The framework of consumer theory we developed in class is general, but also quite restricted in the sense that there is still a distance between the framework and real economy. For example, for luxury goods we usually observe that higher price sometimes leads to higher demand in the data. This is because price in this case is not only the cost a consumer pays to consume the product, it is also a signal of \social status" associated with the product|so consumers derive satisfaction directly from the higher price.

Could you develop a model of consumer demand for luxury products considering the aforementioned signaling role played by price|by extending the basic model we dis-cussed in class? In particular, consider a consumer with income m consumes two goods: luxury good x1 with price p1, and other good x2 with price p2.

(a) Write down an appropriate objective function (utility function) for luxury goods. Explain why it should take the form you proposed.

(b) Write down the budget constraint and consumer optimization problem in this case.

(c) Given the consumer optimization above, do you think that the (Marshallian) demand function is still homogeneous of degree zero in price and income (p1; p2;m)? Explain why.