This preview shows page 1. Sign up to view the full content.
Unformatted text preview: ) + w (18  L) FOCL 2 PC L1/2  w = 0 L1/2 = w 2 PC L1/2 = 2 PC w L = 4 PC2 w2 Now, profit is maximized at the point where total revenue is equal to total cost (i.e. marginal profit is zero). PC C = w PC C = w (18 L) Let w = 1 (Labour = numeraire) PC (4 L1/2) = 18 4 PC2 4PC (2PC) = 18 4 PC2 8PC2 + 4 PC2 = 18 12PC2 = 18 PC2 = 3 / 2 PC = 3 / 2 Now, remember that we had L* = 4 PC2 = 4(3 / 2)2 = 6 w2 (1)2 C = 4L C* = 46 = (18 L) * = (18 6) = 12 How do we know that the Consumption / Leisure choice is, in fact, the one which maximized Robinson's utility? 151 Let's see how! Recall, Robinson's utility was defined as... URC = URC (C , ) = C1/2(18  L)1/2 C = f(L) = 4 L1/2 1. First, we figure out the supposed maximum utility level based on the optimal quantities of leisure and consumption that we determined above. We had... C* = 46 * = (18 6) = 12 URC* = (46)1/2(12)1/2 URC* = (3.13016916)(3.464101615) URC* = 10.84322404 2. Next, we choose a value for L that is slightly higher than the supposed optimum, say 6.1, (and calculate the associated C and ). We then use this point on the production function to see if utility is higher or lower. When L = 6.1 we get... C* = 46.1 and using this point... URC1 = (46.1)1/2(11.9)1/2 URC1 = (3.1431308)(3.449637662) URC1 = 10.84266238 which is less than URC* = 10.84322404. 3. Next, we choose a value for L that is slightly lower than the supposed optimum, say 5.9, (and calculate the associated C and ). We then use this point on the production function to see if utility is higher or lower. When L = 5.9 we get... C* = 45.9 and using this point... 152 * = (18 5.9) = 12.1 * = (18 6.1) = 11.9 URC2 = (45.9)1/2(12.1)1/2 URC2 = (3.117044472)(3.478505426) URC2 = 10.84265611 which is less than URC* = 10.84322404. Now let's do one that is even more challenging. Suppose Robinson Crusoe has a utility function and a production function defined as: URC = URC (C , ) = C1/2(16  L)1/2 C = f(L) = 8 L1/2 Now remember Robinson wants to maximize his leisure subject to his maximum profit. First, l...
View
Full
Document
This note was uploaded on 05/25/2010 for the course ECON 301 taught by Professor Sning during the Spring '10 term at University of Warsaw.
 Spring '10
 sning
 Economics

Click to edit the document details