# Walrasian Equilibrium in A Simple Assignment (Matching) Model

I am reading Acemoglu 1996 and the Walrasian allocation in section II makes me confused. The setting is following.

The economy lasts for two periods and consists of two types of agents, firms and workers, with a continuum of equal measure (normalized to 1) of both types.

At $$t=0$$ workers choose their education level and firms choose their capital stock. At $$t=1$$ productive relations are formed and output is divided between the firm and the worker.

The production function for pair $$(i, j)$$ is $$y_{i j}=A h_{i}^{\alpha} k_{j}^{1-\alpha}$$ , with cost of capital constant $$\mu$$.

Worker $$i$$ maximizes his utility $$V_{i}\left(c_{i}, h_{i}\right)=c_{i}-\frac{1 h_{i}^{1+\Gamma}}{\delta_{i} 1+\Gamma}$$ with $$\Gamma>0$$, where $$c_{i}=W_{i}$$. (I guess here is a typo and the denominator should be $$\delta_{i} (1+\Gamma)$$)

The author states the frictionless Walrasian system as following.

"At $$t=0$$, the auctioneer calls out wage and rate of returns schedules ($$w(h, k)$$, $$r(h, k)$$), and trade stops when all markets clear. The economy will be in Walrasian equilibrium, if and only if (a) given the distribution of investments, the allocation of workers to firms and the wage and rate of return functions are in equilibrium, and (b) given the final rewards, the ex ante investment decisions are privately optimal."

"With Walrasian markets, a worker is allocated to the firm where his marginal product is highest, and since human and physical capital are complements, the most skilled worker will be working for the most productive firm."

"This allocation is a Walrasian equilibrium if and only if all agents are paid their marginal products in their pairings. Therefore, $$w\left(h_{i}, k_{j}\right)=\alpha A h_{i}^{\alpha-1} k_{j}^{1-\alpha}$$ $$r\left(h_{i}, k_{j}\right)=(1-\alpha) A h_{i}^{\alpha} k_{j}^{-\alpha}$$ for all equilibrium pairs $$\left(h_{i}, k_{j}\right)$$. The total equilibrium income of the worker will therefore be given as $$W_{i}=w\left(h_{i}, k_{j}\right) h_{i} .$$ The total income of the firm, $$R_{j}$$, is similarly defined."

"In the Walrasian allocation it must be the case that $$r\left(h_{i}, k_{j}\right)=$$ $$\mu$$, for all equilibrium pairs $$\left(h_{i}, k_{j}\right)$$ which, therefore, must have the same physical to human capital ratio $$\frac{h_{i}}{k_{j}}=\left(\frac{\mu}{(1-\alpha) A}\right)^{1 / \alpha}$$."

"The optimal human capital of worker $$i$$ is given by maximizing $$\alpha A h_{i}^{\alpha-1} k_{j}^{1-\alpha}=h_{i}^{\Gamma} / \delta_{i}$$."

Question 1: To see "the ex ante investment decisions are privately optimal", we consider the worker maximizes utility $$V_{i}\left(c_{i}, h_{i}\right)= w\left(h_{i}, k_{j}\right) h_{i} -\frac{1 h_{i}^{1+\Gamma}}{\delta_{i} 1+\Gamma}$$ and the firm maximizes profit $$r\left(h_{i}, k_{j}\right)k_{j} - \mu k_{j}$$. From the derivation above, it then seems that the agents take the price function as given and thus do not need to consider the effect of their choices on the wage profiles? I cannot get round this as I think the wage profile is a function of agents' choices.

Questions 2: (This continues the first question.) I understand the statement "allocation is a Walrasian equilibrium iff all agents are paid their marginal products in their pairings" as that under competitive equilibrium, all agents take the price schedules as given and then optimize respectively. However here it seems that there are two choices for each agent, the investment choice and the pair choice. It seems to me that there should be some sequential game and we should consider the Nash equilibrium. Why is this not necessary here?

Questions 3: How can we show it more formally and mathematically that under these price schedules, the markets clear? Also it seems that under the Walrasian equilibrium, we obtain a positive assortative matching (which makes sense as we have complementary inputs in the production). I wonder that when the measure of two types of agents are not equal, is it that there would not be Walrasian equilibrium anymore, but is there still positive assortative matching?