Answers to Profit Max Critical thinking

[Guneet]

Hello, 

Can someone post the solutions the Profit Max Critical Thinking (Subs and Comps) worksheet in the Final Tab?

(Some of the other don't have solutions as well)

[J]

This is my approach to the problem using what I've been told and some intuition of my own so take it with a grain of salt.

• F(L,K) = 2L+5K
  
• P=3 w=3 r=10
  

We know this is a perfect substitute function so we're going to want all of either L or K. 

If we compare the MRTS to the price ratio which looks like 

MPl/MPk vs. w/r

2/5 > 3/10

So, we choose all L. But how much L?

We're not being asked to produce a certain fixed quantity, so in theory, demand would be infinite. Plugging F(infinity,zero) into the production function gets us Q=2(infinity) as our profit maximizing quantity. But what about the shutdown condition?

We shut down in the long run if price is less than average total cost in the long-run. (In the short run the condition is if price is less than average variable cost, but recall that in the long-run, all costs are variable. Besides, demand for capital is 0 anyway). Intuitively, this means that we want to be making money, or have a positive profit.

• F(L,K) = Q = 2L+5K
  
• ATC = (wL + rK)/Q
  
• K = 0, L = infinity, w=3, Q = 2(infinity)
• ATC = w(infinity)/Q
• = 3(infinity)/2(infinity) = 3/2 = ATC
• p = 3, ATC = 1.5
• p > ATC
• So the shutdown condition is not in effect here. 
• Let's also look at the first order condition pMPl = w
  
• P = 3, W= 3, MPl = 2
  
• 3(2) = 6 > 3
  
• We get $3 (6-3=3) of profit for every new unit of labor. We call this marginal profit. It's constantly positive so again, we don't shut down. 
  
• L = infinity, K = 0
• Demand for labor is the same in the short and long run, given that it is the only input since K is no good. 

Let's take a look at the perfect complements problem.

• F(L,K) = Q = 2 min (L,2K)
• This function has constant returns to scale. This is true of most if not all perfect complement functions, if I recall correctly. But take that with a grain of salt as well.
• P=3,w=1,r=3
• Let's set L and K equal to each other
• L = 2K
• K = L/2
• We can play around with the production function so it has the variables we need later.
• Q = 2 min (L,2K)
• Q = 2 min (L,L)
• Q = 2L
• Here's the profit function:
• profit = P*Q-wL-rK
• =P(2L)-wL-r(L/2)
• If we want to profit max for labor, we take the derivative of this function with respect to labor.
• d/dL = 2p-w-r/2
• We can't set this equal to zero and solve for L because L ends up canceling out, but what we can do is plug in our known values for p,w, and r. We get our marginal profit of labor from doing this (as done earlier), and if it's positive, then we don't shut down, rather, demand would be infinite instead.
• P=3,w=1,r=3
• d/dL = 2(3)-1-(3/2) = 5-(1.5) = 3.5 > 0, so we don't shut down.
• L = infinity, K = infinity/2