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 $P_x=P_y=5$.

  • $\begingroup$ 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 $y$? $\endgroup$ – Alecos Papadopoulos May 7 '15 at 2:32
  • $\begingroup$ "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? $\endgroup$ – Kitsune Cavalry Mar 18 '16 at 16:16