Question: Cars equipped with Volkswagen's new "clean diesel" TDI engine are in high demand both in Europe and the United States. This has forced \(\mathrm{VW}\) to consider the optimal way to allocate a limited number of engines between cars sold in the two markets. For purposes of this analysis assume that all money amounts are in dollars, that \(\mathrm{VW}\) is concerned with maximizing profits and

- That the fixed costs of the engine plant are 2

- That the marginal costs are 5 per engine

- The marginal cost of producing a car net of the engine is 10 for cars sold in either market (thus the marginal cost of the car is the cost of the car plus the cost of the engine).

- The demand for cars in each market is described by the inverse demand function

\(P_{u s}=16-0.5 Q_{u s}\) and \(P_{o u}=18-Q_{e u}\)

1. If VW has the capacity to produce 3 units of engines per period, determine the optimal number of vehicles they should sell in each period and the price they should charge. (Note: a "unit" of engines might be more than a single engine-for example a unit could be 1000 engines per week. Thus your answer need not be an integer.)
   \(\mathrm{P}_{\text {us }}=\underline{\mathrm{Q}_{\mathrm{us}}}=\)
   \(\mathrm{P}_{\mathrm{eu}}=\)
   \(\mathrm{Q}_{\mathrm{ou}}=\)
2. If $V W$ is can produce a maximum of 2 engine units per period, will this change your answer in part a? If so how? (That is, give the price and quantities you think will be optimal.)
   \[\mathbf{P}_{\mathrm{us}}=\mathrm{Q}_{\mathrm{us}}=\mathrm{Q}_{\mathrm{eu}}=\]
3. If the fixed costs increase from 2.0 to 2.1, will your answer in part a or part b change? If so, how? If not, why not?

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