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Deriving the Dividend Discount Model in the Intermediate金融学金融学 跨期消费模型研究不同时间人们的消费行为，涉及时间因素的的消费和储蓄。即消费者不必将每期的收入用光，他可以选择储蓄，也可以选择贷款。其预算约束线的斜率与利率有关，为-(1+R)，R是利率，然后再此预算约束现的情况下研究消费者偏好。分析的结果一般为：利率提高，人们选择少消费，更倾向于储蓄，利率降低，人们会多消费，倾向于借贷。那它的方程是什么样子的？假设只有两期,C是消费，M是收入：C2=M2+(1+R)(m1-c1)1，2都是角标。we employed a standard two-period model with consumers maximizing utility U(C1,C2).Income(I)in both periods is known,the price of a unit of consumption(C)in both periods is equal to one,and the individual can borrow or lend at the same risk-free interest rate(rf).The maximizing agent has two budget constraints over the two periods.In the first period,I1=C1+Bwhere B is the dollar value of the bonds purchased.In the second period,I2+(1+rf)B=C2FIGURE 1 Intertemporal consumption model.From the first budget constraint,B=I1 C1 and substituting that into the second budget constraint,this gives the equation for the budget constraint line as C2=I2+(1+rf)(I1 C1)The equilibrium condition is then straightforward to derive as follows:The marginal rate of substitution(MRS)is the tradeoff of current consumption for future consumption that leaves the individual indifferent;put differently,the minimum amount necessary to receive in the future to give up a unit of consumption today.As long as the return(1+rf)is greater than the MRS of current and future consumption,the individual will postpone consumption and increase his utility.The individual will continue giving up current consumption precisely to the point where the return of the bond is just sufficient to compensate him for waiting.The important aspect is the concept of diminishing MRS,that with each additional unit of postponed current consumption the individual requires a higher return(slope of indifference curve gets steeper as one moves from right to left).In a finance class,the student typically learns the net present value(NPV)criteria for investment projects as evaluating all projects,ranked from highest to lowest rate of return(ROR),against the opportunity cost of capital,which is assumed to be constant.To directly compare the intertemporal consumptionmodel to concepts in finance requires consideration of a standard concave investment opportunity set that ranks investment projects from right to left by highest to lowest rates of return.The slope of the investment opportunity set is 1 plus the ROR of a project.Investment is viewed as foregone current consumption.As such,the cost of capital is the slope of the indifference curve.This is shown in figure 2.FIGURE 2 Investment opportunity and declining rate of return.The individual will continue to invest as long as the projects ROR is greater than the MRS(cost of capital);in other words,NPV 0.Utility is maximized when the ROR is just equal to the MRS,where NPV=0.In this model,profit maximization is the same as utility maximization,and zero economic profit(NPV=0)is where the return of a project is just equal to the value of foregone current consumption.the two-period model has two constraints that are combined into one to represent the graphs budget constraint line.The constraint for period 1 is I1=C1+P1N(where N is the number of shares of a stock and P1 is the price of a share.)The budget constraint for period 2 is I2+(d2+P2)N=C2(where d2 is the per-share dividend and P2 is the price of the share in the second period.)From the first budget constraint,so substituting that into the budget constraint for period 2 gives The equilibrium is then The latter term is 1 plus the ROR on the stock which is equal to 1 plus the dividend yield plus the capital gain From the equilibrium condition,Rearranging and then by substitution and substituting indefinitely,assuming the stockpays dividends every period,
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