Need A Solid Translation , English Into Armenian

[Ani]

money back guarantee must be done in 7 hours, damn i should've take ECONOMICAL COURSES!! but anyways ... IF ANYONE can do a PART! i would appriciate ..

(9)where σ[C(τ)]≡-U′[C(τ)]/[C(τ)U″[C(τ)]]>0 is the intertemporal substitution elasticity (see Chapter 14) and β(τ)≡f(τ)/[1-F(τ)]>0 is the so-called “hazard rate” or instantaneous probability of death at time τ. Compared to the case of an infinitely lived consumer, the hazard rate is the additional term appearing in the Euler equation. This is the first lesson from Yaari (1965, p. 143): the uncertainty of survival leads the household to discount the future more heavily, i.e. the subjective discount rate in the presence of lifetime uncertainty is ρ+β(τ) rather than just ρ. This makes intuitive sense. If there is a positive probability that you will not live long enough to enjoy a given planned future consumption path, then you tend to discount the utility stream resulting from it more heavily.Up to this point we have studied the optimal behaviour of the consumer when no insurance possibilities are available. But in reality various forms of life insurance exist so a relevant question is how this institutional feature would change the consumer’s behaviour. Yaari (1965, pp. 140-141) suggests a particular kind of life insurance based on so-called actuarial notes issued by the insurance company. An actuarial note can be bought or sold by the consumer and is cancelled upon the consumer’s death. The instantaneous rate of interest on such notes is denoted by r(τ) and non-zero trade in such notes only occurs if r(τ) exceeds r(τ). A consumer who buys an actuarial note in fact buys an annuity which stipulates payments to the consumer during life at a rate higher than the rate of interest. Upon the consumer’s death the insurance company has no further obligations to the consumer’s estate. Reversely, a consumer who sells an actuarial note is getting a life-insured loan. During the consumer’s life he/she must pay a higher interest rate on the loan than the market rate of interest , but upon death the consumer’s estate is held free of any obligations, i. e. the principal does not have to be paid back to the insurance company.In order to determine the rate of return on actuarial notes, Yaari makes the (simplest possible) assumption of actuarial fairness. To derive the expression for r(τ) implied by this assumption, assume that one dollar’s worth of actuarial notes is bought at time τ. These notes are either redeemed with interest at time τ+dτ (if the consumer survives) or are cancelled (if the consumer dies between τ and τ+dτ).Actuarial fairness then implies: xxxwhere the equality holds as dτ→0. The right-hand side of (10) shows the yield if the dollar is invested in regular market instruments whereas the left-hand side shows the yield on the actuarial note purchase. The term in round brackets is less than unity and corrects for the fact that the consumer may pass away between τ and τ+dτ. By solving for r(τ) and taking the limit as dτ→0 we obtain the following- rather intuitive- no-arbitrage equation between the two kinds of financial instruments: xxxRecall that τ is not only the time index but also stands for the age of the consumer so (11) has the sensible implication that r(τ)→∞ as τ→T. The closer the consumer gets to the maximum possible age T, the higher will be the instantaneous probability of death and thus the higher will be the required excess yield on actuarial notes.Let us now return to the consumer’s choice problem. As Yaari (1965, p. 145) points out, the consumer will always hold his/her financial assets in the form of actuarial notes, i. e. he/she will fully insure against the loss of life and the budget identity will be: xxx Hence the restriction on the terminal asset position is trivially met as all actuarial notes are automatically cancelled when the consumer dies. The intuition behind this full-insurance result is best understood by looking at the two cases. If the consumer has positive net assets at any time then they will be held in the form of actuarial notes because these yield the highest return (which is all the consumer is interested in in the absence of a bequest motive). Conversely, if the consumer had any negative outstanding net assets in other than actuarial notes, he/she would be violating the constraint on terminal assets mentioned above (i. e. the requirement that Pr{A(T)≥0}=1).We are not out of the forest of complications yet as we also need to ensure that the consumer is unable to beat the system by engaging in unlimited borrowing (sales of actuarial notes) and covering the ever increasing interest payments with yet further borrowings. This prompts the consumer’s solvency condition (see Yaari, 1965, p. 146 for a detailed derivation): xxxIntuitively, the condition says that the present value of the consumption stream must be equal to the sum of initial financial assets plus the present value of current and future non-interest income (i. e. “human wealth”), using the rate on actuarial notes for discounting.The consumer maximizes expected lifetime utility (EΛ(T) in (3)) subject