Ralph’s Bundled Budgets
Ralph is dealing with a subordinate (“sub”) who likes to “pad the budget.” The sub brings
projects to Ralph, who decides whether they should be funded. The complication is Ralph is
informationally disadvantaged relative to the sub in the following sense. Ralph believes the
probability the project’s cost will be 90 is .5, and the probability it will be 60 is .5. The
project will return a certain revenue which is denoted by R (explained below). Ralph is
busy, so he is not sure whether he wants the sub to report on the details of each project. To
simplify, assume the sub will run across two projects. Ralph is considering one of two
options. First, the sub could propose an individual budget request for each project and
Ralph would treat them independently (“independent evaluation”). This will require that
the sub provide details. Second, the sub could propose a single budget for the two projects
combined and Ralph than would take both projects or neither project (notice this is
different than the assumption in SSY 2013). Assume Ralph and his sub are risk neutral.
Finally, initially assume the project costs are independent, so the probability one project’s
cost is high or low is independent of the cost of other project.
1. Suppose Ralph treats the projects independently, so called “individual evaluation”.
Ralph will offer a (take-it-or-leave-it) hurdle contract that provides the sub with y if
(and only if) the sub submits a budget request that is less than or equal to y, and
otherwise will pay the sub nothing and so the project is rejected. The sub, in turn, makes
his request with full understanding that Ralph cannot renege on the contract. Find
Ralph’s optimal hurdle y* and Ralph’s and the sub’s expected utility for each of the
following values of R: 95, 100, 110, 110, 130, 140, 155. Remember there are two
projects and they are being treated independently, so Ralph’s decision concerning one
project is independent of what happens on the other.
2. Now suppose Ralph will weigh the two projects together before deciding which to fund.
First, to simplify, suppose Ralph asks the sub to make a single budget request that will
cover both projects combined. That is, Ralph chooses a hurdle y such that both projects
are accepted and Ralph provides funds equal to y if the sub makes a request that is less
than or equal to y and otherwise Ralph rejects both projects. You might think of this as
“aggregate evaluation”. Find Ralph’s optimal hurdle and Ralph’s and the sub’s expected
utility under each of the conditions on R described in part 1.
3. For each of the values of R compare “individual evaluation” to “aggregate evaluation”.
4. Consider the joint probabilities in the table below . Repeat part 3 for each case below.
Comment on what you observe.
60
90
pLL
pLH
pHL
pHH
60
pLL
pLH
case 1:
.25
.25
.25
.25
90
pHL
pHH
case 2:
.5
0
0
.5
case 3:
0
.5
.5
0
case 4:
.2
.3
.3
.2
Based on Schwartz, Sudbury and Young, Journal of Management Accounting Research 2013.