Question

2. Consider an entrepreneur who has a business item. It requires an initial investment of $10 million. The probability of the success of the business is determined by the efforts that the entrepreneur puts in the business. For simplicity, let the probability of success be ??=?? where 0≤??≤1 is her effort level. However, efforts generate a disutility (i.e., a negative utility). Her utility function is ??(??,??)=??−25??2 where ?? is her earnings from the business in million dollars. She can earn $50 million if the business succeeds with probability ?? and earn nothing if the business fails with probability 1−??.

(a) Suppose that the entrepreneur has $10 million. She can immediately start the business by investing her entire wealth into the business. What is her expected utility?

(b) Under the condition of (a), which effort level should she choose to maximize her expected utility?

(c) Suppose that the entrepreneur currently has nothing. She needs to borrow $10 million from an “angel investor.” An angel investor is an investor who provides capital for a start-up, in exchange of high returns if the business succeeds. Let the return that the angel investor will receive be ?? (in million dollars). However, if the business fails, the angel investor will not require anything. What is the entrepreneur’s expected utility?

(d) A rational investor will not invest unless the entrepreneur promises enough returns. Say, the angel investor in (c) requires the half of the earnings, i.e., ??=25. Under the condition of (c), which effort level should the entrepreneur choose to maximize her expected utility? (e) Compare the results in (b) and (d). What is the implication of the difference in two results?

Answer 1

Part (a)

U = w - 25e²

Her expected utility = E(U) = probability of success x U_success + (1 - probability of success) x U_failure

= e x (50 - 25e²) + (1 - e) x (0 - 25e²)

= 50e - 25e³ - 25e² + 25e³ = 50e - 25e²

Part (b)

In order to maximize, expected utility, let's differentiate E(U) w.r.t e and equate it to zero.

d[E(U)] / de = 50 - 50e = 0

Hence, e = 50 / 50 = 1

Hence, she should choose e = 1 to maximize her utlity.

Part (c)

The utility function will now be:

U = w - R - 25e² where R is 0 in case of fialure

Her expected utility = E(U) = probability of success x U_success + (1 - probability of success) x U_failure

= e x (50 - R - 25e²) + (1 - e) x (0 - 0 - 25e²)

= (50 - R)e - 25e³ - 25e² + 25e³ = (50 - R)e - 25e²

Part (d)

R = 25

Hence, E(U) = (50 - R)e - 25e² = (50 - 25)e - 25e² = 25e - 25e²​​​​​​​

Hence, d[E(U)] / de = 25 - 50e = 0

Hence, e = 25/50 = 0.5

(e) Compare the results in (b) and (d). What is the implication of the difference in two results?

e = 1 in part (b) while it has fallen down to e = 0.5 in part (d)

Effort taken by promoter is higher in case of self funding or internal funding. The moment there is an external funding, the effort by the promoter sees a decline. External funding or support does build in degree of effortlessness by the promoter or owner.