You are the division manager of a large company that uses a participative budgeting process in which division managers submit budget requests to corporate headquarters each month. Assume monthly revenues are $100,000. Both you and corporate headquarters know that monthly costs can range between $65,000 and $90,000, with the following probabilities for each of the possible cost values:

Cost Amount	Probability
$65,000	1%
$70,000	6%
$75,000	1%
$80,000	1%
$85,000	1%
$90,000	90%

Before submitting your budget request, you will learn your division’s actual costs for the month. However, corporate headquarters will only know the information in the table above. Corporate headquarters is considering whether to use a rationing policy, in which you will receive funds equal to your request only if your request is equal to or below some cutoff point, or a trust policy, in which you will receive all requested funds.

1. In Example #1 from Notes 3-3, we found that corporate headquarters preferred the rationing policy over the trust policy. Do you reach the same conclusion here? In particular, what would the rationing policy look like if corporate headquarters was going to use one? That is, what is the cutoff point? Would corporate headquarters prefer that rationing policy over the trust policy? Show supporting calculations.

2. Suppose now that corporate headquarters can install an information system that refines the range of potential costs into two smaller subsets. Specifically, the information system will indicate (with certainty) whether the actual costs are in the set {$65k, $70k, $75k} or {$80k, $85k, $90k}. Both you and the corporate headquarters will know whether your division’s actual costs are in one of these subsets before you submit your request. However, as before, you will know the actual cost, but corporate headquarters will not. For example, you might learn that the actual cost is $70,000, but corporate headquarters will only know that the actual cost is either $65,000, $70,000, or $75,000. If this information system is installed, what would the rationing policy look like, and would corporate headquarters prefer this over a trust policy?

Hint: you will need to incorporate conditional probabilities in your calculations. To illustrate, let’s calculate headquarters’ expected payoffs if it sets the cutoff point at $65k if the information system says the actual cost is in the subset {$65k, 70k, $75k}. In the problem, we are told the probability that the actual cost is $65k, $70k, or $75k is 0.01, 0.06, and 0.01, respectively. So, the probability that the information system (correctly) says the actual cost is in the subset {$65k, 70k, $75k} is 0.01 + 0.06 + 0.01 = 0.08.

Now, imagine the information system says the actual cost is in the subset {$65k, 70k, $75k}. If the rationing policy sets the cutoff point at $65k, then you (the division manager) will submit a budget request equal to the actual cost (similar to the example we did in class). This implies that the only time your budget request is approved – when the information system says the actual cost is in the subset {$65k, 70k, $75k} – is when you ask for $65k. So, the probability that your budget request is approved, conditional on the information system saying the actual cost is in the subset {$65k, 70k, $75k}, is equal to the probability that the actual cost is $65k divided by the probability that the information system says the actual cost is in the subset {$65k, 70k, $75k}, which is 0.01 0.08 = 0.125. This implies the probability that your budget request is rejected is 1 – 0.125 = 0.875. So, headquarters’ expected payoffs if it sets a cutoff point of $65k is 0.125*($100k - $65k) + 0.875*($0) = $4,375.

You can apply the steps above to figure out headquarters’ expected payoffs for any cutoff point within each subset of potential costs.

3. Information asymmetry plays a crucial role in the relative merits of using rationing versus a trust policy. Thus, many argue reducing that information asymmetry should improve production efficiency. Specifically, total welfare should increase as information asymmetry decreases. Is this argument borne out in the problem? That is, does installing the information system increase total welfare in this problem?