Ed's Utility from Vacations (V)and Meals (M)is Given by the Function

Question

Examlex · Accepted Answer

The Answer is a.Ed's optimization problem is
Max V²M
Subject to p_MM + 200V = 1500
where p_M is the price of meals.Using the Lagrangian,we derive the demand for meals:
M* = 500/p_M
The change in consumer surplus is found from the integral:
∆CS = ∫₅₀⁷⁵ 500/p_Mdp_M = -500 ln(p_M)|₅₀⁷⁵ = -500 [ln(75)- ln(50)] = -202.7
So the change in consumer surplus is -$202.7.
b.The CV is the amount of money needed to offset a consumer's harm from a price increase.Ed's utility before the price change is based on his optimal consumption bundle.
M₁ = 500/50 = 10
and V₁ = 1000/200 = 5.His utility is U(5,10)= 5²10 = 250.Now we look at the expenditure function when p_M = 75.The Lagrangian is:
L = 75M + 200V + [250 - V²M]
The optimization conditions are:
L_M = 75 - V² = 0
L_V = 200 - 2MV = 0
L = 250 - V²M = 0
The first two conditions yield
3/4 = V/M
So V = 3M/4.Plug into the utility function
250 = (3M/4)²M
Solving for M = 7.63.Solve for V = 5.72.The expenditure required to purchase this bundle is:
75M + 200V = 75(7.63)+ 200 (5.72)= 1717.25
Thus the CV is $1,500 - $1,717.25 = $217.25.
c.The EV is the amount of money Ed will pay to prevent the price increase.To find this,we start by finding his utility after the price change with an income of $1500.From above,V = 5 and M = 500/75 = 6.67.His utility from this bundle is U = (5)²(6.67)= 166.75.The Lagrangian to find the expenditure function is:
L = 50M + 200V + [166.75 - V²M]
The optimization conditions are:
L_M = 50 - V² = 0
L_V = 200 - 2MV = 0
L = 166.75 - V²M = 0
Solve for the optimal bundle:
M = 8.74,V = 4.37.
The expenditure is:
50(8.74)+ 200(4.37)= 1311.
Ed would pay up to 1500 - 1311 = $189 to avoid the price change.This is the EV.

Ed's Utility from Vacations (V)and Meals (M)is Given by the Function

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