Household size and productivity

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Household size and productivity José
Anchorena February 2006
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• Three pieces
• Household size and productivity (aggregates and
  long series)
• Household size and productivity (microdata
  Brazil)
• Simulating Lucas (2002) endogenous growth model
  (fertility and growth). Replicating an industrial
  revolution.

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• Road map for this presentation
• 1. 30 minutes
• Idea and motivation
• Model
• Simulation
• 2. 10 minutes
• Idea and data
• 3. 15 minutes
• Model
• Simulation

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Figure 1 Across countries 139 countries
Correlation -0.54 Simple regression an
increase in 1 member in the average household of
country is accompanied by a reduction in 3500
dollars per capita
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Figure 2 Across households in one country
(Brazil) Aprox. 100.000 households Correlation
-0.15 - Simple regression an increase in 1
member in a household is accompanied by a
reduction of 75 dollars per capita.
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Figure 3 Time series for UK Correlation -0.93
for UK (-0.75 for Brazil)
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Question Is there any functional relationship
between these two variables? Direction of
causality? Or, more generally, how is the
underlying process that relates them?
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Traditional answer (necessary consequence) Yes,
they are related, and causality goes FROM
increase in income per capita TO decrease in
household size. Example 1 through womens wages
(Becker (1960), Mincer (1963)) Example 2 through
demand for privacy Paradigmatic case Greenwood
and Seshadri (2005) Technological progress and
Economic Transformation
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New answer (necessary condition) Yes, there is a
relationship, but causality goes in both
directions. Example 1 through less fertility and
more human capital, quantity-quality trade-off
(Becker, Murphy and Tamura (1990), Lucas
(2002)). Example 2 through development of
markets, because of substitution of home for
market production. (Hinted by Locay (1990), and
developed here).
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• Idea
• A smaller household size diminishes the returns
  of producing inside the house
• Household production is substituted by market
  production
• Market production creates incentives for
  technological innovation
• Insert graph of Locay
• Examples McDonalds
• So, question has the atomization of the
  household provided incentives for further
  technology improvement?

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Womens wages increase
Traditional Explanation
Household size diminishes
New Explanation
Market production substitutes home production
Market technology improves
Income per capita increases
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• A simple model
• Many agents
• Preferences over consumption and company
  (household size)
• Two technologies, household and market, to
  produce the same consumption good
• Human capital can only be used in the market
  sector
• Technology levels may depend on household size
• Household production function with diminishing
  returns on only input NH
• Market technology with constant returns to scale

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max
s.t.
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1.a. n exogenous, zs independent of
n Proposition 1 In steady state, if n decreases
(increases), then NM, NH, m and K increase
(decrease). 1.b. n exogenous, zs dependent on
n Proposition 2 In steady state, if n decreases
(increases), then NM, m and K increase
(decrease). NH can increase or decrease depending
on the differential effect of household size on
productivity. 2.a. n endogenous, zs independent
of n. Usual problem to explain fertility decline.
With an increase in z, we can obtain in steady
state an increase in n. Two solutions in the
literature utility of children enters utility of
parents in dynastic models (Becker, Lucas)
preferences non-separable in time (Miller) 2.b. n
endogenous, zs dependent on AVERAGE n. Once we
obtain that n diminishes with increasing z, then
we can obtain two steady state equilibria if z
depends on average n.
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Solving the model I solved the model 1.b, with
exogenous n and zs dependent on n. I postulate
that n follows the following process And the
productivities are determined by Replacing
constraints, I got a recursive problem with two
state variables, k and n, and two decision
variables, nH and nM.
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• Calibration
• Period length 5 years
• Fix depreciation of human capital equal to 0.25
  (in 5 years)
• Using law of motion for human capital, I
  calibrated e to 0.5
• Using data for 2000 on use of time, I calibrated
  l to 0.036.
• Production functions aH equal to 0.7, aM equal
  to 0.35 (Weil).
• Fix ß equal to 0.8 (for five years)
• Regressing household size in itself lagged and
  obtain ?-0.35 and ?1.05
• Fix ? to 0.75
• Use productivities and household sizes, in 1900
  and 2000, to calibrate a, b, ? and f (the
  response of productivities to household size)

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Results
Household and market work as functions of human
capital
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Household and market work as functions of
household size
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Time invested in acquiring human capital as a
function of human capital and of household size
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Consumption as a function of human capital and
household size
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Simulation UK 1900-2000. Given initial capital
and actual household sizes
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• Conclusion
• Results are mixed. There is too much action at
  the start of the century, too little at the end.
• Need to endogeneize household size and make the
  zs dependent on the AVERAGE household sizes.
• Need to determine a way of getting fertility
  decline with increase income.
• Might be useful to impose a constant return
  technology on human capital not to obtain
  diminishing investments in human capital and to
  obtain endogenous growth (this would be a mix
  between this model and Lucas (2002)).
• Take care of functional forms and calibration.

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• Household size and productivity (microdata Brazil
  1976-2002)
• Objective get more detailed functional forms and
  a better idea of parameter values
• In particular
• Functions relating zs with n
• Process for n
• Preferences
• Overlapping generations
• Include mortality?

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Data Household surveys for Brazil between 1976
and 2002 Near 100.000 households, 380.000
persons, per year
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• I have explored the data
• Contemporary correlations between three sets of
  variables household size, development of markets
  (number of services, autarchic production,
  isolation) and income per capita.
• Contemporary regressions of income per capita
  on development of markets, household size and
  other regressors (education, race, age, etc.)
• Three levels state, municipality, individual.
• To be done
• Write a more detailed (micro) dynamic model
• Estimate the model

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Replicating an industrial revolution simulating
Lucas (2002) Model
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• Lucas finds at least two equilibria for this
  economy
• One with hr0, n1, Malthusian
• One with h growing, permanent growth
• I tried to understand transition by simulating
  the movement from one to the other

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Simulation
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• Issues
• I am solving the problem numerically iterating
  on the value function. But the constraints are
  not homogenous of degree one!
• I can not match data on population growth I do
  not obtain a demographic transition in this
  exercise.