# when a=0, would economy accumulate more physical capital? Why?

Utility function is

$U(c_t,l_t)=(1-a)ln(c_t)+aln(l_t)$

$l_t$ is leasure time

$c_t$ is consumption

Production function is $y_t=k_t^e(1-l_t)^{1-e}$

$k_{t+1}=i_t+(1-\delta )k_t$

where k is capital delta is capital depreciation rate. i is investment.

My question is

when a=0, would economy accumulate more physical capital? Why?

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I think that, when a=0, agents get utility only from consumption. They don’t get utility from leisure time so they have a more tendency to work. So output increases, which leads to capital accumulation.

Does this make sense? How can more correctly interpret this?

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In addition to interpretation, I have essentially following question

I derive optimal physical capital equation for delta=1and for the maximization problem

$$v(k_t)=max\sum B^tu(c_t,l_t)$$

$$k_{t+1}=Bey^*_t$$

But I could not demonstrate this is locally stable.

Please give me a hint. Thanks.

If you want, I can write my solution in detail.

## 2 Answers

From the looks of this question, it seems like a typical consumption-saving problem with possibly endogenous leisure. I am assuming the production function is constant returns to scale. I am assuming that $a\in[0,1]$ The problem is:

$\underset{\{c_t,k_{t+1},l_t\}} {Max} \sum_{t=0}^{\infty} \beta^t U[c_t,l_t] = (1-a)ln[c_t]+aln[l_t]$

subject to $k_{t+1} = f[k_t,l_t]+(1-\delta)k_t-c_t$

The Bellman Equation:

$V[k_t] = U[c_t,l_t]+\beta V[k_{t+1}]$

Differentiating w.r.t $k_{t+1}$ and using the envelope condition you will find the Euler as:

$\frac{c_{t+1}}{c_t} = \beta (f_{k_{t+1}}+(1-\delta))$

Which at steady state gives you:
$k^* =(1-l^*)(\frac{e}{\frac{1}{\beta}-(1-\delta)})^{\frac{1}{1-e}}$

If a = 0, there is no labor selection problem and $l^*=0$ leading to full use of available time for labor. Clearly, there is more capital in the steady state equilibrium.

Does this make sense? How can more correctly interpret this?

It does not. Where did you get those formulations/definitions? I still have to educate myself on utility theory, but I see several issues with these definitions. The inconsistencies and insufficiency as outlined below impede reaching a robust interpretation of what (if any) impact parameter $a$ has on the levels of capital.

1. $a$ has to be in [0,1]. Otherwise, leisure time and/or consumption would decrease utility, which is absurd. But the ranges of values other variables can take are unclear.
2. Unless you establish positive, lower bounds --though greater than 1-- for $l_t$ and $c_t$, the utility function would be undefined or unbounded (meaning that it could be minus infinite).
3. In the production function, I assume that $e$ is in $[0,1]$. In that case, $l_t$ cannot be greater than 1. But that leads to the absurdity that leisure time decreases utility. Moreover, the condition $l_t<=1$ contradicts the previous item (requiring $l_t$ to be greater than 1), unless one introduces positive constants in $U(c_t,l_t)$ to come up with an adequate definition of utility that is compatible with the definition of production function.
4. It seems to me that the production function might have to explicitly or implicitly involve $c_t$ (so as to represent production inputs).

I think that, when a=0, agents get utility only from consumption. They don’t get utility from leisure time so they have a more tendency to work.

1. How do $i_t$ and $c_t$ relate? That conjecture about $a$ "implies" that it should be thought of as a function of $l_t$ and/or $c_t$, but that is not evident from the formulations/definitions.

Further scrutiny (and knowledge of utility theory) might help identifying additional issues.