t
Tt
1−α
LM
t
Kt
(16)
1−α
(17)
wt = (1 −
α)PtT AA
t
Tt
LA
t
α
= (1 −
α)PtM AM
t
KtM
LM
t
α
.
(18)
Notice that wages are equalized between workers in agriculture and manufacturing as labor can move freely between them. However, the rates of return
on land and capital are not necessarily equalized. If women are disincentivized
from investing in capital due to coverture, there might be an under accumulation
of capital leading to excessive returns. This point is crucial as it is the source of
economic inefficiency under the NR regime.
3.2
Household Optimal Choice
We begin by analyzing the household choice given a portfolio and political regime.
Maximizing (8) subject to (6) gives the following optimal choices
θI
,
1+γ
(19)
(1 − θ)I
,
1+γ
(20)
γI
.
(1 + γ)
(21)
cf =
cm =
and
b=
Notice that this formulation is general. That is, the political regime will affect
both I and θ, but once they have been determined, these equations dictate the
solution to the household problem.
3.3
Individual Portfolio Optimal Choice
Individuals’ portfolio choices depend on the political regime as the latter impacts
the assets over which men and women have control.
13
3.3.1
The No Right Regime (NR)
Substituting household’s optimal choices: (19), (20) and (21); individual’s budget
constraint (5) and individual’s share in household decision under the N R regime,
(9) into individual’s utility function, (4) gives an optimal behavior that can be
derived from maximizing the following problem for women:
n
o
Ttf = argmax log[Ttf rtT (1 − λ)] + γ log[Ttf (rtT − rtK ) + Ttm rtT + (Ktm + bt−1 )rtK + w] ,
(22)
and the corresponding problem for men:
n
Ttm = argmax log[Ttm (rtT − rtK ) + λTtf rtT + (Ktf + bt−1 )rtK + w]
o
f T
f
m T
K
K
+ γ log[Tt (rt − rt ) + Tt rt + (Kt + bt−1 )rt + w] .
(23)
The solution to women’s maximization problem, (22) and men’s maximization problem, (23) depend on returns on land, rtT , the returns on capital, rtK and
the budget constraint, (5). This optimal choice is summarized in the following
Lemma
Lemma 1 Women’s optimal investment is given by
(i) Ttf = bt−1
(ii) Ttf = min{bt−1 ,
rtT T m +(bt−1 +K m )rK +w
}
(1+γ)(rtK −rtT )
if
rtT ≥ rtK ,
if
rtT < rtK .
And men’s optimal investment is given by
(i) Ttm = bt−1
if
rtT > rtK ,
(ii) Ktm = bt−1
if
rtT < rtK ,
(iii) any combination such that Ttm + Ktm = bt−1
if
rtT = rtK .
Proof: Follows directly from the first order conditions and the constraint (5).
2
14
3.3.2
The Right Regime (R)
Substituting household’s optimal choices: (19), (20) and (21); individual’s budget
constraint (5) and individual’s share in household decision under the R regime,
(10) into individual’s utility function, (4) gives an optimal behaviour that can be
derived from maximizing the following problem for women:
n
Ttf = argmax log[Ttf (rtT − rtK ) + bt−1 rtK ]
o
+ γ log[Ttf (rtT − rtK ) + Ttm rtT + (Ktm + bt−1 )rtK + w] ,
(24)
and the corresponding problem for men:
Ttm = argmax log[Ttm (rtT − rtK ) + bt−1 rtK + w]
o
f T
f
m T
K
K
+ γ log[Tt (rt − rt ) + Tt rt + (Kt + bt−1 )rt + w] .
(25)
The solution to women’s maximization problem, (24) and men’s maximization problem, (25) depend on returns on land, rtT , the return capital, rtK , and the
budget constraint, (5). This optimal choice is summarized in the following lemma
Lemma 2 Women’s optimal investment is given by
(i) Ttf = bt−1
if
rtT > rtK ,
(ii) Ktf = bt−1
if
rtT , < rtK ,
(iii) any combination such that Ttf + Ktf = bt−1
if
rtT = rtK .
And men’s optimal investment is given by
(i) Ttm = bt−1
if
rtT > rtK ,
(ii) Ktm = bt−1
if
rtT < rtK ,
(iii) any combination such that Ttm + Ktm = bt−1
if
rtT = rtK .
Proof: Follows directly from the first order conditions and the constraint
2
given in (5).
15
3.4
Market Clearing
We need to verify that the goods markets clear, that the capital and land supplied
by the household are equal to those demanded by the firms, and that the labor
market clears.
