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Answer to Question #48586 in Microeconomics for Joe Shields

Question #48586
Emily's utility from watching a movie, M, and going clubbing, C, is given by the following formula U(M,C) = 0.2 x logM + 0.8 x logC

A. Find the marginal utility of each good and the marginal rate of substitution between them.
B. Let Y be Emily's budget and let Pm and Pc denote the prices of M and C respectively. Using the Lagrangean solve Emily's utility maximisation problem and write down the general form of the demand equations for movies and clubbing.
C. Find the income elasticity, the own price elasticity and cross - price elasticity of the demand for movies
D. Suppose that Emily has an entertainment budget of Y = £240. If the price of a movie night is £16 and the cost of a clubbing night is £48, identify the equilibrium values of number of times she goes to the movies and clubbing. Does it make economic sense that Emily chooses more of the expensive rather than cheap entertainment?
E. Assume that Emily's entertainment budget doubles. Find the new equilibrium values of M and C.
Expert's answer
A. Find the marginal utility of each good and the marginal rate of substitution between them.
MU(M) = U'(M) = 0.2/(M*ln10) = 0.087/M
MU(C) = U'(C) = 0.8/(C*ln10) = 0.347/C
B. Let Y be Emily's budget and let Pm and Pc denote the prices of M and C respectively. Using the Lagrangean solve Emily's utility maximisation problem and write down the general form of the demand equations for movies and clubbing.
Utility is maximized, when MUm/Pm/MUc/Pc
Y = Pm*M + Pc*C
0.087/(M*Pm) = 0.347/(C*Pc)
Pm*M = 0.25Pc*C
Y = 0.25Pc*C + Pc*C = 1.25Pc*C
C = 0.8Y/Pc
Pm*M = Y - Pc*0.8Y/Pc = 0.2Y
M = 0.2Y/Pm
C. Y = £240, Pm = £16, Pc = £48.
The equilibrium values of number of times she goes to the movies and clubbing will be:
M = 0.2*240/16 = 3 times
C = 0.8*240/48 = 4 times
If Emily chooses more of the expensive rather than cheap entertainment, than utility of consuming the more expensive is higher, than more cheaper. 
D. Y = 480. The new equilibrium values of M and C will be:
M = 0.2*480/16 = 6
C = 0.8*480/48 = 8

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