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In competitive markets without frictions, the marginal buyer/seller determine the price. To what extent is this argument true in markets where a fraction of the sellers are constrained?

Think about the labor market. Some workers are optimally supplying labor given their marginal rate of substitution and the wage rate. Some workers are borrowing-constrained such that they have to supply their full labor endowment.

Assume perfect competition on the labor demand side. Under which conditions can we determine the wage rate using first-order condition of the firms and the first type of workers?

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If I have understood correctly, the question asks what should we know so that we can determine the wage using only information on the unconstrained workers. Here is a toy static model:

Let's say we have $N_u$ unconstrained workers and $N_c$ constrained workers. Each has a total labor endowment $t$. Worker population is denoted $N_c+N_u = N$. The unconstrained workers will solve a utility maximization problem

$$\max u(c,\ell_u)\;\; \text{s.t.}\;\; c_u = w\ell_u,\;\; u_c>0, u_l<0$$ Subscripts in functions denote derivatives. The above will give

$$\ell_u^s: wu_c+u_l = 0 \Rightarrow \ell_u^s = h(c,w)$$

The constrained workers will supply each $t$. So total labor supply will be

$$L^s = N_ct + N_uh(c,w) = (N-N_u)t + N_uh(c,w),\;\; h_w>0$$

(note that some simple utility function forms lead to the supply of labor being independent of the wage. We assume this is not the case here. Also, $h_w>0$ assumes away backward-bending of the individual supply curve).

Since we assume perfect competition (and price taking behavior) on the labor demand side, the firms do not explore possible benefits from the existence of two types of workers. They just go on and equate the marginal product of labor to the market wage

$$\ell^d: MP_L = w \Rightarrow \ell^d = g(w, k_j,T), \;\; g_w <0$$

where $k_j$ is firms capital, and $T$ is technology. If there are $m$ firms we will have the equilibrium condition

$$L^s = L^d \Rightarrow (N-N_u)t + N_uh(c,w)= mg(w, k_j,T)$$

or writing $n_u = N_u/N$

$$(1-n_u)t + n_uh(c,w)= \left(\frac mN\right)g(w, k_j,T) \tag{1}$$

So if we know the proportion of unconstrained workers, and the total worker population, we can determine the wage from $(1)$, using the first-order conditions related to labor demand, and conditions related to the unconstrained workers.

If we further assume a Cobb-Douglas production function (constant returns to scale) for the firms, then labor demand will be linear in capital, and so $g(w, X) = \xi(w,T)k_j$. Then $(1)$ becomes

$$(1-n_u)t + n_uh(c,w)= \xi(w,T)\frac {K}{N} \tag{2}$$

Here we need to know only the proportion of unconstrained workers, and the capital per worker.

Is this what the question asked?

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