to the solvency condition (13) and the non-negativity constraint on consumption (C(τ)≥0). The interior solution to this problem is characterized by the following Euler equation: xxxwhere we have used (11) in going from the first to the second line. The striking thing to note about (14) – and thus Yaari’s second lesson – is the fact that the Euler equation with fully insured lifetime uncertainty is identical to the Euler equation when no lifetime uncertainty exists! It should be observed, however, that the consumption levels will differ between the two scenarios as the lifetime consumption possibility frontier will differ between the two cases.Turning lessons into a workhorseYaari’s crucial insights lay dormant for twenty years until Blanchard (1985) made them the core elements of his continuous-time overlapping-generations model which subsequently became one of the workhorse models of modern macroeconomics. Blanchard simplified the Yaari setup substantially by assuming that the probability density function for the consumer’s time of death is exponential , i. e. f(T) in (1) is specified as: xxxso that 1-F(τ)≡f(T)dT=f(τ)/β and β(τ)≡f(τ)/[1-F(τ)]=β. Hence, instead of assuming an age-dependent instantaneous death probability – as Yaari did – Blanchard assumes that the hazard rate is constant and independent of the consumer’s age. This approach has several advantages. First and foremost, it leads to optimal consumption rules that are easy to aggregate across households (see below). We are thus able to maintain a high level of aggregation in the model despite the fact that the underlying population of consumers is heterogeneous by age. Second, it follows from (15) that the expected remaining lifetime of any agent is equal to 1/ β. By setting β=0, the Blanchard model thus coincides with the representative-agent model studied extensively in Chapters 14 and 15 above.Individual householdsThe first task at hand is to derive the expressions for consumption and savings for an individual household at an arbitrary time during its life. Assume that the utility function at time t of a consumer born at time v<t is given by EΛ(v,t): xxxwhere we have used the property of the exponential distribution in (15) to deduce that 1-F(τ-t)=e. Furthermore, in going from (3) to (16) we have assumed a logarithmic felicity function (featuring a unit intertemporal substitution elasticity), and we have added indexes for the agent’s date of birth (v) and the time to which the decision problem refers (t). Consequently, C(v,τ) stands for planned consumption at time τ by an agent born at time v. The agent’s budget identity is:A(v,τ)=[r(τ)+β]A(v,τ)+W(τ)-T(τ)-C(v,τ), (17)where r(τ), is the interest rate, W(τ) is the wage rate, T(τ) is the lump-sum tax levied by the government, and A(v,τ) are real financial assets. Equation (17) incorporates the Yaari notion of actuarially fair life-insurance contracts and is a straightforward generalization of (12) with (11) substituted in. Specifically, during life agents receive βA(v,τ) from the life-insurance company but at the time of the agent’s death the entire estate A(v,τ) reverts to that company. To avoid the agent from running a Ponzi game against the life-insurance company, the following solvency condition must be obeyed.By combining (17) and (18) the household’s lifetime budget restriction is obtained: xxxwhere H(t) is the human wealth of the agents consisting of the present value of lifetime after-tax wage income using the annuity factor, R(t,τ), for discounting: xxxEquation (19) is the counterpart to (13) above. Intuitively, it says that the present value of the household’s consumption plan must be equal to the sum of financial and human wealth.Intermezzo .Intuition behind the household’s solvency condition. The intuition behind the household’s solvency condition (18) can be explained as follows. We note that (17) can be premultiplied by e and rearranged to:where we have used the fact that (Leibnitz’s rule) in going from the first to the second line. By integrating both sides of (a) over the interval [t,∞) we obtain: xxxwhere we have used (20) and have noted that . The insurance companies will ensure that the limit on the left-hand side of (b) (loosely referred to as “terminal assets”) will be non-negative. Similarly, it is not in the best interest of the consumer to plan for positive terminal assets as he/she has no bequest motive and does not get satiated from consuming goods (as the marginal felicity of consumption remains strictly positive- see (16)).Hence, planned terminal assets will be strictly equal to zero. This yields the solvency condition (18). By using it in (b) the expression for the household’s lifetime buget restriction (19) is obtained. See Chiang (1992, pp. 101-103) for a more formal discussion of the transversality condition in an infinite-horizon optimal control problem.The consumer maximizes expected lifetime utility (17) subject to its lifetime budget restriction (19). The first-order conditions are (19) and: xxxwhere λ(t), the Lagrange multiplier associated with the lifetime budget restriction (19), represents the marginal expected lifetime utility of wealth. Intuitively, the