# Pareto allocations and Competitive equilibrium

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 $$a\neq1$$

• 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 Apr 9 at 11:41
• @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 Apr 9 at 11:48