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Consider the following economy: A representative consumer allocates a fixed 24‐hour time endowment between labor and leisure. Leisure provides direct utility, while labor earns income used for consumption. A single firm hires labor to produce output, but production emits pollution, generating an external cost. A benevolent planner internalizes this cost when choosing labor. This exercise explores private versus social optima, the role of taxation, and comparative statics when preferences or technology change.  Consumer:

In the two firm environment described above,  after trading, firm 2 will 

If investors are risk‐neutral, what is the effective yield on the corporate bond they would expect, assuming they were only concerned about the risk of heatwaves?

In the two firm environment described above, after trading, firm 1 will 

At

Imposing

The difference in the planner’s FOC and the consumer’s FOC can be explained by:

Which algebraic expression is the consumer’s FOC, assuming no income taxes?

The equimarginal principle states that in order to get the greatest reductions in _______ emissions for a given _________ abatement cost, all firms should emit at a point that offers the same marginal abatement cost for each firm.

Now consider transition risk: a carbon‐tax has a 25% chance of raising costs by 5%.  What is the expected transition premium for a risk‐neutral investor?