optimality condition (21) instructs the consumer to plan consumption at each time to be such that the appropriately discounted marginal utility of consumption (left-hand side) and wealth (right-hand side) are equated (see also the discussion following). By using (21) for the planning period (τ=t) we see that C(v,t)=1/λ(t). Using this result and (19) in (21) we can express C(v,t) in terms of total wealth: xxxOptimal consumption in the planning period (τ=t) is proportional to total wealth, and the marginal propensity to consume out of total wealth is constant and equal to the “effective” rate of time preference, ρ+β.Aggregate householdsNow that we know what the consumption rules for individual households look like, the next task at hand is to describe the demographic structure of the Blanchard model. To keep things simple, Blanchard assumes that at each instant in time a large cohort of new agents in born. The size of this cohort of newborns is P(τ,τ)=βP(τ), where P(τ) stands for the aggregate population size at time τ. These newborn agents start their lives without any financial assets as they are unlinked to any existing agents and thus receive no bequests, i. e. A(τ,τ)=0. Of course, at each instant in time a fraction of the existing population dies. Since each individual agent faces an instantaneous probability of death equal to β and the number of agents P(τ) is large, “frequencies and probabilities coincide” and the number of deaths at each instant will be equal to βP(τ) . Since births and deaths exactly match, the size of the population is constant and can be normalized to unity (P(τ)=1).Another very useful consequence of the large-cohort assumption is that we can exactly trace the size of any particular cohort over time. For example, a cohort born at time v will be of size βe at time t≥v, because β[1-e] of the cohort members will have died in the time interval [v,t]. Since we know the size of each cohort it is possible to work with aggregate variables. For example, by aggregating the consumption levels of all existing agents in the economy we obtain the following expression for aggregate consumption at time t: xxxOf course, (23) is simply a definition and is not of much use in and of itself. But because the optimal consumption rule (22) features a propensity to consume out of total wealth which is independent of the generations index v, equation (23) gives rise to a very simple aggregate consumption rule: xxxwhere aggregate financial wealth is defined analogously to aggregate consumption (given in (23)). It cannot be overemphasized that the aggregation property follows from the assumption that each agent faces a constant instantaneous death probability ( see (16)). If instead the hazard rate varies with age- as in the Yaari (1965) model- then the optimal household consumption rule no longer features a generation-independent marginal propensity to consume out of total wealth and exact aggregation is impossible.What does the aggregate asset accumulation identity look like? By definition we have that A(t)≡β from which we derive (by application of Leibnitz’s rule): xxxwhere the first term on the right-hand side represents assets of newborns (A(t,t)=0), the second term is the wealth of agents who die, and the third term is the change in assets of existing agents. By substituting (17) into (25) and simplifying we obtain the aggregate asset accumulation identity: xxxWhereas individual wealth attracts the actuarial interest rate, r(t)+β, for agents that stay alive (see 17), equation (26) shows that aggregate wealth accumulates at the rate of interest, r(t). The amount βA(t) does not represent aggregate wealth accumulation but is a transfer- via the life-insurance companies- from those who die to those who remain alive.In the formal analysis of the model it is useful to have an expression for the “aggregate Euler equation”. It follows from (23) that: xxxAccording to (22) newborn agents consume a fraction of their human wealth at birth, i. e. C(t,t)=(ρ+β)H(t). Equation (24) shows that aggregate consumption is proportional to total (human and financial) wealth, i. e. C(t)=(ρ+β)[A(t)+H(t)]. Finally, it follows from (21) that individual households’ consumption growth satisfies C(v,τ)/C(v,τ)=r(τ)-ρ for τ [t,∞] (see footnote 7). By using all these results in (27) we obtain the aggregate Euler equation modified for the existence of overlapping generations of finitely lived agents: xxxEquation (28) has the same form as the Euler equation for individual households except for the correction term due to the distributional effects caused by the turnover of generations. Optimal consumption growth is the same for all generations (since they face the same interest rate) but older generations have a higher consumption level than younger generations (since the former generations are wealthier). Since existing generations are continually being replaced by newborns who hold no financial wealth, aggregate consumption growth falls short of individual consumption growth. The correction term appearing on the right-hand side of (28) thus represents the difference in average consumption and consumption by