<p>Consider the one-consumer one-firm economy.
The consumer has preferences over leisure <span class="math-container">$l\in(0,L)$</span> and consumption good <span class="math-container">$x ≥ 0$</span> represented by utility function <span class="math-container">$u(x, l) = ax + l$</span>, where <span class="math-container">$a &gt; 0$</span> is a parameter. Moreover, the consumer is endowed with <span class="math-container">$(0, L)$</span>, that is, has zero units of the consumption good and <span class="math-container">$L &gt; 0$</span> units of leisure as an endowment.</p>
<p>The firm uses the production function <span class="math-container">$f: R_{\geq0}\rightarrow R$</span> to produce the consumption good out of labor. If the firm uses <span class="math-container">$z ≥ 0$</span> units of labor it produces <span class="math-container">$f(z)$</span> units of the consumption good, where:</p>
<p><span class="math-container">$$f(z)=\begin{cases}
0 &amp;\text{if } z=0\\
z+1 &amp;\text{if } z\in(0,L)\\
L &amp;\text{if } z\geq L
\end{cases}$$</span></p>
<p>The price of the consumption good is p and the price of labor is <span class="math-container">$w$</span>. The
consumer owns the firm.</p>
<p>The question asks for the Pareto efficient allocations in the case where <span class="math-container">$a=1$</span></p>
<p>I guess we need to solve for <span class="math-container">$$\text{max}\ u(x,l)\ \text{s.t}\ x\leq f(L-l) $$</span>
and we should consider 3 cases.</p>
<p>When <span class="math-container">$l=L$</span> we have <span class="math-container">$f(L-l)=0$</span>  and <span class="math-container">$u(0, L) = L$</span></p>
<p>When <span class="math-container">$l = 0$</span> we have <span class="math-container">$f(L − l) = L$</span> and <span class="math-container">$u(L, 0) = L$</span></p>
<p>When <span class="math-container">$l \in (0, L)$</span> we have <span class="math-container">$f(L − l) = L − l + 1$</span> and <span class="math-container">$(L − l + 1, l) = L + 1$</span></p>
<p>But based on these how can we state the PO allocations? and if we had that <span class="math-container">$a\neq1$</span> what would be the competitive equilibria in this case?</p>
<p>Because when again <span class="math-container">$a=1$</span> profit allocations would be stated as :</p>
<p>when <span class="math-container">$z=0$</span> <span class="math-container">$\pi=p*0-w*0=0$</span></p>
<p>when <span class="math-container">$z\in(0,L)$</span> <span class="math-container">$\pi=p*(z+1)-wz=z(p-w)+p$</span></p>
<p>when <span class="math-container">$z\geq$</span> <span class="math-container">$\pi=L(p-w)$</span> only in the case when <span class="math-container">$z=L$</span> otherwise it won't be optimal.</p>
<p>and we need to solve the consumer’s utility maximization problem as well in order to find CE</p>
<p>But I'm stuck with the other case when <span class="math-container">$a\neq1$</span></p>

CC BY-SA 4.0
2021-04-09T11:41:34.720Z
The consumer's budget constraint is $px + wl \leq \pi + wL$. You have disregarded the optimization problem of the firm altogether. The solution of the utility maximization problem is either no consumprion, or no leisure, or the consumer is indifferent between all bundles on the budget constraint
Grada Gukovic
CC BY-SA 4.0
2021-04-09T11:48:15.753Z
@GradaGukovic yeah for the first part I already calculated the equilibrium based on the b.c where from $px+wl\leq wl+\pi$ when p=w>0 we would get $x+l\leq L+1$ thus $x=L+1-l$. Correct me if I am wrong though
Maybeline Lee
<microeconomics><pareto-efficiency><competitive-equilibrium>
Pareto allocations and Competitive equilibrium
Maybeline Lee
# Pareto allocations and Competitive equilibrium

(Asked by: username_0 on 2021-04-09)

Consider the one-consumer one-firm economy.
The consumer has preferences over leisure $l\in(0,L)$ and consumption good $x ≥ 0$ represented by utility function $u(x, l) = ax + l$, where $a > 0$ is a parameter. Moreover, the consumer is endowed with $(0, L)$, that is, has zero units of the consumption good and $L > 0$ units of leisure as an endowment.

The firm uses the production function $f: R\_{\geq0}\rightarrow R$ to produce the consumption good out of labor. If the firm uses $z ≥ 0$ units of labor it produces $f(z)$ units of the consumption good, where:

$$f(z)=\begin{cases}
0 &\text{if } z=0\\
z+1 &\text{if } z\in(0,L)\\
L &\text{if } z\geq L
\end{cases}$$

The price of the consumption good is p and the price of labor is $w$. The
consumer owns the firm.

The question asks for the Pareto efficient allocations in the case where $a=1$

I guess we need to solve for $$\text{max}\ u(x,l)\ \text{s.t}\ x\leq f(L-l) $$
and we should consider 3 cases.

When $l=L$ we have $f(L-l)=0$ and $u(0, L) = L$

When $l = 0$ we have $f(L − l) = L$ and $u(L, 0) = L$

When $l \in (0, L)$ we have $f(L − l) = L − l + 1$ and $(L − l + 1, l) = L + 1$

But based on these how can we state the PO allocations? and if we had that $a\neq1$ what would be the competitive equilibria in this case?

Because when again $a=1$ profit allocations would be stated as :

when $z=0$ $\pi=p\*0-w\*0=0$

when $z\in(0,L)$ $\pi=p\*(z+1)-wz=z(p-w)+p$

when $z\geq$ $\pi=L(p-w)$ only in the case when $z=L$ otherwise it won't be optimal.

and we need to solve the consumer’s utility maximization problem as well in order to find CE

But I'm stuck with the other case when $a\neq1$

**Tags:** microeconomics, pareto-efficiency, competitive-equilibrium
> **Comments:**
> **username_1**: The consumer's budget constraint is $px + wl \leq \pi + wL$. You have disregarded the optimization problem of the firm altogether. The solution of the utility maximization problem is either no consumprion, or no leisure, or the consumer is indifferent between all bundles on the budget constraint  
> **username_0**: @GradaGukovic yeah for the first part I already calculated the equilibrium based on the b.c where from $px+wl\leq wl+\pi$ when p=w>0 we would get $x+l\leq L+1$ thus $x=L+1-l$. Correct me if I am wrong though  

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