448 NILS H. HAKANSSON Puy probability of death in period j(j s n), pj probability of death in period 3(2m), given that the individual is alive at the beginning of period m capital position at decision point estate at the end of period j-l, given that death occurs in period y, non-capital income received at the end of period 3 if the individual is alive at the beginning of period 3 Y present value at decision point j of the potential non-capital income tream rs-1 interest rate in period j, s amount lent at decision point 3 M; number of investment opportunities in period 3, number of investment opportunities which may be sold short in period 3 net proceeds realized at the end of period j from each unit invested in opportunity i,i=2,., M,, at the beginning of period j, Fs joint distribution function of B2; ,.. AM it zi amount invested in opportunity i, i=l,., Mi, at the beginning of t insurance premium paid at the beginning of period 3 for insurance in period t contractual insurance premium payable at the beginning of period 3 if individual is alive at that point, T present value at decision point j of potential premiums t;, tj+l,., tn-1 amount of consumption in period 3 one-period utility function of consumption utility function of bequests a, patience factor linking periods j and j+I if the individual remains alive at the end of period j patience factor linking periods j and 3+l if the individual passes 3. DERIVATION OF THE BASIC MODEL Te shall now identify the relation which determines the amount of capital (debt)on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: =∑2计+721+y x+=∑1+T1+y+tP √=1 where This content downloaded from 202.115.118.13 on Wed, II Sep 2013 02: 34: 55 AM448 NILS H. HAKANSSON pj5 probability of death in period j(j ? n), Pmi probability of death in period j(2 m), given that the individual is alive at the beginning of period m, Xj capital position at decision point j, Xi estate at the end of period j - 1, given that death occurs in period j-l, yi non-capital income received at the end of period j if the individual is alive at the beginning of period j, Y3 present value at decision point j of the potential non-capital income stream, rj- 1 interest rate in period j, Z1j amount lent at decision point j, M6 number of investment opportunities in period j, Si number of investment opportunities which may be sold short in period j, 13ij net proceeds realized at the end of period j from each unit invested in opportunity i, i = 2, ***, Mj, at the beginning of period j, Fj joint distribution function of 2,j, *, 19M6j, zi6 amount invested in opportunity i, i = 1, ***, Mj, at the beginning of period j, tj insurance premium paid at the beginning of period j for insurance in period j, ti contractual insurance premium payable at the beginning of period j if individual is alive at that point, T3 present value at decision point j of potential premiums t6, t6+1, *.* *, cj amount of consumption in period j, Umk utility function at the beginning of period mn of consumption and bequests given that the individual passes away in period k ? m, u one-period utility function of consumption, g utility function of bequests, Ja, patience factor linking periods j and j + 1 if the individual remains alive at the end of period j, ai patience factor linking periods j and j + 1 if the individual passes away in period j. 3. DERIVATION OF THE BASIC MODEL We shall now identify the relation which determines the amount of capital (debt) on hand at each decision point in terms of the amount on hand at the previous decision point. This leads to the pair of difference equations: Mj (12) xj+l = E fiijzij + rjzlj + yj, -,*** n -1 i=2 and M j (13) xj+1 = X Iijzij + rjzlj + yj + tjlpjj, j- = * *, n i=2 where This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:34:55 AM All use subject to JSTOR Terms and Conditions