In period t, a parental household (indexed by i) equipped with human capital hit earns a labour income of wh^i_t, where w > 0 represents a constant wage rate. This household derives utility out of own consumption c^i_t, the number of children nit and their level of human capital h^i_(t+1). Education is provided by teachers who are equipped with the economy’s average level of human capital h^T_t . Human capital per child evolves from one period to another according toh^i_(t+1) =(e^i_t +e ̄)^η(h^i_t)^τ(h^T_t )^(1−τ), 0 < η, τ < 1 (1)where e ̄ > 0 is a constant parameter and e^i_t represents the level of education per child.The households’ utility function is specified asU_t^i = ln(c^i_t) + γ[ln(n^i_t) + βln(h^i_(t+1))] (2)with γ,β > 0.Raising one child to adulthood requires a share of 0 < z < 1 units of time. Moreover, education is subsidised at a rate 0 ≤ s_e < 1, such that education costs per child amount to wh^T_t e^i_t(1 − s_e).Solve household i’s optimisation problem and explain the economic rationale of your results. 1) In period t, a parental household (indexed by i) equipped with human capital hiearns a labour income of whi, where w > 0 represents a constant wage rate. Thishousehold derives utility out of own consumption c, the number of children nand their level of human capital h+1. Education is provided by teachers who areequipped with the economy's average level of human capital h. Human capital perchild evolves from one period to another according toht+1 = ( + ≥)"(h) (hể)1-7, 0 0 is a constant parameter and et represents the level of education perchild.The households' utility function is specified asUį = ln(ci) + y[ln(n) + ß ln(ht+1)]with 7,3 > 0.Raising one child to adulthood requires a share of 0 < z < 1 units of time. Moreover,education is subsidised at a rate 0 ≤ se < 1, such that education costs per childamount to whe(1 — se).a. Solve household i's optimisation problem and explain the economic rationale ofyour results.

Utility FunctionBudget ConstraintHousehold Income - Cost of educating children =

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Chapter1: Making Economics Decisions

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In period t, a parental household (indexed by i) equipped with human capital hit earns a labour income of wh^i_t, where w > 0 represents a constant wage rate. This household derives utility out of own consumption c^i_t, the number of children nit and their level of human capital h^i_(t+1). Education is provided by teachers who are equipped with the economy’s average level of human capital h^T_t . Human capital per child evolves from one period to another according to

h^i_(t+1) =(e^i_t +e ̄)^η(h^i_t)^τ(h^T_t )^(1−τ), 0 < η, τ < 1 (1)

where e ̄ > 0 is a constant parameter and e^i_t represents the level of education per child.
The households’ utility function is specified as

U_t^i = ln(c^i_t) + γ[ln(n^i_t) + βln(h^i_(t+1))] (2)

with γ,β > 0.
Raising one child to adulthood requires a share of 0 < z < 1 units of time. Moreover, education is subsidised at a rate 0 ≤ s_e < 1, such that education costs per child amount to wh^T_t e^i_t(1 − s_e).

Solve household i’s optimisation problem and explain the economic rationale of your results.

1) In period t, a parental household (indexed by i) equipped with human capital hi
earns a labour income of whi, where w > 0 represents a constant wage rate. This
household derives utility out of own consumption c, the number of children n
and their level of human capital h+1. Education is provided by teachers who are
equipped with the economy's average level of human capital h. Human capital per
child evolves from one period to another according to
ht+1 = ( + ≥)"(h) (hể)1-7, 0<n, T<1
(1)
where ē> 0 is a constant parameter and et represents the level of education per
child.
The households' utility function is specified as
Uį = ln(ci) + y[ln(n) + ß ln(ht+1)]
with 7,3 > 0.
Raising one child to adulthood requires a share of 0 < z < 1 units of time. Moreover,
education is subsidised at a rate 0 ≤ se < 1, such that education costs per child
amount to whe(1 — se).
a. Solve household i's optimisation problem and explain the economic rationale of
your results.

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for all agents at ei,t epub
Assume now that education is provided by public schools fixing the level of education
const.. Assume further that agents are not able to
provide additional education privately, i.e. they treat the level of education as given.
Public schooling is financed by a proportional tax, 0 ≤ v < 1 on labour incomes such
that the budget constraint of a household i reads as
=
=
Ct
(1 − v) whub (1 – zn² pub) = ² pub
(5)
a. Derive the desired number of children under the assumption that education is
provided by public schooling. Explain very briefly the difference of your result
as compared to 1).

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