Specifically, the goods market clearing involves production to be equal to consumption, as shown by
f
Y t = cm
t + ct + 2b
(26)
The capital market clears, as shown by
Kt = Ktm + Ktf ,
(27)
where Kt is the capital used by the manufacturing sector, as in Equation 15, and
Ktm and Ktf are the capital choices by men and women, respectively.
The land market clears, as shown by
Tt = Ttm + Ttf ,
(28)
where Tt is the land used by the agricultural sector, as in Equation 14, and Ttm
and Ttf are the land choices by men and women, respectively.
The last equilibrium condition is labor market clearing:
A
LM
t + Lt = 1.
3.5
(29)
General Equilibrium
We now define the general equilibrium of the economy.
Definition 1 General equilibrium in the economy is a set of prices {PtT , PtM , wt , rtk , rtt },
allocations in the production side of the economy {Yt , YtM , YtT , Tt , Kt , LTt , LM
t }, portfolio choices of the household {Ttf , Ttm , Ktf , Ktm }, household allocation {cft , cm
t , bt }, and a
series of political regimes for each date t, such that:
16
1. Given prices and a rights regime, YtA and YtM solve Equation (12), Yt is given by
f
f
M
m
m
Equation (1), Tt and LA
t solve (14), Kt and Lt solve (15), {Tt , Kt , Tt , Kt } are
given by Lemma 1 when there are no women’s rights and Lemma 2 when women
do have property rights. Finally, {cft , cm
t , bt } solve Equations (19), (20), and (21),
where θ is determined according to Equation (9) when women do not have property
rights and Equation (10) when they do have property rights.
2. Given allocations, PtT and PtM are set by Equation (13). wt is set by Equation (18),
rtT is set by Equation (16), and rtK is set by Equation (17).
3. Budgets balance and markets clear. Individual portfolio choices of men and women,
{Ttf , Ktf , Ttm , Ktm }, are subject to Equation (5) and the household allocation
a
M
{cft , cm
t , bt } are subject to Equation (6). The labor market allocation Lt and Lt
clears as described in Equation (29). The goods market clears as in Equation (26).
The capital market clears as in Equation (27), and the land market clears as in
Equation (28).
4. The political regime at each time t is determined by Equation (11).
The solution to the equilibrium is relegated to Appendix A. We next describe
intuitively the various phases of development of the economy, before showing a
numerical example in the following section.
In our exercise, we will study development by increasing the relative productivity of the manufacturing sector. The economy experiences three phases along
its development path.
1. For AM sufficiently low, the manufacturing sector is small enough that returns between land and capital can be equalized under the NR regime. Accordingly, men do not give rights to women as there is no distortion in the
economy.
2. When AM is large but not too large, there begins to be a wedge between
the returns to capital and the returns to land. The economy operates below
17
potential, but not so much so that men are willing to grant women rights.
3. Finally, after AM grows high enough, the distortion in the economy becomes great enough that men grant women rights.
4
A Numerical Example
In this section, we solve a numerical example of the model in order to illustrate
how it works. As mentioned above, there are three phases of development. First,
as AM is low, there is no distortion caused by coverture. After a certain point,
there is insufficient capital provided to the manufacturing sector, causing an increasing degree of inefficiency. When the inefficiency grows, men eventually give
women rights. First we describe the solution method for the numerical example,
along with parameters chosen, and then we show various results from the model
along with the economics of these results.
4.1
Numerical Solution and Parameters
For the example, we create an evenly spaced grid of AM from 0.5 to 5, while
holding AT constant. This allows for the example to illustrate what happens in
the model as manufacturing grows in relation to agriculture, as happened historically. For each grid point, we first solve the model for the case where women
are not given property rights as follows. First, assume that rk = rt , solve for the
general equilibrium as outlined in Appendix A. If indeed there is a solution with
rk = rt , then the economy is operating without any distortions. Otherwise, we
solve the model under the assumption that returns are not equalized. To do so,
we perform an iterative process as follows:
1. Guess w, rk , rt , and infer portfolio allocations for men and women using
Lemma 2, and thus K and T .
2. Using equations (18) and (29), solve for LM and LA .
18
3. Using K, T , LM , and LA , infer w, rk , and rt using equations (18), (17), and
(16).
4. Update guess and iterate until convergence.
Additionally, we solve the model at each grid point under the assumption
that women have property rights. This uses the same process as when returns
are equalized under the no-rights regime, and is detailed further in Appendix A.
