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:
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?