Problem 4: Utility Maximization
Utility Function: U = B•x•y
B = 40
x, y, Px, and Py are quantities and prices of product X and product Y
Px = 6
Py = 9
Disposable income of the consumer:
A = 200
Requirements:
a) Write the budget constraint function for the consumer.
b) Find x and y to maximize the consumer’s utility. Compute this utility.
c) Suppose the price for X falls by: 2
Also, the consumer’s income falls by: 150
Compute the new maximized utility.
SOLUTION
a)
Budget constraint: x*Px + y*Py = A
=> 6x + 9y = 200
b)
U = maxU <=> MUx/Px = MUy/Py
<=> x*Px = y*Py
<=> x = A/(2*Px) = 16.6667
y = A/(2*Py) = 11.1111
U = Bxy = 7407.41
c)
∆Px = -2 => Px = 6 + ∆Px = 4
∆A = -150 => A = 200 + ∆A = 50
U = maxU:
<=> x = A/(2*Px) = 6.25
y = A/(2*Py) = 2.7778
B = 40
U = Bxy = 694.44
Problem 3: Price Control, Tax, Deadweight loss, and Total surplus
Suppose a product has the following market information:
(D): Qd = a1 – a2•P
(S): Qs = b1 + b2•P
a1 = 100
a2 = 3
b1 = -5
b2 = 6
Requirements:
a) Compute the current equilibrium price and quantity
b) Suppose the government imposes a ceiling price:
Pceiling = 5.5
b1) The quantity of shortage.
b2) Compute deadweight loss.
b3) Compute comsumer surplus, producer surplus, and total surplus.
c) Suppose the government imposes a floor price:
Pfloor = 13
c1) The quantity of surplus.
c2) Compute deadweight loss.
c3) Compute comsumer surplus, producer surplus, and total surplus.
d) Suppose the government imposes a tax size on the product.
T = 8.5
d1) Compute the price paid by the buyers Pd and the price received by the sellers Ps.
d2) Compute the tax revenue and the deadweight loss
d3) Compute comsumer surplus and producer surplus, and total surplus.
e) Compute the tax size that maximize tax revenue. Compute the maxed tax revenue.
SOLUTION
a)
Market equilibrium <=> Qd = Qs
<=> P = (a1–b1)/(a2+b2) = 11.6667
Q = b1+b2*P = 65
b)
b1)
Pceiling = 5.5
Qd = a1–a2*Pceiling = 83.5
Qs = b1+b2*Pceiling = 28
Qshortage = Qd–Qs = 55.5
b2)
Q = 65
Qcontrol = min(Qd,Qs) = 28
Pd = (a1–Qcontrol)/a2 = 24
Ps = (Qcontrol–b1)/b2 = 5.5
DWL = 0.5*(Pd–Ps)*(Q–Qcontrol) = 342.25
b3)
Pmax = a1/a2 = 33.3333
