This problem set covers material from the Solow Growth Model.  The relevant chapter in the textbook is chapter 8.  You can achieve a maximum of 100 points.  The problem set is due in class on  Wednesday, April 5.  You are allowed to work in groups, but please hand in your individual copy of the solution and indicate who you worked with.

Question 1 (15 points) – The Black Death

In the middle of the 14th century, a plague killed about a third of the population of Europe (the ’Black Death’). Assume that the economy of Europe was well described by the Solow model without technology growth and that the economy was in steady state before the plague occurred. How would the following variables change in response to the Black Death in the short run (i.e. immediately) and in the long run (i.e. once the economy reaches steady state)? Assume that total factor productivity At , the population growth rate n, household’s patience β, and production parameter α are all unaffected by the Black Death.

(a) GDP

(b) GDP per capita

(c) wages

(d) interest rates

Explain intuitively.

Question 2 (28 points) – Korean Reunification

In this question, you will study the economic consequences of unifying North and South Korea through the lens of the Solow growth model.

Consider North Korea and South Korea as two countries which each produce output in period t according to

Yi,t  = Ai,t Kt(i)N,      i = {north, south}     (1)

or, in per capita terms,

yi,t  = Ai,t kt     (2)

where Ki,t is the capital stock in country i at time t, Ni,t is the number of people in country i at time t, and ki,t  =  is the capital stock per worker in country i at period t. Assume that α = 0.5. The two countries differ with respect to their total factor productivity, Ai,t . More specifically, Asouth,t  = 2 and Anorth,t  = 1. Furthermore, South Korea is larger in population size than North Korea: Nsouth,t  = 2 and Nnorth,t  = 1.  The population growth rate in both countries is assumed to be zero, i.e.  ni,t  = 0.  The utility function of a household born in period t is the same in both North and South Korea and given by

U = ln(ci,y,t ) + βi ln(ci,o,t+1)                                           (3)

where ci,y,t  is the consumption of the household when young in country i, and ci,o,t+1  is the consumption of the household when old in country i.  Assume that βi  = 1 for simplicity. Households supply one unit of labor inelastically when young and consume their savings when old.  The lifetime budget constraint of a household born in period t in country i is thus given by

ci,o,t+1 

1 + Ri,t

where Ri,t  is the interest rate on capital in country i and wi,t  is the wage rate in country i.  The capital depreciation rate is zero, i.e.  δ = 0.  Capital per worker in both economies evolves according to

ki,t+1  = si,t                                                                                           (5)

(a) Derive the household’s optimal savings si,t  in each country.

(b) Derive wages wi,t  and interest rates Ri,t  in each country.

(c) Derive steady-state output per worker in each country y and calculate its value.

Suppose North and South Korea unify and that there is a ’technology transfer’from South to North. Total factor productivity in the unified country is then given by Aunified,t  = 2. Assume that North and South Korea each were in their respective steady state before unification.

(d) Assume that right after unification people can freely move between North and South while capital is immobile.  Will people move between the North and the South?  In which direction? Explain intuitively.

(e) Provide a formal condition for when the flow of people within the unified country will stop (assuming that capital is immobile).

(f) Consider the South Korean economy.  What is the impact of unification on wages, interest rates and GDP per capita in South Korea in the short run (i.e.  right after unification when capital is immobile)? Explain.

(g) Will unification hurt the long-run living standards of South Korea? Explain.

Question 3 (27 points) – Fertility in the OLG Model

In this exercise, you will study fertility within an overlapping generations model. Note that this will be different from what we have seen in class since the Solow model does not model fertility explicitly. This question is meant to sharpen your understanding of how overlapping generations models (of which the Solow model is one example) more broadly work.