The exercise that follows is a comparative static. We change AM , while holding
all other parameters, including the assets people have to invest, bt−1 , constant.22
We solve the model using the following parameter values:
bt−1 = 5.5;
γ = 10;
λ = 0.7;
ρ = 0.5;
α = 0.5;
At = 1;
These parameter values do not come from a serious calibration exercise, and
are just meant to be part of an illustrative example.
4.2
Results
We now show graphically the results of the numerical exercise and discuss the
economic intuition behind the model. For all graphs, unless otherwise speci22
While we have also solved the model with full dynamics, that is, a constant exogenous
growth rate of technology and allowing bequests to be endogenously determined. The results
are qualitatively the same. However, we believe that the comparative static model makes the
point clearer. Specifically, this way of doing the exercise allows us to focus on the effect of technological growth without interference from growth of bequests received from parents (bt−1 ).
19
20
0.5
No Rights
Rights
With Change
0.45
No Rights
Rights
With Change
18
0.4
16
0.35
14
Income
theta
0.3
0.25
0.2
12
10
0.15
8
0.1
6
0.05
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
4
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Am
Am
Fig. 1: Women’s bargaining power (θ) and Household Income as functions of Am
fied, the line ’No Rights’ shows the evolution of these variables if women are
never given rights, the line ’Rights’ shows these variables if women always have
rights, and the line ’With Change’ shows the evolution of these variables if men
optimally choose to switch political regimes.
Figure 1 shows, in the left panel, how women’s bargaining power (θ) and,
in the right panel, how household income evolve with AM . First notice that θ
remains constant when women have rights. That is, economic growth does not
differentially affect bargaining power by gender in the model. However, when
women do not have rights and rk begins to increase relative to rt , θ begins to
decrease. This because women are getting lower returns on their asset relative to
men, resulting in them losing intra-household bargaining power. When men give
women rights, θ jumps, as women’s portfolio incentives become undistorted, and
they can realize the same returns as men.
Turning to income, when returns are equalized between the assets, income is
not affected by the political regime. This can be seen by the two curves being
the same for low values AM . However, when there is a wedge, income grows
slower under the no-rights regime. This is as women are forced to receive lower
returns on their assets, and underinvestment in capital yields lower wages for
20
5
1.1
10
No Rights
Rights
With Change
9
rt = rk , no rights
1
r < r , no rights
t
k
rt = rk , rights
0.9
Return to capital
Capital
8
7
6
0.8
0.7
0.6
0.5
5
0.4
4
3
0.5
0.3
1
1.5
2
2.5
3
3.5
4
4.5
0.2
0.5
5
1
1.5
2
2.5
3
3.5
4
4.5
Am
Am
Fig. 2: Capital and Returns to Capital as functions of Am
men. When rights are given, household income increases immediately. These
two graphs thus show the tradeoff for men when considering granting rights. On
one hand, women’s rights imply higher total income, on the other, they reduce
men’s intra-household bargaining power.
Figure 2 shows, in the left panel, the capital stock and, in the right panel, the
returns to capital as a function of AM . We begin with the capital stock. First
note that the capital stock grows continuously when women always have rights.
This is standard in growth models. Turning to the example without rights, notice
that at some point, the capital stock stops accumulating. This is when men are
only buying capital and women are only buying land. This specialization of asset
portfolios persists until the wedge in returns is sufficiently large in order to entice
women to buy an asset that they won’t fully appreciate due to coverture laws.
This interpretation can be seen clearly in the return to capital graph, which is
growing sharply during the period in which women do not buy capital, and then
slows down after women start buying capital.
Figure 3 shows, in the left panel, the fraction of labor in manufacturing and,
in the right panel, the fraction of portfolios in capital. Turning to the fraction
of labor in manufacturing, the curve with rights is as expected- continuous and
21
5
0.9
1
No Rights
Rights
With Change
0.8
0.9
0.7
0.7
% Capital
Labor-Manufacturing
0.8
0.6
0.6
Women
Men
With Rights
No Rights
0.5
0.4
0.5
0.3
0.2
0.4
0.1
0.3
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Am
Am
Fig. 3: Fraction of Labor in Manufacturing and Portfolios functions of Am
smooth growth. The no-rights curve follows the same intuition as the capital
curve described above. When there is no distortion, it follows the same path as
when women have rights. When there is a distortion, it first grows slowly as
there is underinvestment in capital, and then slightly more quickly when women
start buying capital.
