14.74
Lecture 12—Inside the household: Is the household an e
ffi
cient
unit?
Prof. Esther Du
fl
o
March 17, 2004
Last lecture, we saw that if we test the unitary model against a slightly more general model where the household maximizes:
MaxµAuA(qA, qB) +µBuB(qA, qB)
we could reject the unitary model.
This representation of the household problem is, however, also specific. This assumes that the household is Pareto efficient.
Definition of a Pareto efficient allocation: An allocation is Pareto efficient when it is not possible to improve the utility of one member without reducing the utility of another member.
Examples: Are the following allocations pareto-efficient?
-Giving all the money member A, and letting him choose what he wants to eat and what he will give to member B.
-The allocation decided by a dictator. -Letting only A consume (qB = 0).
How do we characterize Pareto efficient allocations? Fix the utility of one household member, and let the other member maximize his utility under this constraint and the budget constraint.
MaxuA(qA, qB)
such thatp(qA+qB) =yA+yB+y.
Maximizing this problem is equivalent to maximizing a weighted sum of both members’ utility under the budget constraint.
Note on Pareto-efficiency: It does not say anything about the equality of the distribution. Any weighted sum is a Pareto efficient allocation, for any weight. In particular it does not say anything about discrimination by gender, by age, etc.
1
Is the Household Pareto-efficient?
1.1
Why would it not be e
ffi
cient?
Household members can be thought of as playing a game. The theory of games captures inter-action across individuals when each individual’s inter-actions have effect on the utility of the other individuals.
Each player has a strategy: This is a rule for deciding what action (or actions) to take as a function of the actions of the other people who play the game.
Two people who play a game do not necessarily achieve a Pareto optimum. Following is a very well-known example: the prisoner’s dilemma.
Two prisoner’s have committed a crime for which they are being interrogated by the police. Each has two options:
1. cooperate with the other prisoner (refuse to reveal evidence) 2. defect: reveals evidence.
-If both prisoners cooperate, there is no evidence, they are released and enjoy the booty, worth 10 to each of them.
-If both defect, they go to jail for some time and then share the booty, worth 5 each.
-If one defects and one cooperates, the one who defects spends a bit of time in prison but is released earlier, and he can take the entire booty, he gets the whole booty (worth 20), and has spent 5 in prison so his utility is 15. The other prisoner is sent to jail for a long time, his utility is 0.
Cooperate Defect Cooperate (10,10) (0,15) Defect (15,0) (5,5)
Definition of a Nash equilibrium: A pair of strategies such that no individual can do better by choosing another strategy, assuming that the other individual chooses the strategy of the equilibrium.
What is the Nash equilibrium of this game? Look at the payoffs:
- If prisoner A defects, what is better for prisoner B? - If prisoner A cooperates, what is better for prisoner B? - If prisoner B defects, what is better for prisoner A? - If prisoner B cooperates, what is better for prisoner A? What is the Nash equilibrium of this game?
Is it a Pareto optimum? Why?
So you can see that there are circumstances where you cannot achieve Pareto-optimality when people are taking into account other people’s strategies and their consequences on their utility.
1.2
When could Pareto optimality be achieved in the prisoner’s dilemma
• Imagine that the two prisoners can communicate and take their action at the same time?
• Imagine that they can make a pact ex-ante (before being held by the police), which pact will they make?
• How can they enforce this pact?
Last example: If two people play the prisoner’s dilemma game over and over again, they can achieve Pareto optimum if they are not too impatient?
How: The strategy is – cooperate in each period, unless one defected in one period. After this, always defect.
• What do they gain each period if they follow this strategy?
1.3
Why would a household be Pareto-e
ffi
cient?
-2
Is the household Pareto-efficient? A simple test
Agricultural households do two things: home production and consumption. Intuitively, a Pareto-efficient household wouldfirst maximize the size of its total income, andthen share this income according to the set of weights that are specific to the household. This test of Pareto efficiency is based upon this idea, it is implemented in the paper by Chris Udry “Gender, Agricultural Production and the Theory of the Household”, which you should read.
Setting: Burkina-Faso. Very poor, semi-arid area. There is on average 1.8 wives for each head of the household. Important characteristic: Women and men each control their own plots. Model: Keep thinking about Ahmad and Bijou. Ahmad controls plot A, and Bijou controls plot B.
Plot A has characteristics XA: size, fertility, distance from the house, etc. Plot B has characteristicsXB.
Production function: hA=f(IA;XA), whereIAare the inputs that are applied to plotA(labor of A, labor of B, labor of the children, fertilizer, etc.).
To simplify, we are going to assume that the only inputs are the labor of A and B. So
hA=f(LAA, LAB;XA),
whereLji is the labor that household member iapplies on plotj. Likewise,hB=f(LB
A, LBB;XB).
Imagine a Pareto-efficient household (with the same utility function as in the previous lecture). They maximize:
under the following constraints:
hB=f(LBA, LBB;XB) (2)
LAA+LBA=LA (3)
(4)
p(qA+qb) =p(hA+hB) (5)
• Note that the problem is the same as in the previous lecture, except now individual incomes are determined by the production decisions of the household.
• Note that, once the weights are fixed, how the production is achieved is irrelevant for the household’s total welfare: what matter is total production.
The household can solve this problem sequentially:
• First, maximize production.
• Second, choose the individual consumption levels.
Therefore, the household should apply labor on each plot until the marginal product of labor is equalized across plots.
What does this imply for hA and hB:
-If they have the same XA and XB they should be .... -Once we control forX they should be ....
Therefore, the yield (production divided by size) of each plot should be independent of the gender of the owner of the plot.
Test: for a given year, household and crop, is the yield a function of the gender of the person who owns the plot?
Regression:
Qhtci=Xhtciβ+γGhtci+λhtc+²htci
where: -h: -t: -c: -i:
Qhtci:
Xhtci:
λhtc:
Test: is γ equal to zero?
Results in the tables. The summary is that the household could achieve an increase of 5.8% of the production just by reallocating inputs across plots.
• Can we reconcile these results with efficiency?
— Women do other activities (child rearing): No
— Unobserved differences between plots: No
— Non-convex production technologies: No
• Why is the household not Pareto efficient?
— The “labor market” within the household is not perfect, because of a lack of secure property rights on the land. Men have more labor, but women do not want men to work on their plots because they fear that the plots will then be confiscated by the husband.
— Remember the cosmetic surgery example, from last time: What may happens if A invests in B’s plot?
— Some investment may not be contractible within the households: If it is going to affect my income share, I cannot commitex-ante to not let it affect my bargaining power. As in the case of the education decision, investments are inefficiently low because of incomplete contracts within the family.
— Other evidence: In Cote d’Ivoire, Chris Udry and I find that when women’s income istemporarily high, they consume more food, and when men’s income is temporarily
high they consume more alcohol and tobacco.
∗ Would we expect that in the unitary model? (easy...)
∗ Would we expect that in the efficient bargaining model? (harder....)
— Given this, the household would produce more if they gave all the land to the man, and compensated the woman with a promisedflux of income: However, it is of course very difficult for the man to commit to give the money to the woman!
