Microeconomics Practice Information & Externalities
Master Info Externalities: Solved Problems
Step-by-step solutions based on Varian’s approach.
Problem 1: Uncertainty and Risk Premium
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Question: A consumer has a utility function u(w) = w. Her initial wealth is $100. She faces
a lottery: with probability 50%, she loses $36; with probability 50%, she loses nothing.
1. Calculate the Expected Value (EV) of her wealth.
2. Calculate the Expected Utility (EU).
3. Calculate the Certainty Equivalent (CE).
4. Calculate the Risk Premium.
Solution
Step 1: Expected Value
Outcomes: w1 = 100 (no loss), w2 = 100 − 36 = 64 (loss).
EV = 0.5(100) + 0.5(64) = 50 + 32 = 82
Step 2: Expected Utility
√ √
EU = 0.5 100 + 0.5 64 = 0.5(10) + 0.5(8) = 5 + 4 = 9
Step 3: Certainty Equivalent (CE)
The CE is the wealth wce such that u(wce ) = EU .
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CE = 9 =⇒ CE = 92 = 81
Step 4: Risk Premium
The amount she is willing to pay to avoid the risk beyond the actuarially fair price.
RP = EV − CE = 82 − 81 = 1
Final Answer: EV=82, EU=9, CE=81, Risk Premium=1.
Problem 2: Externalities and Pigouvian Tax
Question: A steel plant produces steel (s) with cost Cs (s) = s2 . A nearby fishery faces a cost
function Cf (f, s) = f 2 + 2s. The market price of steel is ps = 12 and fish is pf = 10.
1. Find the competitive equilibrium output of steel (private optimum).
2. Find the socially optimal output of steel.
3. Determine the Pigouvian tax required to achieve efficiency.
1
Master Info Externalities: Solved Problems
Step-by-step solutions based on Varian’s approach.
Problem 1: Uncertainty and Risk Premium
√
Question: A consumer has a utility function u(w) = w. Her initial wealth is $100. She faces
a lottery: with probability 50%, she loses $36; with probability 50%, she loses nothing.
1. Calculate the Expected Value (EV) of her wealth.
2. Calculate the Expected Utility (EU).
3. Calculate the Certainty Equivalent (CE).
4. Calculate the Risk Premium.
Solution
Step 1: Expected Value
Outcomes: w1 = 100 (no loss), w2 = 100 − 36 = 64 (loss).
EV = 0.5(100) + 0.5(64) = 50 + 32 = 82
Step 2: Expected Utility
√ √
EU = 0.5 100 + 0.5 64 = 0.5(10) + 0.5(8) = 5 + 4 = 9
Step 3: Certainty Equivalent (CE)
The CE is the wealth wce such that u(wce ) = EU .
√
CE = 9 =⇒ CE = 92 = 81
Step 4: Risk Premium
The amount she is willing to pay to avoid the risk beyond the actuarially fair price.
RP = EV − CE = 82 − 81 = 1
Final Answer: EV=82, EU=9, CE=81, Risk Premium=1.
Problem 2: Externalities and Pigouvian Tax
Question: A steel plant produces steel (s) with cost Cs (s) = s2 . A nearby fishery faces a cost
function Cf (f, s) = f 2 + 2s. The market price of steel is ps = 12 and fish is pf = 10.
1. Find the competitive equilibrium output of steel (private optimum).
2. Find the socially optimal output of steel.
3. Determine the Pigouvian tax required to achieve efficiency.
1
