ISI PEB Q4

[dream theater]

4. (a) 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(X,Y) = root(XY) ; and his income is Rs. 100/-. If 10 units of labour are available, find out the equilibrium wage in the competitive labour market.

[L]

N1 + N2 = 10 ........N1- Labor in X, N2 - Labor in Y
Now, X and Y are produced by identical technologies and if N1 are in sector X,

So, X = 2N1  
Y = 2N2 as one unit of labor can produce one unit each of X and Y.

Profits = P1 N1 + P2 N2 - W (N1+ N2)

Total Profits will be zero due to competitive markets.

And P1 X + P2 Y = 100
P1 N1 + P2 N2 = 50

So, Profits = 0
=> w*10 = 50
W = 5

[dream theater]

shouldn't the profit equation be P1.X+P2.Y-w(N1+N2)
and by saying that X=2N1 and Y=2N2 you are implying that both X and Y require half a unit of labour each
but it could also be the case,for example,that one unit of X can be produced with 0.25 units of labour and a unit of Y can be produced with 0.75 units of labour,in this case also one unit of labour produces one unit each of X and Y

[L]

I am not totally sure about that, but, I think identical technology should mean both require equal labor.

In your example, technology can't be identical as less labor is required to produce X as compare to Y.

[dream theater]

ok,but what about the consumer mentioned in the question? you haven't used that info at all
won't the demand for both goods affect the equilibrium wage rate?

[Rajat]

Yes, i also got w=5. The way i have solved it is as given below. Please confirm.

the budget constraint is PxX + PyY = 100 ---------------------------- (1)
Px = price of X
Py = price of Y

Also, in a competitive labour market, we have PxMPLx = w   and PyMPLy = w   -------------------------(2)
where MPLi = Marginal product of labour for commodity i
           w = wage rate

Qx = Lx   and Qy = Ly are the production functions for goods X and Y respectively.
therefore, MPLx = 1 and MPLy = 1.
Putting this in equation (2)
we get Px = Py = w
Putting this equation (1)
w(X + Y) = 100

Also, X + Y = 20 (because Lx + Ly = 10)
therefore, w = 5