<p>Disclaimer: I still have to educate myself on utility theory (same as I disclosed <a href="https://economics.stackexchange.com/questions/23995/when-a-0-would-economy-accumulate-more-physical-capital-why/24000#24000">here</a>).</p>

<blockquote>
  <p>Would production possible without the government: t= 0?</p>
</blockquote>

<p>It cannot be determined with the information given. The problem's definition of $T$ is unclear or incomplete, as the only component of $T$ mentioned in the problem is the physical capital tax rate $t$.</p>

<p>Assuming that the only source of government revenue is the physical capital tax, then no, production would not be possible if $t=0$ because --as you point out--, the latter implies that $T$, $H$, and therefore $Y$ are zero. </p>

<p>But that prompts the question: Does the definition of production $Y$ even make sense if  = ? Obviously not: Production should still be possible because private physical capital $K$ is not affected by the condition $t=0$.</p>

<blockquote>
  <p>Do households have incentives to rent the physical capital to the firm
  if t= 1?</p>
</blockquote>

<p>Yes, they may have an indirect incentive even if $t=1$: The real wage $w$. If nobody rented physical capital, there would be no demand for labor. Consequently, no household would earn income from labor even if a household's utility function indicates a preference for work over leisure.</p>

<p>I gather that the problem is a textbook exercise, and/or that not all the assumptions are reflected in the inquiry. Thus, a portion of my answer possibly goes beyond the scope of that chapter or section.</p>

CC BY-SA 4.0
2018-08-07T10:41:38.450Z
2018-08-07T10:41:38.450Z
Iñaki Viggers
<p>Let us consider a perfectly competitive one-period production economy consisting of one firm, N>0 household(s) and one government. Each household is endowed with x unit(s) of physical capital and with h unit(s) of time that can be allocated to work: e and/or turn into leisure time: l at a rate of one for one. The preferences of the representative household are represented by the following utility function:</p>

<p><a href="https://i.sstatic.net/uHv2S.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uHv2S.jpg" alt="enter image description here"></a></p>

<p>commodity is produced by the single perfectly competitive firm using the following Cobb-Douglas production function:</p>

<p>$$Y=AK^{\alpha}H^bL^{1-\alpha-b}$$
where Y denotes the aggregate level of market production, K represents the aggregate level of private physical capital, H represents the aggregate level of public physical capital, L stands for the aggregate labour input defined as the total amount of time allocated to work, and ,b ∈ (0,1), A>0 are parameters. The firm rents from households both labour at the real wage rate w and physical capital at the real rental rate . Each household earns a real dividend income that corresponds to a fraction 1/N
of the firm’s profit P , pays a physical capital tax t $\in$(0,1) proportional to his real
physical capital income. Let us assume that a fraction  ∈ (0,1) of the total tax revenue T is invested into the public physical capital:  =  and a fraction 1 −  ∈ (0,1) is consumed by the government:  = (1 − ).</p>

<p>My question</p>

<blockquote>
  <p>Would production possible without the government: t= 0? Do households have incentives to rent the physical capital to the firm if t= 1? </p>
</blockquote>

<p>I wrote the governemnt’s revenue = $T= t(NρΧ)$</p>

<p>And then I think to use laffer curve to answer this question. </p>

<p>For example </p>

<p>when t=0, T=0, so H=0 so Y=0. There is no production. </p>

<p>When, t=1, T=NρΧ so H>0 and Y>0</p>

<p>And hh’s phisical capital revenue=(1-t )ρX. So when t=1, there is no revenue for households. So they don’t want to rent their physical capitals. 
In order to find optimal $t^*$</p>

<p>I need to maximize $T= t(NρΧ)$ w.r.t $t$. But I could not get any reasonable value. </p>

