COMP90088 (2022) Tutorial Week 4 Solutions
Bitcoin difficulty
When we want one block to be mined on average every 600 s, the expected number of hashes beforeablockisfoundis1.65×1020s−1×600s=9.9×1022 =276.4.
On average, 275.5 = 5.3 × 1022 hashes need to be performed every 600 s, so the network hash rate is 275.5 H/600 s = 8.9 × 1019 H/s.
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According to the difficulty recomputation equation, the new difficulty is 274.02 × 14 d/12.5 d = 2.15 × 1022 = 274.18.
Bitcoin mining economics
The expected time until a block is mined on this rig is
276.3 H = 1.15 × 109 s = 437 months.
$420000 = $958/month. 437 months
The monthly electricity cost is
3.4 kW × $0.10/(kW h) × 24 h d−1 × 365 d/month = $247/month. 12
For difficulty d, solving
$420000 × 81 × 1012 H/s = $247/month d
gives d = 278.25 as the break-even difficulty. After M months:
1.055M = 278.25−76.3 gives M = 25.3 months of mining life.
Revenue divides by 1.055 every month, so the instantaneous monthly revenue is $958 × 1.055−t = $958e−t log(1.055), where time t is measured in months. Total mining profit is given by the integral of the difference with the electricity cost over 25.3 months:
($958e−t log(1.055) − $247)dt = $7.03 × 103.
Income $x at time t months in the future has a present value of $x×1.008−t = $xe−t log(1.008). Adding this decay factor to the integral gives a maximum acceptable purchase price of
e−t log(1.008)($958e−t log(1.055) − $247)dt = $6.63 × 103.
Evidently not a good investment, since the mining rig is overpriced at $10k even assuming peak block rewards as we have in our calculations. Changing block rewards (including minting reward, collected transaction fees, BTC-dollar exchange rate), change in cost of electricity, mining variance, potential pool fees, potential future legal regulations, and the moral cost of being a miner are some things to consider. You may wish to repeat the above calculations, where the block reward is $250000 and the difficulty is 276.6 as of March 2022.
81 × 1012 H/s Average monthly revenue is
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