We now turn to the portfolio graph. When women have rights, they and men
face the same incentives, and thus the ’With Rights’ line represents men’s portfolio, women’s portfolio, and the aggregate economy’s portfolio, which follows
the same trajectory as the aggregate capital stock. The curve ’Women’ shows
women’s portfolio when there are no rights, ’Men’ shows the men’s portfolio
when there are no rights, and ’No Rights’ shows the aggregate economy’s portfolio. Notice that men quickly put all their money into capital, as they maximize
their individual returns. Women start by not putting any money into capital, as
it gets confiscated by their husband, but when the returns to capital are large
enough, they buy some anyway, and the fraction of capital in their portfolio begins to grow.
Finally, we turn to Figure 4 to study men’s utility and decision to grant rights.
In this figure, we show the difference in men’s utility when women have rights
22
5
0.1
Mens utility with rights - no rights
0.05
0
Um
-0.05
-0.1
-0.15
-0.2
-0.25
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Am
Fig. 4: Difference in Men’s Utility: Rights - No Rights
as compared to when they do not have rights. Thus, when this curve is negative,
men do not grant property rights, while they do when the curve is positive. At
the beginning, men do not give property rights, and this curve is flat. Specifically,
there is no distortion in the economy, so men perceive no benefit to granting
women rights. Once the distortion is present, as rk > rt , it grows rapidly at first
as women do not buy any capital, resulting in the strongly positive slope in the
graph. At some point, women begin to buy capital despite coverture laws, which
slows down the increase of the distortion. Eventually, however, the distortion in
the economy is large enough that men choose to grant women rights.
5
Empirical Evidence
In this section we provide evidence consistent with the mechanism studied above.
Specifically, we establish a two way relationship between women’s liberation
and industrialization. First, we show that in each decade, states that gave prop23
erty rights to women had a larger share of workers in industry than other states.
This is consistent with the model’s prediction that only when inefficiency is sufficiently large do men grant rights to women.
More importantly, the data confirm the model’s prediction for the dynamics of the movement of labor from agriculture to non-agriculture employment.
Specifically, as can be seen in Figure 3, the model predicts that the fraction of
non-agricultural employment is growing concavely with respect to time before
rights are granted, has a discrete jump at the time of granting rights, and then
is concave again after rights are granted. We show here that the data exactly
confirm this sharp hypothesis.
5.1
Data Sources
Data on women’s liberation comes from Geddes and Lueck (2002).23 They coded
the year in which states granted women rights. Following the standard they
began in the literature we consider women’s rights to have been granted in states
when both property rights and control over income had been granted to married
women. The variable is called rights. Data on non-agricultural employment is
taken from census data for the years 1850-1920 (Ruggles, Alexander, Genadek,
Goeken, Schroeder, and Sobek 2010). There is no data for 1890. We deal with this
in two ways. First we linearly interpolate between 1880 and 1900. Secondly we
repeat the analysis without data for 1890.24
5.2
Results
Figure 5 shows the fraction of workers in non-agricultural sectors by census year
for states that have already granted rights and states that have not. The figure
shows that for all census years, states that granted rights to women had a higher
23
24
We thank the authors for making their data available to us.
Results without 1890 are very similar and are not reported for brevity.
24
share of their workforce in non-agricultural sectors. This is consistent with the
view that granting rights was related to the growth of industrialization.
0.80
0.75
0.70
0.66
0.65
Fraction of Workers in the Non-Agricultural Sector
0.60
0.60
0.58
0.58
0.55
0.52
0.50
0.47
0.64
0.60
0.49
0.46
0.52
0.44
0.40
0.40
0.30
0.20
0.10
0.00
1850
1860
1870
1880
1890
1900
1910
1920
Year
Rights
No Rights
Fig. 5: Cross State Comparison of Non-Agriculture Employment
Our model predicts a specific dynamic of changing employment rates with
respect to the date of granting married women property rights, as seen in Figure 3. In order to check these dynamics empirically, we need to control for state
effects, as different states had different levels of agricultural employment, and
year fixed effects, as there was a natural tendency over time to shift towards nonagricultural employment.25 Notice that state effects, year effects, and the difference in time since granting women rights are jointly collinear. Each state granted
rights in a particular year, which implies that controlling for the state implicitly
controls for that year. Including the difference in time since granting rights thus
pins down a year. For example, if rights were granted in 1870 in a particular state,
25
Additionally, the differential effect of the Civil War on southern states needs to be taken into
account, as described below.