newborns, i. e. (28) can be re-expressed as in (29).FirmsThe production sector is characterized by a large number of firms that produce an identical good under perfect competition. Output, Y(t), is produced according to a linearly homogeneous technology with labour, L(t), and physical capital, K(t), as homogeneous factor inputs which are rented from households:Y(t)=F(K(t),L(t)), (30)where F(.) satisfies the usual Inada conditions (see Chapter 14). The stockmarket value of the representative firm is: xxxThe firm chooses labour and capital in order to maximize (31) subject to the production function (30) and the capital accumulation constraint:K(t)=I(t)-δK(t), (32)where I(t) denotes gross investment, and δ is the constant rate of depreciation of capital. There are no adjustment costs associated with investment. The first-order conditions imply that the marginal productivity of labour and capital equal the producer costs of these factors- see, respectively, equations (T1.4) and (T1.5) in Table 16.1. Finally, we recall from Chapter 14 that the market value of the firm is equal to the replacement value of its capital stock, i. e. V(t)=K(t).The government and market equilibriumThe government budget identity is given in (T1.3) in Table 1. The government consumes G(t) units of the good and levies lump-sum taxes on households T(t). Government debt is B(t) so that r(t)B(t) is interest payments on outstanding debt. Like the private sector, the government must remain solvent and obey a no-Ponzi-game condition like: xxxBy using (T1.3) and (33) the government budget restriction is obtained: xxxIntuitively, government solvency means that if there is a pre-existing government debt (positive left-hand side) it must be covered in present-value terms by present and future primary surpluses (right-hand side).At each instant of time, factor and goods markets clear instantaneously. In this closed economy households can only accumulate domestic assets so that, as a result, financial market equilibrium requires that A(t)=K(t)+B(t). Wage flexibility ensures that the aggregate supply of labour (L(t)=1) by households matches labour demand by firms. Goods market equilibrium is obtained when the supply of goods equals aggregate demand, which consists of private and public consumption plus investment: Y(t)=C(t)+I(t)+G(t). For convenience, the key equations of the model have been gathered in Table 1.The phase diagramIn order to illustrate some of the key properties of the model we now derive the phase diagram in Figure 1. We assume for simplicity that lump-sum taxes, government consumption, and public debt are all zero in the initial situation (T(t)=G(t)=B(t)=0). The K(t)=0 line represents points for which the capital stock is in equilibrium. The Inada conditions (see Chapter 14) ensure that it passes through the origin and is vertical there (see point A in Figure 1). Golden-rule (GR) consumption occurs at point A where the K(t)=0 line reaches its maximum: xxxThe maximum attainable capital stock, K , occurs at point A, where consumption is zero and total output is used for replacement investment (F(K,1)/K=δ). For points above (below) the K(t)=0 line consumption is too high (too low) to be consistent with a capital stock equilibrium and consequently net investment is negative (positive). This has been indicated by horizontal arrows in figure 1.The derivation of the C(t)=0 line is a little more complex because its position and slope depend on the interplay between effects due to capital scarcity and those attributable to intergenerational-distribution effects. Recall from Chapter 14 that the “Keynes-Ramsey” (KR) capital stock. K , is such that the rate of interest equals the exogenously given rate of time preference, i. e. r =F(K,1)-δ≡ρ. Since K is associated with a zero interest rate and there are diminishing returns to capital (F<0), K lies to the left of the golden-rule point as is indicated in Figure 1. Furthermore, for points to the left (right) of the dashed line, capital is relatively scarce (abundant), and the interest rate exceeds (falls short of) the pure rate of time preference.When agents have finite lives (β>0) the C=0 line is upward sloping because of the turnover of generations. Its slope can be explained by appealing directly to equations (28) (with A=K as we set B=0), (29), and Figure 1. Suppose that the economy is initially on the C=0 curve, say at point E. Now consider a lower level of consumption, say at point B. With the same capital stock, both points feature the same rate of interest. Accordingly, individual consumption growth, C(v,t)/C(v,t) [=r-ρ], coincides at the two points.Expression (29) indicates, however, that aggregate consumption growth depends not only on individual growth but also the proportional difference between average consumption and consumption by a newly born generation, i. e. [C(t)-C(t,t)]/C(t). Since newly born generations start without any financial capital, the absolute difference between average consumption and consumption of a newly born household depends on the average capital stock and is thus the same at the two points. Since the level of aggregate consumption is lower at B (than it is at