Yt  = At Lt(α)                                                                                            (6)

with α ∈ (0, 1).  That is, labor Lt  is the only input to production.  There is no physical capital. The firm hires labor at the wage rate wt . The firm’s profit maximization problem is then given as

max At Lt(α) − wt Lt                                                                                   (7)

where At  > 0 is total factor productivity. A household born at t lives for two periods and has a lifetime utility function given by

U = lncy,t + β lnco,t+1,   β ∈ (0, 1)                                      (8)

When young, the household inelastically supplies one unit of labor to the labor market and decides how many children to have.  Each children costs p > 0 units of resources.  Denote the number of children a household born at t produces by nt+1 .  We then call nt+1  the fertility rate of a household born at t.1  The number of young people in period t + 1 is thus given by Nt+1  = Ntnt+1  where Nt  is the number of young people born at t.2  When old, the household obtains a fraction q ∈ (0, 1) of each child’s wage income as an intra-household income transfer.  That is, children in this model serve as old-age security.  A household’s budget constraint when young is thus given by

cy,t + pnt+1  = wt (1 − q)     (9)

The same household’s budget constraint when old is given by

co,t+1  = nt+1qwt+1     (10)

(a) Derive the household’s optimal fertility rate nt(∗)+1 , optimal consumption when young cy(∗),t , and optimal consumption when old co(∗),t+1 .  [Note:  These expressions should be a function of the wage rates wt  and wt+1, the cost of children p, the intra-household transfer q, the patience parameter β, total factor productivity At  and labor produc- tivity α . Also be careful not to forget ’inner derivatives’when taking FOCs.]

(b) Labor supply is given by Lt  = Nt . Derive the firm’s labor demand and then determine the equilibrium wage rate w .

(c) Plug the equilibrium wage rate w from (b) into your expression for equilibrium fertility nt(∗)+1  from (a).  Is the equilibrium fertility rate increasing or decreasing in the cohort size Nt ? Explain intuitively.

(d) Use the expression for the equilibrium fertility rate nt+1  from (c) in Nt+1  = nt+1Nt . This gives an equation linking future young cohort size to current young cohort size.

Plot this relationship in a diagram where Nt  is on the x-axis and Nt+1  is on the y-axis. (e) Derive the steady-state expression for young population size N = Nt∗ = N ∗ .

(f) Determine the effect of an increase in the cost of children p on steady-state young population size N ∗ . Explain intuitively.

(g) Determine the effect of an increase in intra-household transfers q on steady-state GDP per capita, y ∗ =  . Explain intuitively.

(h) Suppose you are a World Bank consultant advising a developing country. Given this model, would you recommend policies that reduce the cost of children (subsidies for child care, for example) in order to promote long-run living standards as measured by GDP per capita? Provide some intuition for your answer.

(i) Read the following article:                                                            https://ourworldindata.org/breaking-the-malthusian-trap

Then use our model to determine the effect of an increase in total factor productivity At  on steady-state GDP per capita, y ∗ =  .  Can you relate your findings to the Malthusian Trap?

Question 4 (30 points) – Human Capital in the OLG Model

Consider an overlapping generations model in which households live for two periods.  The utility of a household born in period t is given by

U = lncy,t + β lnco,t+1                                                                           (11)

where cy,t   denotes consumption when young, and co,t+1   denotes consumption when old. β  ∈ (0, 1) is the household’s discount factor.  When the household is young, she has to decide how much time to spend working on the labor market and how much time to spend acquiring education. Denote time spent on the labor market when young by lt . We assume that the household cannot save any income. The household’s budget constraint when young then reads

cy,t  = wt lt                                                                                          (12)

where wt  is the wage rate for unskilled labor.  The household spends her remaining time 1 − lt  acquiring education. The household’s human capital evolves according to

ht+1  = θ(1 − lt )ht(γ),    γ ∈ (0, 1)                                       (13)

where ht  denotes the human capital stock in the economy when the household is young, and ht+1  denotes the human capital stock in the economy after the household has undertaken her human capital time investment 1 − lt . The parameter θ > 0 measures the effectiveness of time spent acquiring education. When old, a household inelastically supplies labor to the labor market where she earns wage rate vt+1  per unit of human capital ht+1 . This implies that the household’s budget constraint when old is given by

co,t+1  = vt+1ht+1        (14)

Finally, there is a representative firm that produces output according to

Yt  = St(α)Ut1 −α ,    α ∈ (0, 1)      (15)

where St  = ht Nt −1  denotes skilled labor, and Ut  = lt Nt  denotes unskilled labor.  The firm pays wage rate wt  for unskilled labor, and wage rate vt  for skilled labor. Population size Nt is constant over time.

(a) Solve the household’s time allocation problem

max   cy,t,co,t+1,lt lncy,t + β lnco,t+1              (16)