25
and the observation is looking at +10 years since granting rights, the observation
must be at 1880. Thus, state fixed effects and differences in time since granting
rights joint imply a calendar year, and are therefore perfectly collinear with year
fixed effects.
We follow Stevenson and Wolfers (2006) approach in estimating the effects of
granting women’s rights over time while still being able to include state and
year fixed effects. Accordingly, we estimate separately the dynamics of nonagricultural employment after and before rights were given, and thus estimate
two regressions, one designated ’After’, and the second designated ’Before’.26
In the ’After’ regression, we have dummy variables for every 10 years (starting
with 0) since rights were granted to married women. The base is thus all the
time periods in which women did not have property rights. Conversely, the ’Before’ regression includes dummy variables for every 10 years before rights were
granted (starting with 0). The base is thus all time periods at least ten years after
women were granted property rights. These two regressions allow us to separately analyze the dynamics of non-agricultural growth before and after rights
were given to women while including state and year fixed effects.
Our specification is of the form:
ln LM
st =
X
αk · rightskst + λs + dt + controlsst + st
k
where ln LM
st is the log of the fraction of workers in non-agricultural sectors
in state s in year t, t ∈ {1850, 1860, . . . , 1920} and rightskst is a series of dummy
variables set equal to one if a state had granted rights k years ago, where k ∈
{0, 10, 20, 30, 40, 50} for the ’After’ regressions and k ∈ {0, −10, −20, −30, −40, −50}
26
Stevenson and Wolfers (2006) studies the effects of unilateral divorce laws on suicide over
time, exploiting cross-state timing of divorce law changes. They also control for state and year
fixed effects, and thus provide an excellent basis for our empirical analysis. While Stevenson
and Wolfers (2006) only studies the ’After’ component, the methodology is generalizable to the
’Before’ case as well.
26
for the ’Before’ regressions.27 λs and dt are state and year dummies and controlsst =
{south × d1870 , south × d1880 , Fraction of Menst , Female Schoolst } are dummies for
south states interacted with postbellum years to account for war destruction, the
fraction of the state’s population that is male, and the fraction of the state’s school
age women who are in school.28
Table 1 shows the results for the ’After’ regressions. The table shows the
dynamic effect of granting rights on non-agricultural employment after rights
were granted. Column 1 shows the estimates for how the fraction of workers in
non-agriculture evolved after the granting of rights. The coefficient on rights0
implies a discrete and statistically significant jump when rights are given, with a
point estimate of 0.092, implying a roughly 10% increase of employment in nonagriculture when rights are granted. We next turn to the coefficients on rights10
through rights30 , which are statistically significant at least at the 10% level. Each
of these coefficients is the change in employment of non-agriculture workforce
relative to a base of when there were no-rights for married women. These estimates are increasing in a concave manner, as predicted by the theoretical model.
The lack of significance of the estimates for rights40 and rights50 implies that the
effects of granting rights dissipates after 30 years. Columns 2 and 3 show these
results are robust to adding controls for the south after the Civil War, to the fraction of women in school, and to the fraction of people in the state who were men.
Table 2 shows the results for the ’Before’ regressions. The table shows the dynamic evolution of non-agricultural employment prior to rights being granted.
27
We use increments of 10 as our data is dependant on the decennial census. For states that
granted rights not in a census year, we round to the nearest decade. For example, California
granted rights in 1872. For our purposes, we round to 1870. Thus, the dummy variable rights0st
takes the value of 1 for California in 1870, while the dummy variable rights1st 0 takes the value 1
for California in 1880. Our results are robust to always rounding up. Using California again as
an example, we code 1880 as the first year rights exist, rather than 1870, as to avoid the case of
assigning rights to California before rights were actually granted.
28
Regressions of the same structure using the fraction of non-agricultural workers (and not the
log of that variable) as the dependent variable are very similar although there are somewhat less
precise. Still all results are significant at less than 10%. The final two variables, Fraction of Men
and Female School, are taken from Geddes and Lueck (2002) and included for completeness here.