<p>——</p>

<p>An edit:</p>

<p>Should I consider X depends on t here. That is X=X(t)</p>

<p>Then    $$ \max tNρΧ(t)$$ subject to $t$.</p>

<p>By FOC. NpX(t)+tNρΧ’(t)=0</p>

<p>There exist $t^*$ such that $NpX(t*)+t^*NρΧ’(t*)=0$ Holds. </p>

<p>Is this way correct? Or </p>

<p>How can I solve this question? </p>

<macroeconomics>
Laffer curve in one period economy
b11bb
# Laffer curve in one period economy

(Asked by: username_0 on 2018-08-06)

Let us consider a perfectly competitive one-period production economy consisting of one firm, N>0 household(s) and one government. Each household is endowed with x unit(s) of physical capital and with h unit(s) of time that can be allocated to work: e and/or turn into leisure time: l at a rate of one for one. The preferences of the representative household are represented by the following utility function:

[![enter image description here](https://i.sstatic.net/uHv2S.jpg)](https://i.sstatic.net/uHv2S.jpg)

commodity is produced by the single perfectly competitive firm using the following Cobb-Douglas production function:

$$Y=AK^{\alpha}H^bL^{1-\alpha-b}$$
where Y denotes the aggregate level of market production, K represents the aggregate level of private physical capital, H represents the aggregate level of public physical capital, L stands for the aggregate labour input defined as the total amount of time allocated to work, and ,b ∈ (0,1), A>0 are parameters. The firm rents from households both labour at the real wage rate w and physical capital at the real rental rate . Each household earns a real dividend income that corresponds to a fraction 1/N
of the firm’s profit P , pays a physical capital tax t $\in$(0,1) proportional to his real
physical capital income. Let us assume that a fraction ∈ (0,1) of the total tax revenue T is invested into the public physical capital: = and a fraction 1 − ∈ (0,1) is consumed by the government: = (1 − ).

My question

> Would production possible without the government: t= 0? Do households have incentives to rent the physical capital to the firm if t= 1?

I wrote the governemnt’s revenue = $T= t(NρΧ)$

And then I think to use laffer curve to answer this question.

For example

when t=0, T=0, so H=0 so Y=0. There is no production.

When, t=1, T=NρΧ so H>0 and Y>0

And hh’s phisical capital revenue=(1-t )ρX. So when t=1, there is no revenue for households. So they don’t want to rent their physical capitals.
In order to find optimal $t^\*$

I need to maximize $T= t(NρΧ)$ w.r.t $t$. But I could not get any reasonable value.

——

An edit:

Should I consider X depends on t here. That is X=X(t)

Then $$ \max tNρΧ(t)$$ subject to $t$.

By FOC. NpX(t)+tNρΧ’(t)=0

There exist $t^\*$ such that $NpX(t\*)+t^\*NρΧ’(t\*)=0$ Holds.

Is this way correct? Or

How can I solve this question?

**Tags:** macroeconomics

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## 1 answers

### Answer by username_1 on 2018-08-07. Score: 3 | Accepted answer

Disclaimer: I still have to educate myself on utility theory (same as I disclosed [here](https://economics.stackexchange.com/questions/23995/when-a-0-would-economy-accumulate-more-physical-capital-why/24000#24000)).

> Would production possible without the government: t= 0?

It cannot be determined with the information given. The problem's definition of $T$ is unclear or incomplete, as the only component of $T$ mentioned in the problem is the physical capital tax rate $t$.

Assuming that the only source of government revenue is the physical capital tax, then no, production would not be possible if $t=0$ because --as you point out--, the latter implies that $T$, $H$, and therefore $Y$ are zero.

But that prompts the question: Does the definition of production $Y$ even make sense if = ? Obviously not: Production should still be possible because private physical capital $K$ is not affected by the condition $t=0$.

> Do households have incentives to rent the physical capital to the firm
> if t= 1?

Yes, they may have an indirect incentive even if $t=1$: The real wage $w$. If nobody rented physical capital, there would be no demand for labor. Consequently, no household would earn income from labor even if a household's utility function indicates a preference for work over leisure.

I gather that the problem is a textbook exercise, and/or that not all the assumptions are reflected in the inquiry. Thus, a portion of my answer possibly goes beyond the scope of that chapter or section.


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