Question

On a given day Adam enjoys the consumption of beer (B) and wine (W) according to the function:

U(B,W)=20B-B^2+18W-3W^2

a) How many glasses of beer and glasses of wine does he consume during a day? How much utility does he get from this consumption? (Cost is no object to Adam)

b) Lately, however, Adam has been advised by his doctor to limit the sum of glasses of wine and glasses of beer consumed to 5. How many glasses of wine and glasses of beer will he consume under these circumstances? How much utility will he get?

Answer 1

la) Gulven U = 203 – B2 + 18 W - 3 W2 -Adam will choose the level of B and W such that his utility is manimised. The first order conditions for utility mascimisation are, du dB. =0 => 20-23 = 0 2B = 20 B = 20 2 =>B_= 10 and, du dw = 18-ow=0 6w=18 WE 18 6 W=3 the given Now put B=10 and W=3 in Whility function U=20B-B² +180 -3w² U = 20x10 - (10)?+ 18x3 - 3x630)3 U=200-100L54 -27 > U - 127

10 glasses of beer his . So Adam cuill consume and 3 glasses of wine to manimise utility and the value of utility from consumption is 127.

(b) Adam's objective is to maximise his utility, U = 2013-B2+18W-3W2 subject to constraint, B+W = 5 - B=5-W - Put (is in utility function, U=20B-B² + 18W-3w2 U = 2015-w) - (5-6) ² + 18W-34² > U = 100- 20W -(25+WPLOW)+18W-3 W2 => U = 100-20w-25-W²+low +18W-3W² => U = 75 +8W - 4W?

Let B* and wt be the number of glasses of beer and wine that will maximise Adam's utility, so first order condition for. Whility másimiscelionis, du dwt =0 3) d (75 + 8W- 4W?). -dwt ² 0 + 8-8W* = 0 8.W* 8 => wt = dolor To W =1

#tw + + y =4 Put wt = 1 in the constraint, B = 5 B* = 5-Wt ² B = 5-1 BT Now put B = 4 and w+ = 1 in function, ů - 208-02 + 18 W*- 3 w*? u= 20x4-(4)?+ 18x1 – 3 x(1) > 4 - 80 - 16+ 18-3 -> 79 in utility

So once Adam has been aduised by his doctor to limit the sum of glasses of wine. and glasses of beer consumed to 5 he consume I glass of wine wal), Land glasses of_beer [3*=4") to maximise his Whility is 79 (Ur. 79) utility and the value of manimum