Need help with question (a). Please also provide formulars on how to calculate E

Question

Need help with question (a). Please also provide formulars on how to calculate E(y1), E(y2), E(s) and E(r) respectively. Thanks!! Problem 2: Adverse Selection & Moral Hazard Javier is a moneylender who lives in the village of Nyabiheke in Rwanda. Half of the farmers in Nyabiheke are SAFE farmers and the other half are RISKY farmers. Both types of farmers need a loan of S200 in order to farm. Farmers will take a loan as long as they can earn at least zero expected income. SAFE farmers have a good harvest in which they earn revenues of S400 with 100% probability. They never have a bad harvest. RISKY farmers have a good harvest in which they earn revenues of $600 with 50% probability. They have a bad harvest in which they earn revenues of SO with 50% probability. Javier has perfect information about the farmers, i.e. he knows who is a SAFE farmer and who is RISKY Javier's opportunity cost is money is 15%. The farmers must repay the full loan plus interest if harvest is good, but nothing if harvest is bad. (a) Let ys and yR denote the incomes of SAFE and RISKY farmers, respectively and s and R denote Javier's profits from loans to SAFE and RISKY farmers, respectively. Derive the expressions for EUs) and EOR), the expected incomes of SAFE and RISKY farmers; and, E(Ts) and E(TR), the expected values of Javier's profits from loans to SAFE and RISKY, respectively, as functions of the interest rate, i.

Explanation / Answer

E(ys ) = 400*100%-(1+i)200 = 200-200i

E(yr ) = 50%(600-(1+i)200) + 50%(0-0) = 200-200i

E(s ) = (1+i)200 - (1+0.15)200 = -30+200i

E(r) = 50%[(1+i)200 - (1+0.15)200] + 50% [0-(1+0.15)200] = -130 +100i

In excel spreadsheet

200-200i 200-200i -30+200i -130 +100i i E(ys ) E(yr ) E(s ) E(r ) 0 200 200 -30 -130 0.1 180 180 -10 -120 0.2 160 160 10 -110 0.3 140 140 30 -100 0.4 120 120 50 -90 0.5 100 100 70 -80 0.6 80 80 90 -70 0.7 60 60 110 -60 0.8 40 40 130 -50 0.9 20 20 150 -40 1 0 0 170 -30 1.1 -20 -20 190 -20 1.2 -40 -40 210 -10 1.3 -60 -60 230 0 1.4 -80 -80 250 10 1.5 -100 -100 270 20 1.6 -120 -120 290 30 1.7 -140 -140 310 40 1.8 -160 -160 330 50 1.9 -180 -180 350 60 2 -200 -200 370 70 2.1 -220 -220 390 80 2.2 -240 -240 410 90 2.3 -260 -260 430 100 2.4 -280 -280 450 110 2.5 -300 -300 470 120 2.6 -320 -320 490 130 2.7 -340 -340 510 140 2.8 -360 -360 530 150 2.9 -380 -380 550 160 3 -400 -400 570 170

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