I have a diamond overlapping model

The question is as follows

Let us consider an infinitely lived production economy populated at time t by $N_t$ identical and perfectly competitive adult individuals. Each agent is born from a single adult parent and lives for two periods. Each youth is part of his parent consumption and each adult is endowed with 1 unit of time that can be allocated between child rearing and market activity. At time t, the budget constraint of the representative adult individual can be written as: $$c_t\le d_t +w_t e_t$$ where e denotes the time allocated to work, w stands for the real wage, d represents the real dividend income that corresponds to a fraction 1/N of the aggregate profit $\pi$, and c is the consumption level. Under the assumption that the cost or raising each child is a fraction $\tau$ of the parental time endowment, the time constraint faced by the representative household at time t is: $$\tau n_t + e_t=1$$ where n denotes the number of children born at time t. The preferences of an adult individual at time t can be described by a logarithmic utility function U which depends on his consumption level and the number of children:$$u(c_t, n_t)=ln c_t + \gamma ln n_t$$ where γ>0. Let us assume that there is a single perfectly competitive firm using the following Cobb-Douglas production function at every time t: $$ Y_t= AX_t^{1-\epsilon}L_t^{\epsilon}$$ where ε∈(0,1) stands for the output elasticity with respect to the land input, A>0 denotes a productivity parameter and Y represents the aggregate output. The stock of land is fixed: $\bar{𝑋}$ = $𝑋_t$ and the labour input $𝐿_t$ is defined as the aggregate amount of time allocated to work. The law of motion of the adult population from time t to t+1 is given by: $$N_{t+1}=n_tN_t$$ The question asks to write down the utility maximization problems for both adults and firm and I need to derive both competitive labor supply and labor demand sechedules respectively.

My answers are respectively:

$d_t$ is dividend income and $d_t=\pi/N$ For agent problem:

$$max [ln c_t + \gamma ln n_t]$$

Subject to $ c_t\le d_t+ w_t (1-\tau n_t)$

Lagrangian $L= ln c_t + \gamma ln n_t +\lambda [c_t- d_t- w_t (1-\tau n_t)]$


(1) W.r.t $c_t$: $(1/c_t)+\lambda =0$

(2) w.r.t $n_t$: $\gamma / n_t +\lambda w_t \tau =0$

(3) w.r.t. $c_t-d_t-w_t+w_t \tau n_t=0$

By (1) and (2):

$$\frac{(1/c_t)}{\gamma / n_t }=\frac{\lambda}{\lambda w_t \tau}$$

So, $c_t= n_t\tau w_t$ …(4)

When substitute (4) into (3)

$n_t\tau w_t-d_t-w_t+w_t \tau n_t=0$

$$n_t=\frac{\pi /N+w_t}{2\tau w_t}$$

which is labor supply schedule.

For firm’s maximization problem

$$max [Y_t-w_tL_t-X_tr_t]$$

$$max [Y_t= A\bar{X}^{1-\epsilon}L_t^{\epsilon}—w_tL_t-\bar{X}r_t]$$

By FOC w.r.t. $L_t$

$\epsilon A\bar{X}^{1-\epsilon}L_t^{\epsilon -1}-w_t=0$$

Then $$L_t= (W_t / \epsilon A)^{1/(1-\epsilon )}$$ which is labor demand sechedule.


So far, I have written all question with my answers in order to give information about the general framework of the question. And I will be glad if you let me know when see any mistakes so far. But my actual question is posted below:

how can I write the equilibrium conditions on the goods’ market at time t and how can I show that it is equivalent to the labor market clearing conditions according to the Walras law?

I do that

For labor market clearing: $N_{t+1}=L_t$

Then, $$N_{t+1}/N_t= L_t/N_t$$

$n_t=e_t={Nw_t-\tau \over 2N}$

But after that, I cannot proceed my solution. Help me to do this part?

Thank you.



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