A study of music consumption in UK showed that in 2011, a typical 18 to 35-year

Question

A study of music consumption in UK showed that in 2011, a typical 18 to 35-year old, Jackie, spent 80% of music budget on digital music (X), and 20% on CDs (Y); Assume his annual music budget was £40, and the price of digital music was £1, and the price of CD was £8. 1) what is the slope f his indifterence urves? Does interior solution exist? wny? de points) 2) Use Lagrangian method to solve for the optimal consumption of digital music and CDs that make him most satisfied; Show your work. (8 points) MRSxy at the optimal solution, does MRSxy - Px/Py? (2)

Explanation / Answer

1).

Consider the given problem here “a typical 18-35 years old” “Jackie” spent “80%” of music budget on digital music (X) and “20%” on CDs(Y).

So, here the given the information the utility function is given by, “U = X^0.8*Y^0.2”.

So, by total differentiation we have.

=> U = X^0.8*Y^0.2, => dU = 0.8*X^(0.8-1)*Y^0.2*dX + X^0.8*0.2*Y^(0.2-1)*dY.

=> 0.8*X^(-0.2)*Y^0.2*dX + X^0.8*0.2*Y^(-0.8)*dY = 0.

=> 0.8*(Y/X)^0.2*dX + 0.2*(X/Y)^0.8*dY = 0, => 0.8*(Y/X)^0.2*dX = (-0.2)*(X/Y)^0.8*dY.

=> (-0.8)*(Y/X)^0.2 / 0.2*(X/Y)^0.8 = (dY/dX), => (-4)*(Y/X) = (dY/dX), => dY/dX = (-4)*(Y/X).

So, here the slope of the “Indifference Curve” is “dY/dX = (-4)*(Y/X)”.

So, here the utility function is negatively sloped and convex to the origin, => here the interior solution is possible, => at the optimum “X, Y > 0 “.

2).

So, here the price of “X” is “Px=1” and the price of “Y” is given by “Py=8” and the level of budget is given by, “m=40”. So, the lagrangian function is given by.

=> L = X^0.8*Y^0.2 + c[m – Px*X – Py*Y], where “c” be the lagrange multiplier and “m” be the level of budget.

=> the FOC for maximization are given by, “dU/dX = dU/dY = 0”.

=> dU/dX = 0.8*X^(-0.2)*Y^0.2 + c[ – Px] = 0, => 0.8*X^(-0.2)*Y^0.2 = c* Px…………...(1).

Similarly, dU/dY = 0, => 0.2*X^0.8*Y^(-0.8) = c*Py…………...(2).

So, by “1” divided by “2” we have.

=> [0.8*X^(-0.2)*Y^0.2]/[ 0.2*X^0.8*Y^(-0.8)] = Px/Py,

=> 4*(Y/X) = Px/Py, => Y = Px*X/4*Py ……………(3). Now, by putting this relation into the budget line we get the optimum solution for “X”.

=> Px*X + Py*Y = m, => Px*X + Py*[Px*X/4*Py] = m, => Px*X + Px*X/4 = m.

=> [1 + 1/4]*Px*X = m, => [5/4]*Px*X = m, => X = 4m/5Px = 4*40/5*1 = 32, => X=32.

Now, by substituting “X” into “3” we get, “Y = m/5Py = 40/5*8 = 1”, => Y=1.

3).

So, here the utility function is given by, “U = X^0.8*Y^0.2” and the “MRS” is the ratio of “MUx” to “MUy”.

=> MUx = 0.8*X^(-0.2)*Y^0.2 and MUy = 0.2*X^0.8*Y^(-0.8), => MRS = MUx/MUy.

=> MRS = [0.8*X^(-0.2)*Y^0.2] / [0.2*X^0.8*Y^(-0.8)] = 4*(Y/X), => MRS = 4*Y/X.

So, here we can see that the utility function is negatively sloped and “convex” to the origin, => there will interior solution and at the optimum “MRS=Px/Py”.

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