E), this point features a larger proportional difference between average and newly born consumption, thereby decreasing aggregate consumption growth (i. e. C(t)<0). In order to restore zero growth of aggregate consumption, the capital stock must fall (to point C). The smaller capital stock not only raises individual consumption growth by increasing the rate of interest but also lowers the drag on aggregate consumption growth due to the turnover of generations because a smaller capital stock narrows the gap between average wealth (i. e. the wealth of the generations that pass away) and wealth of the newly born. In summary, for points above (below) the C(t)=0 line, the capital-scarcity effect dominates (is dominated by) the intergenerational-redistributional effect and consumption rises (falls) over time. This is indicated with vertical arrows in Figure 1.In terms of Figure 1, steady-state equilibrium is attained at the intersection of the K(t)=0 and C(t)=0 lines at point E. Given the configuration of arrows, it is clear that this equilibrium is saddle-point stable, and that the saddle path, SP, is upward sloping and lies between the two equilibrium loci. 3. APPLICATIONS OF THE BASIC MODEL The effects of fiscal policyAs a first application of the Blanchard-Yaari model we now consider the effects of a typical fiscal policy experiment, consisting of an unanticipated and permanent increase in government consumption. We abstract from debt policy by assuming that the government balances its budget by means of lump-sum taxes only, i. e. B(t)=B(t)=0 and G(t)=T(t) in equation (T1.3). We also assume that the economy is initially in a steady state and that the time of the shock is normalized to t=0.In terms of Figure 16.2, the K(t)=0 line is shifted downward by the amount of the shock dG. In the short run the capital stock is predetermined and the economy jumps from point E to A on the new saddle path SP. Over time the economy gradually moves from A to the new steady-state equilibrium at E. As is clear from the figure, there is less than one-for-one crowding out of private by public consumption in the impact period, i. e. -1<dC(0)/dG<0. In contrast, there is more than one-for-one crowding out in the long run, i. e. dC(∞)/dG<-1.The reason for these crowding-out results is that the change in the lump-sum tax includes an intergenerational redistribution of resources away from future towards present generations (Bovenberg and Heidra, forthcoming). At impact, all households cut back on private consumption because the higher lump-sum tax reduces the value of their human capital. Since households discount present and future tax liabilities at the annuity rate (r(τ)+β, see (20)) rather than at the interest rate, existing households at the time of the shock do not feel the full burden of the additional taxes and therefore do not cut back their consumption by a sufficient amount. As a result, private investment is crowded out at impact (K(t)<0 at point A) and the capital stock starts to fall. This in turn puts downward pressure on before-tax wages and upward pressure on the interest rate so that human capital falls over time. So, future generations are poorer than newborn generations at the time of the shock because they have less capital to work with and thus receive lower wages (since F<0).If the birth rate is zero (β=0) there is a single infinitely lived representative consumer and intergenerational redistribution is absent. Crowding out of consumption is one-for-one, there is no effect on the capital stock, and thus no transitional dynamics. In terms of Figure 2, the only effect on the economy consists of a downward jump in consumption from point B to point C.The non-neutrality of government debtThe previous subsection has demonstrated that lump-sum taxes cause intergenerational redistribution of resources in the Blanchard-Yaari model. This suggests that Ricardian equivalence does not hold in this model, i. e. the timing of taxes is not intergenerationally neutral and debt has real effects. Ricardian non-equivalence can be demonstrated by means of some simple “bookkeeping” exercises (see also Chapter 14). The result that must be proved is that, ceteris paribus the time path of government consumption (G(τ) for τ [t,∞)), aggregate consumption (C(t)) depends on pre-existing debt (B(t)) and the time path of taxes (T(τ) for τ [t,∞)) (Buiter, 1988, p. 285).Total consumption is proportional to total wealth (see (24)) which can be written as follows: xxxwhere Ω(t) is defined as: xxxNote that in deriving (36), we have used the definition of human wealth (20) to go from the first to the second line and the government budget restriction (34) to get from the second to the third line. In view of (37) and (34) it follows that Ω(t) vanishes if and only if the birth rate is zero and R(t,τ)=R(t,τ). If the birth rate is positive, Ω(t) is non-zero and Ricardian equivalence does not hold.Recall that in the Blanchard-Yaari model the birth rate of new generations is equal to the instantaneous death probability facing existing generations. As a result it is not a priori clear which aspect of the model is responsible for the failure of