# Multidimensional product differentiation and density functions

I’m having trouble with a model (link) I’m working on for a review.

The author uses a two firms hotelling model with a multidimensional product space (x,y)with orthogonal characteristics, which are uniformly distributed over the unit square. Preferences are standard:

$$U_i = V - p_i -t |x-x_i| - t |y-y_i|$$

I understand the mechanics of the model (find indifferent consumer, derive demand, find profits, best response, and eq.), but I have a problem with the math behind some profit functions and consumer surplus functions. In particular, the integrals over the densities and cdf really bug me.

In the proof (eq. 31) the author finds the demand in the case when $$p_1 > p_2$$ through an integral like the following:

$$D_1 = \int_0^\hat{x} F(y \leq \tilde y(x)) f(x) dx$$

$$\hat{x}$$ is the intercept of the locus of indifferent consumers. $$\tilde{y}$$ is firm 1’s marginal consumer. F() should be the CDF and f(x) the marginal density. My understanding is that $$F(y \leq \tilde y(x)) = \int_0^{\tilde{y}} f(y) dy = y ]_0^\tilde{y}$$

Is this correct?

Then, I don’t understand what $$f(x)$$ is... given that x is the standard uniform, shouldn’t this be just 1 that after integration would become $$x]_0^\hat{x}$$?

Sorry, it’s very late and my brain is out of energy. I have to ask since I am very far from my comfort zone... hoping that you could help a bit.

Probably, the author probably is assuming $$x_i$$ and $$y_i$$ are independent in addition to the stated assumption on pg. 7 that types "are orthogonal" (which merely means uncorrelated.) (However, since they are also uniformly distributed, we have, $$f_{X,Y}(x,y)=f_X(x)f_Y(y)\equiv 1, \forall x,y$$, meaning they are independent.)
Assuming independence, $$\iint_{\{(x,y):0\le y\le \tilde y(x),0\le x\le 1\}} f_{X,Y}(x,y)\,dydx = \int_0^1 f_X(x)\left[\int_0^{\tilde y(x)} f_Y(y)\,dy\right]dx.$$