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In period 1 the consumer of type $\theta$ selects an option contract consisting of an up-front fee, $B>0$, and exercise price, $\bar{R}$. The consumer pays $B$ at the end of the first period. In period 2, he realises his valuation, $\theta$, distributed on $[0,1]$ by CDF $G(\theta)$ with density $g>0$. His expected payoff, from choosing contract $(B, \bar{R})$, should thus be $-B+\int_{\bar{R}}^{1}(\theta-\bar{R})g(\theta)d(\theta).$ However, in the paper I am told that it is instead: $-B+\int_{\bar{R}}^{1}(1-G(\theta))d\theta.$ Are these two expressions equivalent? What does it mean to integrate a CDF in this fashion? Thank you.

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When integrals look different than what pops into your head, often the reason is integration by parts. For your example note that $$\int_R^1 (\theta -R) g(\theta) d \theta + \int_R^1 G(\theta) d \theta = (1-R) - 0,$$ where the right-hand side is equivalent to $\int^1_R 1 d\theta$. Hence, the two expressions you consider are equivalent.

It's of the form $$\int u(x) v'(x) dx + \int u'(x) v(x) dx = \int u(x) v(x) dx.$$

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