0.510.5 Penny\'s Waffle House manufactures frozen waffles and has the following
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Question
0.510.5 Penny's Waffle House manufactures frozen waffles and has the following production function: q= kosp a. Using the Lagrangian method, find the cost minimizing formulas for k and 1 to produce qo units of waffles. (Hint: Set up the Lagrangian, find the FOC's, and solve for k and I as a function of exogenous variables-not of cach other) b. Suppose q,-900, w = 45, and v-5. Find the cost minimizing quantities of capital and labor. Now suppose that Penny's Waffle House only wants to spend 6,750. How much capital and labor would it choose to use? How many units of waffles would it be able to produce? c.Explanation / Answer
A) The long run cost minimization problem can be solved using Lagrangian method.
The cost structure of the firm is given by
C = rK + wL
Setting Lagrangian would imply:
Minimize C = rK + wL – (K0.5L0.5)
Finding the partial derivatives and setting them equal to zero gives
w – 0.5 K0.5L-0.5 = 0
r – 0.5 L0.5K-0.5 = 0
The first two equations give
K/L = w/r
K = L(w/r)
Substitute this value in the production function
Q = K0.5L0.5
Q = (L(w/r))0.5L0.5
L* = Qr0.5/w0.5
K* = Qw0.5/r0.5
b) See that L* = 900*(5/45)^0.5 = 300
K* = 900(45/5)^0.5 = 2700
Cost = 45*300 + 2700*5 = 27000
c) Here C is only 6750. Find the level of output
6750 = 45*L + 5*K
6750 = 45*Qr0.5/w0.5 + 5*Qw0.5/r0.5
6750 = 15Q + 15Q
Q* = 225 units
Required L = 225*(5/45)^0.5 = 75 units and K = 225(45/5)^0.5 = 675 units
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