1. Denote the number of children by n and the survival probability of a child to adult- hood by . A parental household derives utility out of consumption (c), the number of surviving offspring (n) and child expenditures (h). Preferences are specified as follows = uln(cc) B₁ ln(n · π(ñ, h)) + ẞ2 In(h), ẞ1 > B2, (1) where the parameter c> 0 reflects subsistence consumption. Child survival is given by π = π + (1 − µñ)λh, ≥ 0, x > 0. (2) is exogenous to the household and positively affected by average income y such (y). The representative household's budget constraint reads as that y = c + nhy (3) a) Solve the household's utility maximisation problem. b) Show and explain how fertility n responds to changes in households' income y. c) Show and explain how n and h respond to changes in 7. 2. Assume now that all households are identical such that y = y. a) In light of your results obtained in 1., show that the reaction of population growth in response to a change in income can be expressed as dgL dy C 1 = Nπ - Ly(y μπ π 1 - TT / მყ . (4) Hint: Population growth gr is determined by the number of surviving children. The populations growth rate is therefore nл. The total differential delivers the above equation. b) In light of (4), explain how the model can generate a hump-shaped correlation between income y and population growth (No derivations! Just provide an economic rationale based on your findings).
1. Denote the number of children by n and the survival probability of a child to adult- hood by . A parental household derives utility out of consumption (c), the number of surviving offspring (n) and child expenditures (h). Preferences are specified as follows = uln(cc) B₁ ln(n · π(ñ, h)) + ẞ2 In(h), ẞ1 > B2, (1) where the parameter c> 0 reflects subsistence consumption. Child survival is given by π = π + (1 − µñ)λh, ≥ 0, x > 0. (2) is exogenous to the household and positively affected by average income y such (y). The representative household's budget constraint reads as that y = c + nhy (3) a) Solve the household's utility maximisation problem. b) Show and explain how fertility n responds to changes in households' income y. c) Show and explain how n and h respond to changes in 7. 2. Assume now that all households are identical such that y = y. a) In light of your results obtained in 1., show that the reaction of population growth in response to a change in income can be expressed as dgL dy C 1 = Nπ - Ly(y μπ π 1 - TT / მყ . (4) Hint: Population growth gr is determined by the number of surviving children. The populations growth rate is therefore nл. The total differential delivers the above equation. b) In light of (4), explain how the model can generate a hump-shaped correlation between income y and population growth (No derivations! Just provide an economic rationale based on your findings).
Chapter7: Uncertainty
Section: Chapter Questions
Problem 7.8P
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can you solve 1a 2a and 2b can you send solutions through pictures as it will be much easier to follow thanks.
Transcribed Image Text:1. Denote the number of children by n and the survival probability of a child to adult-
hood by . A parental household derives utility out of consumption (c), the number
of surviving offspring (n) and child expenditures (h). Preferences are specified as
follows
=
uln(cc) B₁ ln(n · π(ñ, h)) + ẞ2 In(h),
ẞ1 > B2,
(1)
where the parameter c> 0 reflects subsistence consumption.
Child survival is given by
π = π + (1 − µñ)λh,
≥ 0, x > 0.
(2)
is exogenous to the household and positively affected by average income y such
(y). The representative household's budget constraint reads as
that
y = c + nhy
(3)
a) Solve the household's utility maximisation problem.
b) Show and explain how fertility n responds to changes in households' income y.
c) Show and explain how n and h respond to changes in 7.
2. Assume now that all households are identical such that y = y.
a) In light of your results obtained in 1., show that the reaction of population
growth in response to a change in income can be expressed as
dgL
dy
C
1
= Nπ
-
Ly(y
μπ π
1 -
TT / მყ .
(4)
Hint: Population growth gr is determined by the number of surviving children.
The populations growth rate is therefore nл. The total differential delivers the
above equation.
b) In light of (4), explain how the model can generate a hump-shaped correlation
between income y and population growth (No derivations! Just provide an
economic rationale based on your findings).
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