Consider an environment of overlapping generation model that is similar to the model we have
studied in the class. Suppose time is discrete and time horizon is innite, t = 0; 1; 2; :::The
economy is populated with 2-period lived individuals. In period t, Lt new individuals are born
and Lt1 old individuals are dying if they havent died earlier due to covid-19. Assume population
grows at rate n. The lifetime utility of an individual born at t is given by U (c1t ; c2t+1) where
c1t and c2t+1 denote the consumption when young and when old. Utility of an initial old is
simply u (c2; 0). Suppose U is additively separable and the periodic utility, u(c), is an increasing
and concave function of consumption and satises the inada condition. Individuals discount the
future with the discount rate (cid:12) where 0 < (cid:12)(cid:12)< 1: The initial capital stock K0 owned equally by all
initial old individuals. In each period, rms hire labour and rent capital from individuals to
produce output, sell the output in good market, and pay labour and capital. The CRS production
function is Yt = F Kt ; AtL Dt. The intensive form of production: yt = f (kt) has the following
properties: f0(k) > 0, f00 (k) < 0 and limk!0 f0(k) = +1: Suppose At+1 = (1 + gc) At and there is
a drop in the growth of productivity, gc < g. Markets are perfectly competitive. Real interest rate
in period t is rt & the real wage rate per unit of eective labour is wt . Initial conditions (A0, L1
(number of initial old)), K0 are giv
Below are the list of events
(cid:15) (i) Young individuals are born with labour endowments, which is supplied inelastically to rms.
Young individuals may die due to the getting Covid19. Assume that the utility of dead is zero.
With a probability p1 they survive and get to the second period of their life. [Hint: This
probability was one in the model in the class]
(cid:15) (ii) Old individuals are those young individuals that survive the rst period of life. With a
probability p2; they survive to the end of the second period of life. Again the utility of dead is
assumed to be zero. [Hint: This probability was one in the model in the class. Recall that in the
model in the class, all old people die and exit the economy at the end of the second period. Now
they may die before that]. For simplicity we assume even if they die at the middle of the second
period they will consume capital income and existing wealth.
(cid:15) (iii) Firms hire labour from young individuals and rent capital from old individuals to produce
output. Capital and labour are paid.
(cid:15) (iv) Young individuals divide labour income between rst-period consumption and saving to
period t + 1 which form new capital for period t + 1 production. Here for simplicity we assume
that if a young individual dies her savings are kept in an account and will be given to the rm.
Carefully formulate an individual problem for nding the optimal saving. What are the impact of
p1 and p2 on optimal saving?.
Home assignment II on Spectroscopy 2024 Answers.pdf
Consider an environment of overlapping generation model that is simi.pdf
1. Consider an environment of overlapping generation model that is similar to the model we have
studied in the class. Suppose time is discrete and time horizon is innite, t = 0; 1; 2; :::The
economy is populated with 2-period lived individuals. In period t, Lt new individuals are born
and Lt1 old individuals are dying if they havent died earlier due to covid-19. Assume population
grows at rate n. The lifetime utility of an individual born at t is given by U (c1t ; c2t+1) where
c1t and c2t+1 denote the consumption when young and when old. Utility of an initial old is
simply u (c2; 0). Suppose U is additively separable and the periodic utility, u(c), is an increasing
and concave function of consumption and satises the inada condition. Individuals discount the
future with the discount rate where 0 < < 1: The initial capital stock K0 owned equally by all
initial old individuals. In each period, rms hire labour and rent capital from individuals to
produce output, sell the output in good market, and pay labour and capital. The CRS production
function is Yt = F Kt ; AtL Dt. The intensive form of production: yt = f (kt) has the following
properties: f0(k) > 0, f00 (k) < 0 and limk!0 f0(k) = +1: Suppose At+1 = (1 + gc) At and there is
a drop in the growth of productivity, gc < g. Markets are perfectly competitive. Real interest rate
in period t is rt & the real wage rate per unit of eective labour is wt . Initial conditions (A0, L1
(number of initial old)), K0 are giv
Below are the list of events
(i) Young individuals are born with labour endowments, which is supplied inelastically to rms.
