Question

A firm faces uncertain revenues and uncertain costs. Its revenues may be $120,000, $160,000, or $175,000, with probabilities .2, .3, and .5, respectively. Its costs are $150,000 or $170,000 with chances .6 and .4, respectively. (Revenues and costs are independent).

a. How many possible profit outcomes exist? Draw a decision tree listing these profit outcomes at the branch tips. Compute the firm’s expected profit by folding back the tree (It does not matter which uncertainty, demand or cost, is resolved first in the tree).

b. Without a decision tree, calculate separately the firm’s expected revenue and expected cost. What is the firm’s expected profit? (This result underscores a great computational convenience of the expected-value criterion. Expected profit is equal to expected revenue minus expected cost; that is, expectations can be taken separately.)

Answer 1

A] Profit = Revenue - Cost. There are 3 revenue possibilities, 2 cost possibilities. So, total 3 x 2 = 6 cases of profit are possible. All the corresponding 6 cases of profit, are shown in the below tree.

Firm Revenue 120000 16 0000 17500g 1 ༦ ༧༠༠༠ 170000 170000 Cost 170000 150000 150000 > 150000 L 1 . 175000 160000 Profit 175oOo 170000 160000 150 000 120000 - 17000 120000 150000 I Soooo - 170000 sooo CR- 1000 10000 25 000 - 30000 (hoss) soooo (hos) (hoss) CS Scanned with CamScanner

B] Expected Value = Σ [ Value . Probability (Value) ]

Expected Revenue = 0.2 (120000) + 0.3 (160000) + 0.5 (175000) = 24000 + 48000 + 87500 = 159500

Expected Cost = 0.6 (150000) + 0.4 (170000) = 90000 + 68000 = 158000

Expected Profit = Expected Revenue - Expected Cost = 159500 - 158000 = 1500