Breakeven Probability: Suppose you are selling a product for $5 per unit; your competitor is also charging $5 per unit. Variable costs are $2 per unit; there are no fixed costs. Your demand is for 100,000 units per year; at the $5 price, your total contribution to overhead is $300,000 per year.   You are considering lowering the price to $4 per unit for next year. If your only competitor’s price remains at $5, your demand will be 200,000 units next year and your total contribution to overhead will be $400,000. If, on the other hand, your competitor matches your price, your demand will be for only 125,000 units next year, and your contribution will be $250,000. You do not know how likely it is that your competitor will match your price.   For instance, if the probability of matching is 0.1, your expected contribution is 0.1* $250,000 + 0.9 * $400,000 = $385,000, and you will be better off dropping your price. If the probability of matching is 0.8, on the other hand, your expected contribution is 0.8 * $250,000 + 0.2 * $400,000 = $280,000, and you would be better off leaving the price at $5. At what probability would you be indifferent?   Let p be the probability that your competitor matches your price. Then your expected contribution if you price at $4 is p * $250,000 + (1 — p) * $400,000. You break even when this expected value equals $300,000, or when p = 2/3. If you believe that the probability of your competitor’s matching your price exceeds 2/3, you should not lower your price; if you think it is less, you should lower the price. Analyze the problem for the case in which the probability that your competitor will match your $4 price is 0.3. Hint: This problem introduces two concepts:   First, when thinking of breakeven the concern is not volume since the possible levels of volume are given.  Rather, the breakeven question can be recast as a question of probability: “at what matching probability are we indifferent between the old price and the new price?”    Second, since we are comparing an existing decision to a potential alternative, we are not concerned with fixed costs in the traditional sense, but rather with the return that we must forgo if we abandon the existing decision (price at the current $5). Since the matching probability of the new price is uncertain, the concept of expected value is again relevant to the decision to lower price.  In this case, it is the expected contribution of the new price. EC match = Volume match * $4.00 - Current Total Contribution EC no match = Volume no match * $4.00 - Current Total Contribution EV = EC match * P match + EC no match * P no match Where: ECi: The Expected Contribution from the i th competitor behavior Pi: The Probability of the i th competitor behavior EV: The Expected Value of the decision.