This a repost. In my first try at the problem here I was not very clear with my question and I was also troubled by my lack of knowledge of MathJax. And I probably lost the attention of you guys while trying to make things clearer. So here comes a second try: I'll post the problem:
10.4 A RBC model with generations
We will study and solve analytically a simple model with generations where, as in reality, investment and consumption can react quite differently to productivity shocks. The model is adapted from Huffman (1995), who studies the more general case where agents have a positive probability of death. We assume that the population grows at the rate n, and the supply of labor also grows at the rate n. Each generation of households lives forever, works only in the rst period of its life, and consumes in all periods by saving under the form of capital. The household of generation has a utility function:
$$\sum_{t=\tau}^\infty\beta^t \log C_t $$
which it maximizes subject to the budget constraints:
$$ C_t+K_{t+1}=L_t w_t, \qquad\text{for } t=\tau, $$ $$ C_t+K_{t+1}=R_t K_t, \qquad\text{for } t>\tau. $$
Finally the production function is Cobb-Douglas:
$$ Y_t = A_t K_t^\alpha L_t^{1-\alpha}. $$
10.4.1 Questions
1. Compute the values of consumption and investment as functions of capital and output.
2. Show how consumption and investment respond to technology shocks.
Now maximizing and manipulating the first order conditions you will get these answers for question 1:
$$C_{t}=(1-\beta)[Y_{t}+(1-\delta)K_{t}] \tag{1}$$
$$K_{t+1}=\beta[Y_{t}+(1-\delta)K_{t}]\tag{2}$$
$$I_{t}=\beta Y_{t}-(1-\beta +\delta\beta)K_{t} \tag{3}$$
Now here comes my puzzle with question 2: I'm supposed to substitute the production function in every equation to show the interaction with the technology shocks -- (1), (2) and (3). For instance, Equation (2) would become:
$$K_{t+1}=\beta[A_tK^\alpha_{t}L^{1-\alpha}_t+(1-\delta)K_{t}]$$
Then I should try to solve the dynamics of capital in equation (2) and substitute the result in equations (1) and (3). But how can I do that? Log linearization does not seem give me a solution and it seems strange to assume $K_{t+1}$=$K_t$ (actually, can I do that?). I'll be glad if someone could point me the way.
Observation: There is a solutions book of Benassy's Macroeconomic Theory available on the internet. You can google something like "Benassy Solutions" and you will find it. The solution for this problem is available at pages 83-84, but the book does not give the real answer for the second question of the problem. It says it would be "tedious" to derive the dynamics of capital and substitute in the equations of consumption and investment, so it just points the way (not with enough leads for me, at least) for the answer, as I described above.
This is a standard "modern" macro problem. First, you should have a dynamic equation for labor, L, which typically will be that it grows at rate n. So you would use that information to "guess" that the variable K/L would be constant in a steady state. So you would then define a variable k=K/L and rewrite everything in terms of "small" k rather than capital K. What you get is an equation very much like your non-numbered equation obtained by substituting the definition of Y into (2). A special and easy case is simply to set L=1 and solve for k=K. Now what you will discover is that the equation you have is non-linear. It looks like it can't be solved in closed form, but it can, it's a Bernoulli-type equation, which can be linearized by a change of variable. Or you could log-linearize it instead. The solution is k as a function of time t and the paramaters of the problem, from which you can get a solution for K(t)=k(t)*L(t), and Y(t), and all the rest. I would recommend Blanchard & Fischer's Lectures on Macro, or David Romer's Advanced Macro textbook. Google them.