<p><strong>Two commodities, X and Y, are produced with identical technology and are 
sold in competitive markets. One unit of labour can produce one unit of each of 
the two commodities. Labour is the only factor of production; and labour is 
perfectly mobile between the two sectors. The representative consumer has the 
utility function: U = XY ; and his income is Rs. 100/-. If 10 units of labour are 
available, find out the equilibrium wage in the competitive labour market.</strong></p>

<p>Please tell me if I am going wrong.</p>

<p>My attempt: Solving the consumer's utility maximization problem, I have the demands $$x(p_x)=\frac{50}{p_x},\ y(p_y)=\frac{50}{p_y}.$$ The firm has same production functions for both X and Y: $x=y=f(l)$. Since one unit of labour produces one unit each of X and Y, we consider $f(l)=l.$ Therefore the firm operates with a profit function $$\pi=P_xl+P_yl-wl,$$ producing $l$ units of X and Y respectively. Here, $w$ is the wage in the competitive labour market and $P_x, P_y$ are the respective equilibrium market prices. Since $(l,P_x)$ and $(l,P_y)$ must lie on the respective demand curves, we get $l=\frac{50}{P_x}=\frac{50}{P_y}$. Since the competitive firm makes zero profits, $\pi=100-wl=0.$ If 10 units of labour are available, the equilibrium wage is $w=100/10=10.$ The equilibrium market prices for X and Y are $P_x=P_y=5$.</p>

CC BY-SA 3.0
2015-05-07T02:32:57.530Z
Please clarify "One unit of labor can produce one unit of each of the two commodities" means: $l = x$, and separately, $l=y$, _or_ that if a firm, that produces both goods, employs _one_ unit of labor, it ends up with one unit of $x$ _and_ one unit of $y$?
Alecos Papadopoulos
CC BY-SA 3.0
2016-03-18T16:16:13.797Z
"his income is Rs. 100/-" I'm having a hard time understanding this particular line. What does Rs. mean and what are the things after the 100 supposed to denote?
Kitsune Cavalry
<microeconomics><producer-theory>
undergrad Producer theory problem
saubhik
# undergrad Producer theory problem

(Asked by: username_0 on 2015-04-02)

**Two commodities, X and Y, are produced with identical technology and are
sold in competitive markets. One unit of labour can produce one unit of each of
the two commodities. Labour is the only factor of production; and labour is
perfectly mobile between the two sectors. The representative consumer has the
utility function: U = XY ; and his income is Rs. 100/-. If 10 units of labour are
available, find out the equilibrium wage in the competitive labour market.**

Please tell me if I am going wrong.

My attempt: Solving the consumer's utility maximization problem, I have the demands $$x(p\_x)=\frac{50}{p\_x},\ y(p\_y)=\frac{50}{p\_y}.$$ The firm has same production functions for both X and Y: $x=y=f(l)$. Since one unit of labour produces one unit each of X and Y, we consider $f(l)=l.$ Therefore the firm operates with a profit function $$\pi=P\_xl+P\_yl-wl,$$ producing $l$ units of X and Y respectively. Here, $w$ is the wage in the competitive labour market and $P\_x, P\_y$ are the respective equilibrium market prices. Since $(l,P\_x)$ and $(l,P\_y)$ must lie on the respective demand curves, we get $l=\frac{50}{P\_x}=\frac{50}{P\_y}$. Since the competitive firm makes zero profits, $\pi=100-wl=0.$ If 10 units of labour are available, the equilibrium wage is $w=100/10=10.$ The equilibrium market prices for X and Y are $P\_x=P\_y=5$.

**Tags:** microeconomics, producer-theory

> **Comments:**
> **username_1**: Please clarify "One unit of labor can produce one unit of each of the two commodities" means: $l = x$, and separately, $l=y$, _or_ that if a firm, that produces both goods, employs _one_ unit of labor, it ends up with one unit of $x$ _and_ one unit of $y$?  
> **username_2**: "his income is Rs. 100/-" I'm having a hard time understanding this particular line. What does Rs. mean and what are the things after the 100 supposed to denote?  

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## 0 answers