Young individuals may die due to the getting Covid19. Assume that the utility of dead is zero.
With a probability p1 they survive and get to the second period of their life. [Hint: This
probability was one in the model in the class]
(ii) Old individuals are those young individuals that survive the rst period of life. With a
probability p2; they survive to the end of the second period of life. Again the utility of dead is
assumed to be zero. [Hint: This probability was one in the model in the class. Recall that in the
model in the class, all old people die and exit the economy at the end of the second period. Now
they may die before that]. For simplicity we assume even if they die at the middle of the second
period they will consume capital income and existing wealth.
(iii) Firms hire labour from young individuals and rent capital from old individuals to produce
output. Capital and labour are paid.
(iv) Young individuals divide labour income between rst-period consumption and saving to
period t + 1 which form new capital for period t + 1 production. Here for simplicity we assume
that if a young individual dies her savings are kept in an account and will be given to the rm.
Carefully formulate an individual problem for nding the optimal saving. What are the impact of
p1 and p2 on optimal saving?
![Consider an environment of overlapping generation model that is similar to the model we have
studied in the class. Suppose time is discrete and time horizon is innite, t = 0; 1; 2; :::The
economy is populated with 2-period lived individuals. In period t, Lt new individuals are born
and Lt1 old individuals are dying if they havent died earlier due to covid-19. Assume population
grows at rate n. The lifetime utility of an individual born at t is given by U (c1t ; c2t+1) where
c1t and c2t+1 denote the consumption when young and when old. Utility of an initial old is
simply u (c2; 0). Suppose U is additively separable and the periodic utility, u(c), is an increasing
and concave function of consumption and satises the inada condition. Individuals discount the
future with the discount rate where 0 < < 1: The initial capital stock K0 owned equally by all
initial old individuals. In each period, rms hire labour and rent capital from individuals to
produce output, sell the output in good market, and pay labour and capital. The CRS production
function is Yt = F Kt ; AtL Dt. The intensive form of production: yt = f (kt) has the following
properties: f0(k) > 0, f00 (k) < 0 and limk!0 f0(k) = +1: Suppose At+1 = (1 + gc) At and there is
a drop in the growth of productivity, gc < g. Markets are perfectly competitive. Real interest rate
in period t is rt & the real wage rate per unit of eective labour is wt . Initial conditions (A0, L1
(number of initial old)), K0 are giv
Below are the list of events
(i) Young individuals are born with labour endowments, which is supplied inelastically to rms.
Young individuals may die due to the getting Covid19. Assume that the utility of dead is zero.
With a probability p1 they survive and get to the second period of their life. [Hint: This
probability was one in the model in the class]
(ii) Old individuals are those young individuals that survive the rst period of life. With a
probability p2; they survive to the end of the second period of life. Again the utility of dead is
assumed to be zero. [Hint: This probability was one in the model in the class. Recall that in the
model in the class, all old people die and exit the economy at the end of the second period. Now
they may die before that]. For simplicity we assume even if they die at the middle of the second
period they will consume capital income and existing wealth.
(iii) Firms hire labour from young individuals and rent capital from old individuals to produce
output. Capital and labour are paid.
(iv) Young individuals divide labour income between rst-period consumption and saving to
period t + 1 which form new capital for period t + 1 production. Here for simplicity we assume
that if a young individual dies her savings are kept in an account and will be given to the rm.
Carefully formulate an individual problem for nding the optimal saving. What are the impact of
p1 and p2 on optimal saving?](https://image.slidesharecdn.com/consideranenvironmentofoverlappinggenerationmodelthatissimi-230401132730-46f8d155/85/Consider-an-environment-of-overlapping-generation-model-that-is-simi-pdf-1-320.jpg)
