Final Answer:
Marvin's compensating variation (CV) is $250, the change in consumer surplus (ΔCS) is $125, and the equivalent variation (EV) is $250.
Step-by-step explanation:
Marvin's utility function is U = q₁⁰.⁵ q₂⁰.⁵, and his income is $500. Initially, the prices are p₁ = $2 and p₂ = $1. The consumer maximizes utility subject to the budget constraint, given by p₁q₁ + p₂q₂ = Y. Solving for the initial quantities (q₁₀, q₂₀), we find q₁₀ = 125 and q₂₀ = 250.
When the price of good 1 increases from $2 to $4, the new budget constraint becomes 4q₁ + q₂ = 500. Solving for the new quantities (q₁₁, q₂₁), we find q₁₁ = 83.33 and q₂₁ = 166.67.
Now, we calculate the compensating variation (CV), which is the amount of money Marvin needs to receive to reach the initial level of utility at the new prices. CV = Y - p₁₁q₁₁₀ - p₂₁q₂₁₀, where p₁₁ = $4 and p₂₁ = $1. This gives CV = $250.
The change in consumer surplus (ΔCS) is the difference in the consumer surplus between the initial and new situations. ΔCS = 0.5(p₁₀ + p₂₀)(q₁₀ + q₂₀) - 0.5(p₁₁ + p₂₁)(q₁₁ + q₂₁), resulting in ΔCS = $125.
The equivalent variation (EV) is the amount of money Marvin would be willing to forgo at the initial prices to be as well off as he is at the new prices. EV = p₁₀q₁₁₀ + p₂₀q₂₁₀ - Y, leading to EV = $250.
In summary, the compensating variation is the additional income needed, the change in consumer surplus represents the welfare change, and the equivalent variation is the amount one is willing to pay to achieve the same utility level at the initial prices.