27
Table 1
After
(1)
ln LM
st
0.092+
(0.050)
(2)
ln LM
st
0.083+
(0.047)
(3)
ln LM
st
0.082+
(0.047)
rights10
0.172∗
(0.073)
0.148∗
(0.071)
0.148∗
(0.071)
rights20
0.208∗
(0.100)
0.190+
(0.101)
0.190+
(0.100)
rights30
0.212+
(0.124)
0.221+
(0.123)
0.221+
(0.123)
rights40
0.186
(0.143)
0.207
(0.143)
0.207
(0.144)
rights50
0.150
(0.164)
0.179
(0.163)
0.179
(0.165)
F-test of joint significance
p=0.00
p=0.01
p=0.01
Year dummies
Yes
Yes
Yes
State dummies
Yes
Yes
Yes
South×1870
No
Yes
Yes
South×1880
No
Yes
Yes
Female School
No
No
Yes
No
356
0.938
No
355
0.952
Yes
355
0.952
rights0
Fraction of Men
N
R2
NOTE. All models are weighted by state population. Standard errors, clustered at the
state level, are reported in parentheses. +p < 0.10, *p < 0.05.
28
Each of the coefficients on rights, shown in column 1, is the change in employment of non-agriculture workforce relative to a base of at least 10 years after
rights were granted. The coefficients on rights0 and rights−10 jointly imply a
discrete and statistically significant jump when rights are given, as they are statistically different at a p-value of 0.01 or lower, depending on the model specification. We next turn to the coefficients on rights−10 through rights−50 , which are
statistically significant at least at the 5% level. These estimates are increasing in a
concave manner, as predicted by the theoretical model.29 Columns 2 and 3 show
these results are robust to adding controls for the south after the Civil War, to the
fraction of women in school, and to the fraction of people in the state who were
men.
6
Concluding Remarks
In this paper, we propose and model a novel mechanism through which men
choose to give women rights through a desire to correct capital market imperfections related to women’s portfolio choices. The story is consistent with historical
evidence on how the laws of coverture affected investment decisions by married
women. Furthermore, we show that people were aware at the time of the implications of women’s property rights on financial markets.
We solve a general equilibrium model with endogenous property rights determination, and study a numerical example which illustrates how technological
growth in manufacturing interacts with the laws of coverture in order to induce
inefficiencies. When deciding whether to grant women rights, men face a tradeoff. On one hand, granting rights may increase overall output, and thus household income, while on the other hand, granting rights reduces men’s bargaining
power within the household, reducing their share of household income. At a
To avoid confusion, recall that rights−50 is 50 years before rights are granted while rights−10
is 10 years prior. Moving from 50 years to 10 years prior to granting rights, the estimates become
less negative in a concave manner.
29
29
Table 2
Before
(1)
ln LM
st
-0.136∗
(0.028)
(2)
ln LM
st
-0.113∗
(0.031)
(3)
ln LM
st
-0.113∗
(0.031)
rights−10
-0.235∗
(0.049)
-0.185∗
(0.051)
-0.185∗
(0.052)
rights−20
-0.255∗
(0.064)
-0.216∗
(0.064)
-0.216∗
(0.064)
rights−30
-0.234∗
(0.079)
-0.201∗
(0.079)
-0.201∗
(0.080)
rights−40
-0.315∗
(0.097)
-0.213∗
(0.088)
-0.213∗
(0.088)
rights−50
-0.499∗
(0.090)
-0.392∗
(0.103)
-0.392∗
(0.102)
F-test of joint significance
p=0.00
p=0.00
p=0.01
F-test for a “jump”
p=0.00
p=0.01
p=0.01
Year dummies
Yes
Yes
Yes
State dummies
Yes
Yes
Yes
South×1870
No
Yes
Yes
South×1880
No
Yes
Yes
Female School
No
No
Yes
No
356
0.941
No
355
0.953
Yes
355
0.953
rights0
Fraction of Men
N
R2
NOTE. All models are weighted by state population. Standard errors, clustered at the
state level, are reported in parentheses. +p < 0.10, *p < 0.05.
30
certain point of development, the benefits of women’s property rights dominate
and men give rights.
We show empirically, using cross-state variation in the timing of the granting of married women’s property rights, that the model is consistent with the
dynamics of the growth in non-agricultural employment.
Our findings contribute to a growing literature on how development and
women’s rights are intricately linked. Women are given rights when development reaches a certain level and industrialization begins to decelerate. After
granting rights, there is a feedback from women’s empowerment into growth.
Our empirical evidence support this two-way relationship.
31
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A
Solution to the General Equilibrium
We next calculate the general equilibrium under each one of the two political
regimes.
A.0.1
The No Right Regime (NR)
Under the NR regime two qualitatively different cases are considered: Case A,
when the equilibrium gives rise to an efficient allocations of resources, that is
rtT = rtK ; and Case B, when inefficiency arises , that is rtT < rtK .
Case A, rtT = rtK :
In this case the system could be reduced and described by 14 variables and 15
equations. Notice that once we solve this system we can then calculate Y, I, θ, ci , b.