Ricardian equivalence. The analysis of Weil (1989b) provides the strong hint that it is the arrival rate of new generations which destroys Ricardian equivalence (see Chapter 14 above). This suggestion was formally demonstrated by Buiter (1988) who integrates and extends the Blanchard-Yaari-Weil models by allowing for differential birth and death rates (and) and Hicks-neutral technical change. In his model the population grows at an exponential rate n≡. Buiter (1988, p. 285) demonstrates that a zero birth rate (=0) is indeed necessary and sufficient for Ricardian equivalence to hold. 4. EXTENSIONSIn this section we demonstrate the flexibility of Blanchard-Yaari model- and thus document its workhorse status- by showing how easily it can be extended in various directions. These extensions are by no means the only ones possible- some others are mentioned in the Further Reading section of this chapter.Endogenous labour supplyAs we have seen throughout the book, an endogenous labour supply response often plays a vital role in the various macroeconomic theories. In Chapter 15, for example, it was demonstrated that the intertemporal substitutability of household leisure forms one of the key mechanisms behind most models in the real business cycle (RBC) tradition. The aim of this subsection is therefore to extend the basic Blanchard-Yaari model by allowing for an endogenous labour supply decision of the households. We follow Heijdra and Ligthart (2000) by introducing various taxes and assuming simple functional forms for preferences and technology in order to keep the discussion as simple as possible. We analyse the effects of a consumption tax in order to demonstrate some of the key properties of the model.Extending the modelAssume that the utility function used so far (see (16)) is replaced by: xxxwith 0<1. Leisure is defined as the consumer’s time endowment (which is normalized to unity) minus labour supply, L(v,τ). Note that (16) is obtained as a special case of (38) setting =1. Since labour supply is now endogenous, the agent’s budget identity (17) is replaced by: xxxwhere X(v,τ) represents full consumption, i. e. the sum of spending on goods consumption and leisure, t is a proportional tax on private consumption, t is a proportional tax on labour income, and Z(τ) are age-independent transfers received from the government. The household’s solvency condition is still given by (18).Following Marini and van der Ploeg (1988) we solve the household’s optimization problem by using two-stage budgeting. We have encountered this technique several times before in this book, albeit in the context of static models- see for example Chapters 11 and 13. The procedure is, however, essentially the same in dynamic models. Intuitively the procedure works as follows. In the first stage we determine how the consumer chooses an optimal mix of consumption and leisure conditional upon a given level of full consumption (X(v,τ)). Then, in second stage, we determine the optimal time path for full consumption itself. The procedure is valid provided the utility function is intertemporally separable.In stage 1 the consumer chooses C(v,τ) and [1-L(v,τ)] in order to maximize instantaneous felicity, log[], given the restriction (40) and conditional upon the level of X(v,τ). This optimization problem yields the familiar first-order condition calling for the equalization of the marginal rate of substitution between leisure and consumption and the relative price of leisure and consumption: xxxBy substituting into , we obtain expressions for consumption and leisure in terms of full consumption: xxxSince sub-felicity – the term in square brackets in () – is Cobb-Douglas and thus features a unit substitution elasticity, spending shares on consumption and leisure are constant. To prepare for the second stage we substitute into the lifetime utility functional to obtain the following expression: xxxwhere P() is a true cost-of-living index relating sub-felicity to full consumption: xxxIn stage 2, the consumer chooses the path of full consumption in order to maximize subject to the dynamic budget identity and the solvency condition. This problem is essentially the same as the one that was solved in Section above so it should therefore not surprise the reader that the solution takes the following form: xxxEquation says that full consumption is proportional to total wealth (the sum of financial and human wealth) whereas shows that optimal full consumption growth depends on the difference between the interest rate and the pure rate of time preference. Finally, is the definition of human wealth. It differs from because labour income is taxed at a proportional rate and because the household receives transfers.By aggregating and across surviving generations and making use of , expressions for aggregate consumption growth and labour supply are obtained – see equations and in Table. Compared to the basic Blanchard-Yaari model we have introduced the following simplifications. First, we abstract from government spending and debt and assume that all tax revenues are rebated to households in a lump-sum fashion. As a result, the government budget identity is