This paper addresses two economics problems applied to an oil company operating in Alaska. The first question analyzes how four different tax structures — a flat tax, a per-barrel tax, a proportional income tax, and a progressive income tax — affect the company's extraction decisions and profitability. The second question works through three methods of revenue maximization for a combined market and two discriminating sub-markets, comparing the resulting optimal prices, quantities, and elasticities. The paper concludes by explaining why each method yields a different maximum revenue figure depending on the underlying assumptions used.
A flat tax, by definition, does not vary with the quantity of oil extracted. Therefore, it has no direct effect on the volume of oil that the company extracts in Alaska. However, given that the company likely wishes to maintain the profitability levels it achieved in previous years, the flat tax may indirectly induce an increase in operations and in the quantity of oil extracted, as the company seeks to offset the fixed tax burden through higher output.
Understanding how different tax structures create different incentive effects is a central concern in applied microeconomics, particularly for capital-intensive industries such as oil extraction where output decisions are made over long planning horizons.
A $2 per barrel tax imposed by the state means that the company must pay additional money for every barrel of oil it extracts. The key difference between this per-barrel tax and the flat tax discussed above is that the total tax liability increases directly with output: the more the company extracts, the more it pays. Although per-unit taxes of this kind are generally designed to reduce the level of business activity and decrease the quantity extracted, the effect may differ if the company determines that the revenue gained from additional barrels substantially outweighs the tax cost incurred.
If the state instead levies a proportional income tax, then the higher the company's income, the higher its total tax liability — though the rate itself remains constant. The effect is somewhat similar to the per-barrel tax described above: the company may decide to extract less in order to report a lower income and, consequently, face a lower tax bill.
A progressive tax, by contrast, is a tax that "takes a larger percentage from the income of high-income people than it does from low-income people."[1] Applied to corporations, this would mean that companies with higher incomes are taxed at successively higher rates. In the present scenario, however, we are not dealing with progressive taxation.
Progressive taxation in this context would mean, for example, that the state levied 10% of annual income on earnings up to $1,000,000 and 15% on earnings up to $2,000,000. The likely behavioral effect of such a strategy is that the company would attempt to position its income near the upper boundary of the lower tax bracket. In this example, the optimal income level for the company would be approximately $900,000–$950,000, just below the threshold that triggers the higher rate, and extraction volumes would be adjusted accordingly.
Because the tax in part (d) is applied to overall income rather than to Alaskan output specifically, the effect of a proportional income tax would be the same as described above — the income declared in Alaska would simply form part of the company's consolidated income. However, if a per-barrel tax of the type described in part (b) were levied, the company would likely redirect its extraction activity to the other states in which it operates, in order to avoid the per-barrel charge applied in Alaska.
To evaluate the optimal price and quantity for the combined market without price discrimination, we set PA = PB and define QT = QA + QB.
This gives: QT = 66,000 − 10P, and therefore P = 6,600 − Q/10.
The profit-maximizing condition requires that marginal cost equals marginal revenue. Marginal revenue is the derivative of the total revenue function VT = P × Q:
VT = 6,600Q − Q²/10
Differentiating: dVT/dQ = 6,600 − Q/5 = 3,000 (since marginal revenue = 3,000)
Solving: Q = 16,500 and P = 4,950.
The price elasticity of demand for the combined market is derived from the formula dQT/dP = −E × Q/P. Since dQT/dP = −10, we obtain E = 3.
Total revenue: V = P × Q = 81,675,000.
To find the price that maximizes revenue in each sub-market under price discrimination, we express each revenue function as P × Q and differentiate with respect to P, then set the derivative equal to zero.
Market A:
VA = PA × QA = 24,000PA − 2PA²
dVA/dPA = 24,000 − 4PA = 0 ⟹ PA = 6,000; QA = 12,000
Market B:
VB = PB × QB = 42,000PB − 8PB²
dVB/dPB = 42,000 − 16PB = 0 ⟹ PB = 2,625; QB = 21,000
Elasticity is calculated using the same approach as above. The result is Ec = 4 for Market A and Ec = 64 for Market B.
Total revenue under price discrimination: PA × QA + PB × QB = 103,500,000.
In this case, we use the quantity function Q = 66,000 − 10P and the total revenue function VT = 66,000P − 10P². In order to calculate the maximum of this function, we differentiate it and set it equal to zero.
"Direct optimization of combined revenue function"
"Why three methods yield different revenue values"
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