The 13 variables are: {Y M , Y A , K, K m , K f , T, T m , T f , P M , P A , rK , rT , w}
35
and the 13 equations are:
Y A = AA T α (LA )(1−α)
(30)
Y M = AM K α (LM )(1−α)
(31)
1
−1
ρ
P M = (Y M )ρ−1 (Y A )ρ + (Y M )ρ
1 −1
P A = (Y A )ρ−1 (Y A )ρ + (Y M )ρ ρ
A 1−α
L
T
A A
r = αP A
T
M 1−α
L
rK = αP M AM
K
rT = rK = r
(32)
(33)
(34)
(35)
(36)
1 = LM + LA
(37)
α
T
LA
α
K
M M
w = (1 − α)P A
LM
w = (1 − α)P T AA
(38)
(39)
Kf = 0
(40)
Tf = B
(41)
B = Km + T m
(42)
36
The concept of the solution proceeds in the following steps:
1. The first 4 equations, (27)–(30) solves for the 4 variables {Y M , Y A , P M , P A }.
Of course as a function of the other variables. In particular, equations (29
) and (30) are the inverse demand for intermediate goods {Y M , Y A , } and
equations (27) and (28) are their supply. These 4 equations solve for quantities and prices.
2. The 4 equations, (31)–(34) solves for T = T 1 (r, LM ) and K 1 = K(r, LM ).
3. The 3 equations, (34)–(36) solves for T = T 2 (w, LM ) and K 2 = K(w, LM ).
4. Steps 2 and 3 along with equations (29) and (30) and the supply for assets:
equations (37), (38) and (39) we solve for the five variables {K, T, r, w, LM , LT }.
5. So far we used 12 equations, (27)–(38) to solve for 12 variables. Notice that
{rK , rT } are two variables and we solved for LA .
6. Once T and K are solved individuals’ optimal portfolio choice, equations
(37), (38) and (39) pins down K m , K f , T m , T f .
Solution:
Isolating
Pm
PT
from (31), (32) and (33) gives
PM
AA
= M
PA
A
Isolating
Pm
PT
LA K
T LM
1−α
(43)
from (35) and (36) gives
PM
AA
=
PA
AM
T LM
LA K
α
(44)
From (43) and (44) we get
LA
LM
=
T
K
37
(45)
Substituting (45) into (44)
PM
AA
=
PA
AM
(46)
(46) along with (29) and (30) gives
YM
YA
1−ρ
=
AM
AA
(47)
substituting (27) and (28) into (47) gives
K α (LM )(1−α)
T α (LA )(1−α)
1−ρ
=
AM
AA
ρ
(48)
From (48), (45) and (34) we solve for LM
"
LM = 1 +
AM
AA
−ρ #−1
1−ρ
(49)
From (37)–(39)
T = 2B − K m
(50)
K = K m = 2BLM
(51)
and
Thus {K, T, LM , LA } are solved, from (27) and (28) we solve got {Y M , Y A },
then (29) and (3) solve got {P M , P A }, (31) and (35) solves for r and w.
38
Case B, rtT < rtK :
As in the previous case, in this case the system could be reduced and described
by 13 variables and 13 equations. Notice that the equalization in the interest
rate is replaced by full identification of individuals’ portfolio choice, while in the
previous case men were indifferent between holding real or moveable assets.
The 13 variables are: {Y M , Y A , K, K m , K f , T, T m , T f , P M , P A , rK , rT , w}
and the 13 equations are:
Y A = AA T α (LA )(1−α)
(52)
Y M = AM K α (LM )(1−α)
(53)
1 −1
P M = (Y M )ρ−1 (Y A )ρ + (Y M )ρ ρ
1 −1
PT = (Y A )ρ−1 (Y A )ρ + (Y M )ρ ρ
A 1−α
L
T
T A
r = αP A
T
M 1−α
L
rK = αP M AM
K
1 = LM + LA
(54)
(55)
(56)
(57)
(58)
α
T
LA
α
K
M M
w = (1 − α)P A
LM
2BrK + w
Kf = B − K
(r − rT )(1 + γ)
2BrK + W
Tf =
(rK − rT )(1 + γ)
w = (1 − α)P T AA
(59)
(60)
(61)
(62)
Km = B
(63)
Tm = 0
(64)
39
The concept of the solution proceeds in the following steps:
1. The first 4 equations, (27)–(30) solves for the 4 variables {Y M , Y A , P M , P A }.
Of course as a function of the other variables. In particular, equations (29
) and (30) are the inverse demand for intermediate goods {Y M , Y A , } and
equations (27) and (28) are their supply. These 4 equations solve for quantities and prices.
