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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:

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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 𝛼,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 𝜌. 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 𝛾 ∈ (0,1) of the total tax revenue T is invested into the public physical capital: 𝐻 = 𝛾𝑇 and a fraction 1 − 𝛾 ∈ (0,1) is consumed by the government: 𝐺 = (1 − 𝛾)𝑇.

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.

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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?

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Disclaimer: I still have to educate myself on utility theory (same as I disclosed here).

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 𝐻 = 𝛾𝑇? 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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