1) In period t, a parental household (indexed by i) equipped with human capital h earns a labour income of whi, where w> 0 represents a constant wage rate. This household derives utility out of own consumption, 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 h₁+1 = (e+ē)" (hi)' (h)¹7, 0<1,7 <1 (1) where e > 0 is a constant parameter and e represents the level of education per child. The households' utility function is specified as U=In(c) + [In(n) + 3 ln(hi+1)] with 7,3 > 0. Raising one child to adulthood requires a share of 0 << 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). 2 a. Solve household i's optimisation problem and explain the economic rationale of your results. b. Let's define a variable z capturing the households' relative human capital en- dowment with respect to the economy's average, such that Show that relative human capital of household i, evolves according to zin = (²²4-7 (2x-(1-8e)ē) z-(1-se)ē -(z) (3) (4) c. Suppose the +1-locus (4) intercepts from above at z. = 1 with the 45-degree line. What does this information imply for the evolution of inequality? Explain the impact of the education subsidy on the +1-locus with an appropriate diagram. Remark: No derivations are required here. Just provide a decent rationale and do as asked!
1) In period t, a parental household (indexed by i) equipped with human capital h earns a labour income of whi, where w> 0 represents a constant wage rate. This household derives utility out of own consumption, 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 h₁+1 = (e+ē)" (hi)' (h)¹7, 0<1,7 <1 (1) where e > 0 is a constant parameter and e represents the level of education per child. The households' utility function is specified as U=In(c) + [In(n) + 3 ln(hi+1)] with 7,3 > 0. Raising one child to adulthood requires a share of 0 << 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). 2 a. Solve household i's optimisation problem and explain the economic rationale of your results. b. Let's define a variable z capturing the households' relative human capital en- dowment with respect to the economy's average, such that Show that relative human capital of household i, evolves according to zin = (²²4-7 (2x-(1-8e)ē) z-(1-se)ē -(z) (3) (4) c. Suppose the +1-locus (4) intercepts from above at z. = 1 with the 45-degree line. What does this information imply for the evolution of inequality? Explain the impact of the education subsidy on the +1-locus with an appropriate diagram. Remark: No derivations are required here. Just provide a decent rationale and do as asked!
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Please try answering these attached questions, they are all related to each other.
![1) In period t, a parental household (indexed by i) equipped with human capital h
earns a labour income of whi, where w> 0 represents a constant wage rate. This
household derives utility out of own consumption , 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 hf. Human capital per
child evolves from one period to another according to
h₁+1 = (et+e)" (hi)*(h²) ¹-7,
(1)
where ē> 0 is a constant parameter and et represents the level of education per
child.
The households' utility function is specified as
0<1,7 <1
U = ln(c) + [ln(ni) + 3 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.
(2)
b. Let's define a variable z capturing the households' relative human capital en-
dowment with respect to the economy's average, such that
Show that relative human capital of household i, evolves according to
(za - (1 - 8e)ē)
z-(1-se)ē
- (ai)*
x+1=
"
(3)
(4)
c. Suppose the +1-locus (4) intercepts from above at z = 1 with the 45-degree
line. What does this information imply for the evolution of inequality? Explain
the impact of the education subsidy on the +1-locus with an appropriate
diagram.
Remark: No derivations are required here. Just provide a decent rationale
and do as asked!
2) Assume now that education is provided by public schools fixing the level of education
for all agents at eit = epub = 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 << 1 on labour incomes such
that the budget constraint of a household i reads as
(1-v) whub (1-zn-pub) = c.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).
b. The government has to observe a balanced budget, i.e. expenditures for public
pub
schooling equal tax revenues (whub (1-zn)Nt), where N₁ represents the
number of parental households in the economy. Derive the level of public edu-
cation epub for a given tax and explain this result.
c. Given your observations in 1)c., consider an economy that converges towards
T. When and why would you recommend (if at all) that education should be
provided by public schools?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1fd16735-8754-414f-abc5-79108431741b%2F90547cdf-7885-41d5-b38f-2112cd8b3544%2Fg7b7ck_processed.png&w=3840&q=75)
Transcribed Image Text:1) In period t, a parental household (indexed by i) equipped with human capital h
earns a labour income of whi, where w> 0 represents a constant wage rate. This
household derives utility out of own consumption , 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 hf. Human capital per
child evolves from one period to another according to
h₁+1 = (et+e)" (hi)*(h²) ¹-7,
(1)
where ē> 0 is a constant parameter and et represents the level of education per
child.
The households' utility function is specified as
0<1,7 <1
U = ln(c) + [ln(ni) + 3 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.
(2)
b. Let's define a variable z capturing the households' relative human capital en-
dowment with respect to the economy's average, such that
Show that relative human capital of household i, evolves according to
(za - (1 - 8e)ē)
z-(1-se)ē
- (ai)*
x+1=
"
(3)
(4)
c. Suppose the +1-locus (4) intercepts from above at z = 1 with the 45-degree
line. What does this information imply for the evolution of inequality? Explain
the impact of the education subsidy on the +1-locus with an appropriate
diagram.
Remark: No derivations are required here. Just provide a decent rationale
and do as asked!
2) Assume now that education is provided by public schools fixing the level of education
for all agents at eit = epub = 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 << 1 on labour incomes such
that the budget constraint of a household i reads as
(1-v) whub (1-zn-pub) = c.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).
b. The government has to observe a balanced budget, i.e. expenditures for public
pub
schooling equal tax revenues (whub (1-zn)Nt), where N₁ represents the
number of parental households in the economy. Derive the level of public edu-
cation epub for a given tax and explain this result.
c. Given your observations in 1)c., consider an economy that converges towards
T. When and why would you recommend (if at all) that education should be
provided by public schools?
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