Question

Happiness can be produced with wine and roses according to Q = W ^1/2R ^1/4, where W is bottles of wine and R is bouquets of roses obtained per month. If wine costs $20 per bottle and roses cost $60 per dozen, the happiness-maximizing combination of wine and roses costing $360 in total is:

a. W = 12 bottles, R = 2 bouquets

b. W = 9 bottles, R = 3 bouquets

c. W = 15 bottles, R = 1 bouquets

d. W = 18 bottles, R = 0 bouquets

e. W = 6 bottles, R = 4 bouquets

Can you please share step-bu-step on how to do this problem? Please share forumals needed as well. Thanks!

Answer 1

a. W = 12 bottles, R = 2 bouquets

Explanation:
Q = W ^1/2R ^1/4

The happiness - maximizing combination of wine and roses is that where MRS = price ratio
MRS = MUw/MUr =

Price ratio = Price of wine, Pw/Price of roses, Pr = 20/60 = 1/3

So, MRS = Price ratio gives,
2R/W = 1/3
So, W = 3*(2R) = 6R

Budget constraint: Pw*W + Pr*R = Income
20*(6R) + 60R = 360
So, 120R + 60R = 180R = 360
So, R = 360/180 = 2
W = 6R = 6*2 = 12