I have posted this in the econonomics forum already, but I post it here too since it is more traffic here and my question is primarily a math question.
I am working through Rochet & Stole's chapter on multi-dimensional screening (2003), and I am struggling filling in the blanks between equation (2.1) p. 154, and its simplified form on page 155. In particular, my question concerns:
Consumers are of type $\theta$ $\in$ $\Theta$ = [$\underline\theta$, $\bar\theta$] with associated cdf $F(\theta)$, which is absolutely continous, and associated density function $f(\theta)$ = $F'(\theta)$. This is the distribution of types.
There also is a prefererence relation for each consumer for $q \in Q = [0,\bar q]$:
$u = v(q, \theta) - P$
with the single crossing property $v_{q\theta} > 0 $. I am adding this for context, but all information may not be necessary for my particular question.
This renders an indirect utility function, $u(\theta)$, defined by:
$u(\theta) = \max_{\rm {q\in Q}} \{v(q, \theta) - P(q)\}$
The monopoly firm is using a non-linear tariff, $P(q)$, and wants to maximize its expexted payoff (I am skipping some information not needed for my question on what renders this equation; this is equation (2.1) in text):
$E(\pi) = \int^\bar\theta_\underline\theta [S(q(\theta), \theta) - u(\theta)] dF(\theta)$
subject to:
$(1)$ $du/d\theta = v_\theta(q(\theta), \theta)$
$(2)$ $dq(\theta)/d\theta\geq0 $
$(3)$ $IR\space constraint$
Note that the function $S(q(\theta), \theta)$ is not important for my question, hence, I am not including its definition here.
Now to my question. The authors simplifies the objective function, using constraint (1). The following is the new simplfied function that they are maximizing:
$E(\pi) = \int^\bar\theta_\underline\theta [S(q(\theta), \theta) - \frac {1-F(\theta)}{f(\theta)}v_\theta(q(\theta), \theta) - u(\underline\theta)]dF(\theta)$
subject to:
$(2)$ $dq(\theta)/d\theta\geq0 $
$(3)$ $IR\space constraint$
Now, this to me is the same as saying:
$\int^\bar\theta_\underline\theta u(\theta) dF(\theta) = \int^\bar\theta_\underline\theta [\frac {1-F(\theta)}{f(\theta)}v_\theta(q(\theta), \theta) + u(\underline\theta)]dF(\theta) $
And here is where I need help to fill in the steps. I am unable to get from the original objective function to the updated one. As the authors explain, this is done by integation by parts, and using constraint (1) above. I get something like this:
$\int^\bar\theta_\underline\theta u(\theta) dF(\theta) = u(\theta)F(\theta)|^\bar\theta_\underline\theta - \int^\bar\theta_\underline\theta F(\theta)v_\theta(q(\theta),\theta)d\theta$
I can simplify this abit further, but it does not simplify to the authors' expression. Hence, can anybody help me fill in the blanks?
I hope that I have left sufficient information, otherwise, please let me know what I need to clarify. I also leave the citation for the source if you rather want to read through the set up there.
Rochet, J., & Stole, L. A. (2003). The economics of multidimensional screening. (pp. 150-197). Cambridge: Cambridge University Press