Xiaoxi likes the Porch\'s dollar coffees. But he also really enjoys bubble tea f

Question

Xiaoxi likes the Porch's dollar coffees. But he also really enjoys bubble tea from Rose Tea Cafe. Assume his utility function is u(x_1, x_2) = 48x_1^0.5 + x_2. where x_1 is bubble tea and x_2 is Porch coffee. Suppose his income is 217 (it's rough living off a grad student stipend), bubble tea costs p_1 = 3, and dollar Porch coffee cost of course one dollar (p_2 = 1). At these prices, how many bubble teas will Xiaoxi demand? [You may either use the tangency condition or do an optimization problem.] At these prices, how many Porch dollar coffees will he demand?

Explanation / Answer

U = 48x1^.5 +   x2                        ------------------------ (1)

X1 = bubble tea

X2 = Porch coffee

Differentiation of eq. 1 w.r.t. x1 will give MUx1

MUx1 = dU/dx1 = 48*.5/( x1^.5)    = 24/( x1^.5)   

Differentiation of eq. 1 w.r.t. x2 will give MUx2

MUx2 = dU/dx2 = 1

Also,

MUx1 / MUx2 = Px1 / Px2

24/( x1^.5) /1 = 3/1

24/x1^.5 = 3

Or

X1 = 8^.5 = 2.828 or 3

217 = P1*x1 + P2*x2

217 = 3*2.828   + 1*X2

X2 = 217/3*2.828 = 25.57

a.

Bubble tea demanded = 2.828 units or 3 units

b.

Porch dollar coffee demanded = 25.57 units or 25 units

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