1.consider an economy producing a single good by prodn fn : Y=min{K,L} where Y is output of the final good,suppose the economy is endowed with 100 units of capital and labour supply given by Ls=50w, w is wage rate, all markets are competitive find equilibrium wage and rental rate.
y=min{k,L}
y=L when L<=K
y=100 when L>K
consider y=L
since it's competitive equilibrium
total cost= L*w + K*r
since k is constant and L=Y
Therefore, MC= w
P=w----(1)
MPL= 1, MPK = 0
Therefore, w=1 and r=0.
Therefore Labor hired is 50 and output is 50 as well.
Well, thats what i make of this question. Tell me what you think.
Here If the firm is minimizing cost then it has to be the case that L=K. In that case 50w=100 then w=2...@Lovekesh but as you explained if L<K then MPL is 1 true but in optimal choice of inputs am getting w=2.. Plz help....
it's not essential that all 100 units of capital will be used (since it's not free). so i don't think v can say that w=2. 's' apparently has some answer. plz come 2 our rescue :D
Vasudha here it has given that economy is producing a single good and it is endowed with 100 units of capital in that case it has to use it's resources efficiently otherwise the production will be inside PPF so what i thought was it has to use all 100 units of capital ( it is endowed with 100 units of capital in that case how can we find the rental rate???) I may be wrong..but plz clarify on this..
If wot ram has said is correct then it has to be the case that r=2...otherwise he wud tend to use capital nd labor in a different ratio which wud nt be compatible wid this productn functn....this is wot i think regardng caital price..though nt sure..
This is what i came up with.
Vasudha was right as always K=L=Y.
Let output be Y.
Total cost = Lw+Kr
L=Y and K=Y
total cost = Yw+Yr
L=50w
therefore, Y=50w
w=Y/50
Total cost= (y^2)/50 + Yr
Marginal cost = 2Y/50 + r
P= 2Y/50 + r
But in competitive equilibrium for profits to be zero, P= ATC
ATC= Y/50 + r
if you equate both the equations, answer comes out to be Y/50 = o
Therefore, Y=o
So, nobody produces anything. Tell me where i went wrong.
Lovekesh it was cool till the derivation of the total cost function but here u don't have to have P=MC. u can see that as long as P>w+r profits keep increasing as u increase output..
Vasudha
P>ATC for profits. Supply curve is MC curve only as for profit maximization condition in competitive industry. firms are making money, then they will go on producing till all capital is utilised. Labor wage will be 2. But what about economy endowed with 100 units of capital and competitive markets. And what about rental rate of capital.
hey ..i was thinking in the lines of Ram..that all of 100 units of K is used up and so Y=L=K...thats how i got w=2, labour employed is 100..but im clueless as to how to find r ....:|
hey lovekesh i got what u did in ur last post ,however if labour employed =100=Y then according to what u did ie P= 2Y/50 + r...P=4+r...r=P-4...i may be horribly wrong though
Could be right ritu. since new firms would like to join in and reap the benefits of P>ATC. So, their natural response will be to bid up the wage of labors and rental rates of capital. With more wage, labor force will increase but since amount of capital is fixed, the opportunity cost of capital will rise which is nothing but their rental rates. So, r=p-4. In this way, competitive equilibrium be there and profits would be zero.
I don't think wage can remain at 2 since profit is there. And wage can't come down just because some labor force is not able to find a job coz capital is fixed.
Therefore, in equilibrium, wage will be 4 and rental rate will be p= 4-r. Then only we can attain equilibrium, when there is no tendancy for new firms to enter. And that can happen only when p= 4+r.
P=w+r
Therefore, w=4 and r=p-4
due to nature of production fn: we will have Y=K=L
: MC=w+r
: maximum Y=100 (since capital is fixed at 100)
due to nature of economy: we will have p=w+r (to eliminate profits)
for labor market eq:
L=50w
= Y=50w
this implies... w=Y/50 .. (and since max Y is 100.. Max w=2... if w is more than 2 labor supply>labor demand)
and 'r' can be anything .. as long as p=w+r
since we dont have a demand fn, we cant put any restriction on price or output..from the demand side..
its like ... we can have w=1 .. r=1 .. we will employ 50 L and 50 K.. and remaining K will lie idle and p=2.. Y=50.. .and no profits...
there wont be any problem even if capital is free good... say, w=1, r=0, we still employ 50 units of both... and Y still equals 50.. but the price is now "1" .. so that profits equal=0
and similarly there wont b any problem if r=10 .. we just raise the p accordingly...
...and when w=1.5 OR w=2... same thing goes.. just output increases to 75 and 100 respectively...
(if we had demand fn we would have been able to say something abt price and thus abt r.. but here, NO!)
therefore answers become.. w belongs to [0,2] .. and r belongs to [0,infinity)
this is probably whole wrong but plss tell me than what is the mistake...???????????????
I'm not sure if this is right, but I approached this like in the case of a consumer's budget equation.
The firm is endowed with 100 units of capital. So assuming capital rate is r, budget equation is
wL + rK = 100r
Now it's a question of maximizing output at this cost.
Setting L = K,
L = 100r/(r+w). Also, given L = 50w.
Equating the 2, 100r/(r+w) = 50w
solving for w, I get the equation w = 1/2[(r^2 + 8r)^1/2 + r]
Doesn't look right to me, I must say. But am I making a real big flaw in this calculation?
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