Question

Ben spends $100 per month on playing video games (G) at an arcade or going to movies (M). His utility function is U(G,M)=2ln(G)+M The price of G is $1 and the price of M is $10. Answer the following questions: Ben would like to choose the consumption of G and M so that he maximizes his utility. How many G and M would Ben like to consume? Label the x-axis as G for quantities of G and y-axis as M for quantities of M. Compute his maximum level of satisfaction. Draw the following in the same graph: i. an indifference curve that represents Ben's preferences, ii. his budget line and iii. his optimal consumption bundle. Derive Ben's demand function for M. Is M a normal good? What happens to the demand of M as price of M increases and income is held constant?

Answer 1

U = 2 ln(G) + M

(a) Budget line: 100 = G + 10M

Utility is maximized when MUG/MUM = PG/PM = $1 / $10 = 1/10

MUG = U/G = 2/G

MUM = U/M = 1

MUG/MUM = (2/G) / 1 = 2/G = 1/10

G = 20

Substituting in budget line,

100 = 20 + 10M

10M = 80

M = 8

(b) When G = 20 and M = 8.

U = 2 ln(20) + 8 = 2 x 3 + 8 = 6 + 8 = 14

(c) From budget line,

When G = 0, M = 100/10 = 10 (Vertical intercept) & when M = 0, G = 100 (Horizontal intercept).

In following graph, AB is the budget line and utility is maximized at point X where indifference curve IC0 is tangent to AB with optimal bundle being G0 (= 20) and M0 (= 8).

(d) Generalized budget line: I = G x PG + M x PM

Utility is maximized when MUG/MUM = PG/PM

2/G = PG/PM

G x PG = 2PM

Substituting in budget line,

I = 2PM + M x PM

M x PM = I - 2PM

M = (I - 2PM) / PM = (I/PM) - 2

When I increases, demand for M increases, so M is a normal good.

With I constant, when PM increases, (I/PM) decreases and so, demand for M decreases.