Equilibrium wage and rental rate in an competitive economy

[Asd1995]

Consider an economy producing a single good by a production function Y=
min(L,K) where Y is the output of the final good. K and L are input use of capital and labour respectively. Suppose this economy is endowed with 100 units of capital and labour supply s L is given by the function L(supply)=50*w, where w is the wage rate. Assuming that all markets are competitive find the equilibrium wage and rental rate.

Firstly, efficient productions are K=L, hence L quantity is produced with L units of labour and capital.

Compeitive good market requires 0 profit==>

p=w(L) plus w(K)

Consumer expenditure on good equals their earnings from labour ==

p*L=w(L)*L == w(K)=0

Demand equals supply in capital market==

D(capital)=100

Demand equals supply in labour market==

50*w(l)=D(labour)=D(capital)=100

So w(l)=2

But the answer says w(l)=1 and w(k)=0!? How come?

[Amit Goyal]

[Asd1995]

Sir, why are we assuming the price to be 1, or how is it justifiable? I solved the problem and concluded p=2.

[Asd1995]

We could just as easily choose p=3, and then the w plus l would equal 3, so the labour demand always less than labour supply will not hold.

[Amit Goyal]

This is because labor supply is a function of real wage and not nominal.

[Asd1995]

Oh, understood. So is equating "Money earned by labour"="Money spent on goods"  fallacious?