static – see in Table . Second, we have simplified the production structure of the extended model somewhat by assuming a Cobb-Douglas technology – see. Using this specification in , and yields the expressions and , respectively.Phase diagramThe phase diagram of the model is drawn in Figure 16.3. The endogeneity of the labour supply decision considerably complicates the derivation of the phase diagram. For that reason we report the details of this derivation in a mathematical appendix to this chapter and focus here on a graphical and intuitive discussion.The capital stock equilibrium locus (CSE) represents the (C,K) combinations for which net investment is zero. Apart from the fact that the model now includes various tax rates and government consumption is set equal to zero, the CSE line is identical to the one discussed in detail in Chapter 15. The CSE line is concave and for points above (below) this line consumption is too high (low) and net investment is negative (positive).The consumption equilibrium (CE) locus represents the (C,K) combinations for which aggregate consumption is constant. In the representative-agent model of Chapter 15, aggregate and individual consumption coincide and CE is simply the locus of points for which the interest rate equals the rate of time preference and the output-capital ratio is constant. For convenience, the CE line for the representative-agent model is included in the figure as the dashed line connecting points A3 and A4.In contrast, in the overlapping-generations model, individual and aggregate consumption do not coincide and as a result, the position and slope of the CE curve are affected by two conceptually distinct mechanisms, namely the factor scarcity effect (FS) and the generational turnover effect (GT). The interplay between these two effects ensures that CE has the shape of a rather prominent nose. Along the lower branch, A1A2, consumption is low, equilibrium employment is close to unity, and CE is upward sloping. In contrast, along the upper branch, A2A3, consumption is high, equilibrium employment is low, and CE slopes downward. The dynamic forces at work can be studied by writing (T2.1) as follows: xxx (16.49).where r(C,K) is short-hand notation for the dependence of the real interest rate on consumption and the capital stock. Simple intuitive arguments can be used to motivate the signs of the partial derivatives of the r(C,K) function, which are denoted by rC and rK, respectively. Some simple graphs can clarify matters.Consider Figure 16.4 which depicts the situation in the rental market for capital and the labour market. In panel (a), the supply of capital is predetermined in the short run-say at K0. The demand for capital is downward sloping- due to diminishing returns to capital- and depends positively on the employment level- because the two factors are cooperative in production. Panel (b) depicts the situation in the labour market. There are diminishing returns to labour- so labour demand slopes downwards- and additional capital boosts labour demand. The labour supply curve follows from the optimal leisure-consumption choice. It slopes upwards because isolates the pure substitution effect of labour supply.Let us now use Figure 16.4 to deduce the signs of rC and rK. Ceteris paribus the capital stock, an increase in consumption shifts labour supply to the left so that the wage rises and employment falls. The reduction in employment shifts the demand for capital to the left so that- for a given inelastic supply of capital- the real interest rate must fall to equilibrate the rental market for capital. The thought experiment compares points E and A in the two panels.An increase in capital supply- ceteris paribus consumption- has a direct effect which pushes the interest rate down and an induced effect operating via the labour market. The boost in K shifts the labour demand curve to the right, leading to an increase in wages and employment and thus to an outward shift in the capital demand curve. Although this induced effect pushes the interest rate up somewhat, the direct effect dominates and rK<0. The comparison is between points E and B in the two panels of Figure 16.4.We can now study the dynamical forces acting on aggregate consumption along the two branches of the CE curve in Figure 16.3. First consider a point on the lower branch of this curve. Holding capital constant, an increase in aggregate consumption leads to a small decrease in labour supply and thus a small decrease in the interest rate. At the same time, however, the capital-consumption ratio falls so that aggregate consumption growth increases, i.e. C>0 for points above the lower branch of CE: xxx (lower branch of CE)Now consider a point on the upper branch of the CE curve. Ceteris paribus K, a given increase in C has a strong negative effect on labour supply and thus causes a large reduction in the interest rate which offsets the effect operating via the capital-consumption ratio, i.e. C<0 for points above the upper branch of CE: xxx (upper branch of CE)These dynamic effects have been illustrated with vertical arrows in Figure 16.3.In summary, the CE curve is very similar to the