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!
Chapter3: Scarcity, Trade-offs, And Production Possibilities
Section: Chapter Questions
Problem 3P
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