Example if the prot in case of success is 10 a claim

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inside equity and outside equity. Example: If the pro°t in case of success is 10, a claim of 4 can be interpreted either as a 40 percent equity stake, or as a risky debt claim with nominal value 4 which is defaulted upon in case of failure. Remark: Indeterminacy no longer true if there is a left- over value in case of failure.
1.1 Conclusions In summary: ° Because of moral hazard there is a limit to pledgeable income ° Projects with positive NPV may not be funded ° The entrepreneur needs to have enough assets to be °nanced ° Higher private bene°ts, higher threshold for °nanc- ing This model can also explain the ³credit rationing´puzzle:
° Lenders are not willing to raise interest rates even if the demand for loans exceeds their supply at the prevailing rates ° Loan markets are personalised because of private in- formation Explanation: ° Higher interest rates reduces the stake of the entre- preneur ° Reduced stake may demotivate the entrepreneur and lower the probability of repayment (moral hazard).
2 The variable investment model ° Technology with constant returns to scale: invest- ment I 2 [0 ; 1 ) giving return RI in case of success, and 0 in case of failure. ° The entrepreneur can work or shirk: if shirk, private bene°t BI : BI p L Low 0 p H High Private benefits Probability of success Entrepreneur’ s effort: ° The entrepreneur has internal funds A and needs to borrow I ± A to °nance the project.
A °nancial contract speci°es I and a sharing rule: ´ in case of failure: 0 for both; ´ and in case of success: R b ; R l , for entrepreneur and °nanciers respectively, where R l + R b = RI: De°ne: ° 1 µ p H R; exp. income per unit of investment; ° 0 µ p H ° R ± B ° p ± ; exp. pledgeable income per unit of investment. Assume ° 1 > 1 > ° 0 i.e.: ° 1 > 1 ! the project has NPV > 0 if the entre- preneur behaves ( ° 1 > 1 ); ° 0 < 1 ! the net return per unit of investiment p H R ± 1 is lower than the agency rent p H B ° p .
Financial contract: choose a level of investment I and a sharing rule ( R b ; R l ) that max U b = p H R b ± A = p H ( RI ± R l ) ± A while satisfying the investors±break even constraint (which holds with equality with competitive capital mkt) p H R l = I ± A and the entrepreneur±s IC ° pR b ² BI Solving PC for R l and substituting out in U b and IC b max I ( p H R ± 1) I st " p H R ± B ° p ! ± 1 # I + A ² 0 ! @U b @I = ° 1 ± 1 > 0 : want to maximise I:
How much high? Not in°nitely high, because entrepreneur has °nite assets, and so too much investment relative to his own share would destroy incentives! Analytically, @IC b @I = ° 0 ± 1 < 0 ! cannot raise I too much, otherwise violate IC b : Thus, raise I until IC binds. ! I solves IC b Also IC b binds.
More precisely, I = 1 h 1 ± p H ° R ± B ° p ±i A = 1 1 ± ° 0 A = kA where k = 1 1 ± ° 0 > 1: the manager invests k > 1 times his wealth. How much does he borrow? I ± A = kA ± A ( k ± 1) A = dA d = ( k ± 1) times his wealth, where d = ° 0 1 ± ° 0 The max loan he gets, dA; is his debt capacity (or out- side °nancing capacity, or borrowing capacity) and is a multiple of his assets.
Last, the entrepreneur±s utility coincides with NPV U b = p H R b ± A = ( p H R ± 1) I while the gross utility U g b = U b + A is U g b = ( p H R ± 1) I + A = [( ° 1 ± 1) k + 1] A = vA where v > 1 is de°ned as the shadow value of equity, and expresses the sensitivity of gross utility to changes in A : @U g b @A = ( ° 1 ± 1) k + 1 > 1 An increase in A increases gross utility directly, and indi- rectly through the increase in borrowing capacity.
Remark: k

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