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Ramsey Taxation Problem

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Suppose there are two medical goods. Heart surgery and plastic (cosmetic) surgery. The market price for the heart surgery is $100k, the market price for plastic surgery is $10k. The price elasticity of demand for heart surgery is -0.2. The price elasticity of demand for plastic surgery is -3.0. The government is considering a tax on the surgeries. The tax on heart surgery is denoted th, and the tax on plastic surgery is denoted tp.

a) Why did I choose to make the demand elasticity for plastic surgery large relative to heart surgery?

b) What is the equation for the optimal (Ramsey) value of th in terms of tp.

c) In a clearly written paragraph, comment on the relative size of th compared to tp, and why we are seeing this result.

d) Suppose the government has selected tax levels th and tp using the Ramsey rule. Furthermore, at those taxes, the market sells 300 thousand heart surgeries and 1.5 million plastic surgeries. Finally, the government collects $1 billion in revenue from these taxes. What are the values of th and tp?

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Key Assumptions:

The supply is completely elastic, where a supplier will provide an infinite amount of goods/service at any price point.

The supply market is monopolistic in nature

a) Why did I choose to make the demand elasticity for plastic surgery large relative to heart surgery?
Demand Elasticity can be interpreted as the percentage change in the demand in response to a percentage change in the price.
Heart surgeries are a necessary service for which people would continue to buy in the face of an increase in the prices (due to increase in the supply price or due to taxation). Plastic surgeries are cosmetic surgeries that are not essential commodities and hence people wouldn't continue to buy these services ...

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