Question
In: Economics
2) Mary opens up a shop making flip-flops. Let: q(L,K) = sqrt(L)sqrt(2K) w = price of...
2) Mary opens up a shop making flip-flops. Let:
q(L,K) = sqrt(L)sqrt(2K)
w = price of labor per unit = $5
r = price of capital per unit = $15
p = price of flip-flops per unit= $5
a) Suppose Mary contracts on 50 units of flip-flop making machinery (kapital)… what is her optimal Labor demand in the short run i.e. what quantity of labor (L) should she hire if capital is fixed at K = 50?
b) Suppose wage increases to $10 per unit… how does her optimal short run labor demand change (i.e. what is the new optimal short run L)? Intuitively… what change resulted and why?
c) Mary is now planning for the long run… she must make 1000 flip flops – use the substitution method to write her problem as a single variable optimization (hint she wants to spend as little as possible subject to her production being at least q=1000). Let w=5 and r=15 still. DO NOT SOLVE…
Solutions
Expert Solution
2. The production function is given as
. The cost of production would be 
or 
. The total revenue would be 
or 
.
(a) In the short run, K is fixed at 50. The
production function would be 
or 
. The cost would be 
or 
. The profit would be 
or 
or 
. The profit would be maximum where 
or 
or 
or 
or 
or 
. The optimal labor demand would hence be 25 units.
(b) If the wage increases, then the cost
function changes to be 
, and the profit would be 
in the short run. The profit would be maximum where
or 
or 
or 
or 
. If the wage increases, the optimal labor demand would be 6.25
units. As the wage increases, the labor demand decreases, because
increase in wages increases the cost, and decreases the profit. The
relation between wage and marginal labor product is that 
. As price remains constant, an increase in wage require marginal
product of labor to increase, which would require the labor input
to decrease.
(c) In this case, both K and L are variable, and since output is given, the cost should be minimized with the output constraint. The problem would be as below.
Minimize
Subject to 
The Lagrangian function in this case would be 
, and solving its FOCs would give us the required labor and
capital demand.
Rahul Sunny answered 2 years agoRelated Solutions
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