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Unformatted text preview: Labour which produces income that, in turn, allows the consumer to purchase consumption (this gives the consumer utility). Leisure This produces utility for the consumer at the opportunity cost of foregone work (and thus, foregone consumption). For our model, we will consider 2 goods consumption, c, and leisure, , with consumer tastes represented by a utility function U = U(c , ) 57 Now, we have a utility function with the consumer's preferences over consumption and leisure defined as U = U(c , ). But what is the constraint in this case? The consumer can purchase consumption at a price, P, and receives a given wage, w, for each unit of labour they supply to the labour market. Thus, the consumer's consumption is constrained by their wage income... Pc = w (Tbar ) Pc + w = w Tbar So generally, Max U(c , ) + (w Tbar Pc w) c, MRS = MU MUc FOC FOCc U () w = 0 Uc () P = 0 (1) (2) Now we isolate for (1) and (2) and equate these expressions (3) = (4): U () = w Uc () = P U () = Uc () w P U () = __w__ Uc () P So, the MRS is the real wage! Okay, now that we've established that we can take an example of how to find the labour supply function for a consumer. (3) (4) 58 Suppose a consumer has preferences over consumption and leisure that are represented by the following utility function: U(c , ) = ln c + ln Is this a Cobb-Douglas form of the utility function? Yes! Why? ln c + ln is equivalent to c1 1/2 So, our constraint is still income = cost of consumption... Pc = w (Tbar ) Pc + w = w Tbar and we form the Lagrangian, Max ln c + ln + (w Tbar Pc w) c, MRS = MU MUc FOC FOCc __1__ w = 0 2 __1__ P = 0 c (1) (2) Now we isolate for (1) and (2) and equate these expressions (3) = (4): __1__ = 2w __1__ = Pc __1__ = __1__ 2w Pc __c__ = __w__ 2 P 59 (3) (4) So, we can now find the demand functions for consumption and leisure in terms of P, w, and Tbar. c = __2w__ P (5) Now, we sub (5) into our resource constraint to get our demand function for leisure. P 2w + w = wTbar P 3w = wTbar * = Tbar 3 (6) Now, we can sub * into (5) to get the consumer's demand function for consumption. c* = __2w Tbar __ 3P (7) If this is truly a Cobb-Douglas form of the utility function we should be able to find proportions of income "spent" on lei...
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- Spring '10