2. The 4 equations, (32)–(35) solves for T = T 1 (r, LM ) and K 1 = K(r, LM ).
3. The 3 equations, (35)–(37) solves for T = T 2 (w, LM ) and K 2 = K(w, LM ).
4. Steps 2 and 3 along with equations (29) and (30) and the supply for assets:
equations (37), (38) and (39) we solve for the five variables {K, T, r, w, LM , LT }.
5. So far we used 12 equations, (27)–(38) to solve for 12 variables. Notice that
{rK , rT } are two variables and we solved for LA .
6. Once T and K are solved individuals’ optimal portfolio choice, equations
(37), (38) and (39) pins down K m , K f , T m , T f .
Solution:
Isolating
Pm
PT
from (54) and (55) gives
PM
AA
=
PA
AM
Isolating
Pm
PT
T LM
LA K
α
(65)
from (49) and (50) gives
PM
=
PA
YA
YM
1−ρ
(66)
Substituting (47) and (48) into (66) we get
PM
=
PA
AA T α (LA )(1−α)
AM K α (LM )(1−α)
40
1−ρ
(67)
from (65) and (67) we get
AA
AM
ρ
=
K
T
αρ LA
LM
1−ρ+αρ
(68)
looking at the supply side of factors of production (assets invested by individuals, which is driven by individuals’ portfolio optimal choice) summarized in
equations (56)–(59) gives
K = 2B −
and
T =
2BrK + w
(rK − rT )(1 + γ)
2BrK + w
(rK − rT )(1 + γ)
(69)
(70)
Substituting (69), (70) and (53) into (68) and rearranging gives the solution of
LM as a function of returns to assets {rK , rT }
(
M
L
=
1+
)−1
−ρ
α M 1−ρ+αρ
A
2B(1 + γ)(rK − rT )
−1
2BrK + w
AA
(71)
Now we can get all variables as a function of {rK , rT }. Specifically, equations
(60) and (61) implicitly solve for rK and rT .
A.0.2
The Right Regime (R)
In this case, there is no friction in the model and rK = rT must hold.
In this case the system could be reduced and described by 13 variables and
13 equations. Notice that the equalization in the interest rate is replaced by full
identification of individuals’ portfolio choice, while in the previous case men
were indifferent between holding real or moveable assets.
The 13 variables are: {Y M , Y A , K, K m , K f , T, T m , T f , P M , P A , rK , rT , w}
41
and the 13 equations are:
Y A = AA T α (LA )(1−α)
(72)
Y M = AM K α (LM )(1−α)
(73)
1
−1
ρ
P M = (Y M )ρ−1 (Y A )ρ + (Y M )ρ
1 −1
P A = (Y A )ρ−1 (Y A )ρ + (Y M )ρ ρ
A 1−α
L
T
T A
r = αP A
T
M 1−α
L
rK = αP M AM
K
rT = rK = r
(74)
(75)
(76)
(77)
(78)
1 = LM + LA
(79)
α
T
LA
α
K
M M
w = (1 − α)P A
LM
w = (1 − α)P T AA
(80)
(81)
B = Kf + T f
(82)
B = Km + T m
(83)
K = Kf + Km
(84)
Solution:
Isolating
Pm
PT
from (71), (72) and (73) gives
PM
AA
=
PA
AM
Isolating
Pm
PT
LA K
T LM
1−α
(85)
from (75) and (76) gives
PM
AA
=
PA
AM
42
T LM
LA K
α
(86)
From (43) and (44) we get
LA
LM
=
T
K
(87)
PM
AA
=
PA
AM
(88)
Substituting (45) into (44)
(46) along with (69) and (70) gives
YM
YA
1−ρ
=
AM
AA
(89)
substituting (67) and (68) into (89) gives
K α (LM )(1−α)
T α (LA )(1−α)
1−ρ
=
AM
AA
ρ
(90)
From (90), (87) and (74) we solve for LM
"
LM = 1 +
AM
AA
−ρ #−1
1−ρ
(91)
Given that the individuals are indifferent between investing in two assets, we
substitute T = 2B − K, which yields
1 − LM
LM
=
2B − K
K
2B − K
1
−1=
LM
K
K
LM =
2B
(92)
(93)
(94)
Thus once LM is solved, K and T are solve as well. From here all variables
can be backed out.
43