one for the standard Blanchard model with exogenous labour supply for values of L close to unity. At the same time, it is very similar to the CE curve for the representative-agent model with endogenous labour supply for values of L close to zero. Put differently, on the lower branch of the CE curve the generational turnover effect dominates whereas on the upper branch the factor scarcity effect dominates.It follows from the configuration of arrows that the unique equilibrium E in Figure 16.3 is saddle-point stable. Although we have drawn Figure 16.3 such that the equilibrium occurs on the downward-sloping part of the CE curve, there is nothing to prevent the opposite occurring, i.e. it is quite possible that the structural parameters are such that E lies on the lower branch of CE.Raising the consumption taxWe now illustrate how the model can be used for policy analysis. We focus attention on the effects of an unanticipated and permanent increase in the consumption tax. Using the methods explained in detail in Chapter 15, the model can be loglinearized along an initial steady state. The resulting expressions are collected in Table 16.3.Solving the loglinearized model is child’s play and proceeds along much the same lines as in Chapter 15. First we use (T3.5) to compute the ‘quasi-reduced-form’ expression for output: xxxwhere f summarizes the intertemporal labour supply effects: xxxSecond, we use (16.50) in (T3.2) and impose K(t)=0 to get the loglinearized CSE line: xxxThe CSE curve is upward sloping and an increase in the consumption tax shifts the curve down- see the shift from CSE0 to CSE1 in Figures 16.5 and 16.6. For a given capital stock, an increase in t reduces labour supply, and thus employment and output. To restore capital stock equilibrium, employment and output must return to their former levels, i.e. consumption must fall.Finally, we obtain the loglinearized CE line by substituting (16.50) and (T3.4) into (T3.1) and setting C(t)=0: xxxAs was apparent from our discussion concerning Figure 16.3 above, the slope of the CE line around the initial steady state is ambiguous and depends on the relative strength of the factor scarcity and generational turnover effects. These two effects show up in the denominator of the coefficient for K(t) on the right-hand side as, respectively,(f-1)(r+b) and (r-p). There are thus two cases of interest. First, if (r-p) exceeds (f-1)(r+b) then the GT effect dominates the FS effect, and the CE line is upward sloping as in Figure 16.5. Second, if the reverse holds and (f-1)(r+b) is larger than (r-p) then the FS effect dominates the GT effect so that the CE curve is downward sloping as in Figure 16.6.It turns out that the effect of the consumption tax on the long-run capital stock depends cricically on the relative strength of the GT and FS effects. Indeed, by solving (16.52) and (16.53) we obtain the following expression for the steady-state effect on capital of the consumption tax change: xxxIf the GT effect is stronger than the FS effect, an increase in the consumption tax leads to an increase in the consumption tax leads to an increase in the long-run capital stock. The intuition behind these results can be explained with the aid of Figure 16.5 and 16.6. In Figure 16.5 the GT effect is dominant, the CSE curve shifts down by less than the CE curve does , and the steady state shifts from E0 to E1. At impact the tax shock causes a redistribution from old to young existing generations. The old generations are wealthy and thus have a high consumption level, whereas the young generations consume very little and thus face only a small increase in their tax bill. Since the additional tax revenue is recycled to all generations in an age-independent lump-sum fashion, older generations are hit harder by the tax shock than younger generations are the proportional difference in consumption between the old and young agent falls. In terms of (16.49), r(t) changes hardly at all but the generational turnover term, C(t), falls so that aggregate consumption growth increases at impact. The reduction in aggregate consumption outweighs the fall in production, net investment takes place and the economy gradually moves from point A to the new steady state in E.Matters are quite different if the FS effect dominates the GT effect, a situation which is depicted in Figure 16.6. Now the downward shift in CE dominates the downward shift in CSE and the new steady state,E1, is associated with a lower capital stock. This long-run effect is best understood by nothing that with a dominant FS effect, the long-run capital-labour ratio is more or less unchanged. Since the consumption tax reduces labour supply this can only occur if the capital stock falls also. In the impact period the reduction in consumption is dominated by the fall in output and net investment is negative. At the same time, the reduction in labour supply reduces the capital-labour ratio at impact so that the interest rate falls and the aggregate consumption profile becomes downward sloping. In summary, it follows that both K<0 and C<0 at point A. Over time, the economy gradually moves from point A to the new steady state at E.