Answer

1. Efficient allocation without exploration:
   
   Q = 200
   P = 1000 - 2*Q
   P = 1000 - 2*200 = $600
   R = P - MEC = $600 - $300 = $300
   
   P = $600, R = $300, Q = 200 
2. Minimum price to induce exploration:
   
   EV of drilling = 0.2*( 10*(P-$300) - $400 ) + 0.8*( - $400 )
   EV = 2*P - $600 - $80  - $320
   EV = 2*P - $1000
   Minimum price P* occurs where EV = 0
   
   P* = $500
3. New equilibrium:
   
   P = 1000 - 2*Q
   500 = 1000 - 2*Q
   Q = 250
   R = $500 - $300 = $200
   
   P = $500, R = $200, Q consumed = 250 
4. New units and wells drilled:
   
   New units found: 250 - 200 = 50
   
   Expected units for N wells: 0.2*10*N
   Wells for 50 expected units: 0.2*10*N = 50
   
   N = 50/2 = 25
5. Costs, revenues and profits on exploration:
   
   Exploration costs: 25 wells * $400 per well = $10k
   Extraction costs on new units: 50 units * $300 = $15k
   Total costs: $10k + $15k = $25k
   Revenue on new units: $500 * 50 units = $25k 
   Economic profits on exploration: $25k - $25k = $0 
6.  Changes in social surplus:
   
   Change in CS = $100*(200+250)/2 = $22,500 gain
   Change in PS on original units: -$100*200 = -$20k
   PS on newly-found units (from above) = $0
   Overall change in SS = +$22,500 - $20,000 + $0 = $2500 gain