<p>Assume that you have two firms, firm A and firm B. Firm A faces the
production function:
<span class="math-container">$$
Y_{A}=A_{A}F\left(K_{A},L_{A}\right)
$$</span>
and firm B faces:
<span class="math-container">$$
Y_{B}=A_{B}F\left(K_{B},L_{B}\right)
$$</span>
Factors are paid their marginal products. In other words:
<span class="math-container">$$
w=MP_{L}
$$</span>
for both A and B, such that:
<span class="math-container">$$
w=A_{A}F_{L_{A}}\left(K_{A},L_{A}\right)=A_{B}F_{L_{B}}\left(K_{B},L_{B}\right)
$$</span></p>
<p>Factor markets clear such that the wage level <span class="math-container">$w$</span> solves :
<span class="math-container">$$
L_{A}^{-1}\left(w^{*},A_{A},K_{A}\right)+L_{B}^{-1}\left(w^{*},A_{B},K_{B}\right)=L^{S}
$$</span>
where <span class="math-container">$L^{s}=L_{A}+L_{B}$</span></p>
<p>My question is this: does <span class="math-container">$A_{A}$</span> changing have any effect on the
equilibrium wage? If it doesn't, the comparative static becomes easy
(i.e. <span class="math-container">$L_{A}$</span> goes down). But, if it does, then how can we show it
in this model?</p>

CC BY-SA 4.0
2022-03-02T17:06:04.577Z
Your last two equations imply that $w=L^S$ ?
Bertrand
CC BY-SA 4.0
2022-03-02T17:32:22.450Z
sorry I'll correct that- should be the inverse function.
Kwame Brown
CC BY-SA 4.0
2022-03-02T22:24:14.557Z
This setting is too parsimonious to be a general equilibrium model. What goods do two firms produce? How agents supply labor? Do you allow a firm has 0 production? etc.
Alalalalaki
CC BY-SA 4.0
2022-03-02T22:26:17.023Z
With just 1 labour market, it is a partial equilibrium, indeed. The last equation defines implicitly the equilibrium wage $w^*$ as a function of $L^S,A_A, A_B,K_A, K_B$, which should be taken into account for comparative statics.
Bertrand
<labor-economics><general-equilibrium>
Changes in TFP and wage rates across firms
Kwame Brown
# Changes in TFP and wage rates across firms

(Asked by: username_0 on 2022-03-02)

Assume that you have two firms, firm A and firm B. Firm A faces the
production function:
$$
Y\_{A}=A\_{A}F\left(K\_{A},L\_{A}\right)
$$
and firm B faces:
$$
Y\_{B}=A\_{B}F\left(K\_{B},L\_{B}\right)
$$
Factors are paid their marginal products. In other words:
$$
w=MP\_{L}
$$
for both A and B, such that:
$$
w=A\_{A}F\_{L\_{A}}\left(K\_{A},L\_{A}\right)=A\_{B}F\_{L\_{B}}\left(K\_{B},L\_{B}\right)
$$

Factor markets clear such that the wage level $w$ solves :
$$
L\_{A}^{-1}\left(w^{\*},A\_{A},K\_{A}\right)+L\_{B}^{-1}\left(w^{\*},A\_{B},K\_{B}\right)=L^{S}
$$
where $L^{s}=L\_{A}+L\_{B}$

My question is this: does $A\_{A}$ changing have any effect on the
equilibrium wage? If it doesn't, the comparative static becomes easy
(i.e. $L\_{A}$ goes down). But, if it does, then how can we show it
in this model?

**Tags:** labor-economics, general-equilibrium

> **Comments:**
> **username_1**: Your last two equations imply that $w=L^S$ ?  
> **username_0**: sorry I'll correct that- should be the inverse function.  
> **username_2**: This setting is too parsimonious to be a general equilibrium model. What goods do two firms produce? How agents supply labor? Do you allow a firm has 0 production? etc.  
> **username_1**: With just 1 labour market, it is a partial equilibrium, indeed. The last equation defines implicitly the equilibrium wage $w^*$ as a function of $L^S,A_A, A_B,K_A, K_B$, which should be taken into account for comparative statics.  

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## 0 answers

