Capital accumulation

Question

I have following dynamic optimization problem

$$V(k_0)=\max \sum_{t=0}^{\infty} b^t((1-a)ln c_t+a ln(l_t))$$

Subject to $l_t+e_t=h$ and $y=Ak_t^pe_t^{1-p}$ and $k_{t+1}=i_t+(1-x)k_t$

Where c is consumption, i is inverse,ent, y is output, k is phisical capital per capita. $l$ is leisure time , e is working time. V is value function. x is depreciation rate. A is total factor productivity.

The question I asked to you is that when a is zero, the economy would accumulate more physical capital? Or not? If yes, Why?

I think that

Capital accumulates according to $k_{t+1}=i_t+(1-x)k_t$

Since investment $i_t=y_t-c_t$

So

$k_{t+1}=y_t+(1-x)k_t-c_t$

When a=0, the utility function becomes $u(c_t)=ln(c_t)$

That, househoulds get more utility from consumption with this new utility function in comparision to the previous utility function given. So consumption level increases, which decreases one period forward capital stock.

But the things that I said seems to be wrong to some extent. What do you think ? Do you have any other wise idea about this?

Answer 1

$a=0$ means implies the household does not enjoy leisure. In other words, the trade-off between consumption and leisure is switched off. It's not generally true that with $a=0$, utility is higher for a given value of $c_t$, as this depends on the previous value of $l$. Whether or not consumption rises as a consequence depends also on the parameter $b$, i.e. the utility discount factor. If the household does not value leisure, he might work 80 hours per week and obtain a higher income. If he is very forward-looking however, he might also save a lot for later consumption.

Answer 2

Without labor disutility, the household will work full-time so that $e=1$. The Euler equation will pin down the optimal steady state capital-to-labor ratio, which is just a function of the time preference $b$ and the depreciation rate $x$. If the capital to labor ratio is always a fixed number regardless of $a$, you can immediately see that the numerator $k$ must be higher if the denominator $e$ is at its maximum.

Intuitively, the production factors are q-complements: more of one factor makes the other one more productive as well. With higher labor, capital becomes more productive and it pays to invest more.