Rates of substitution are discussed in many books on Microeconomics; see e.g. Mas-Colell, Whinston and Green (1995) or Varian (1992). The de…nition of the time preference rate is not very explicit in the literature. An alternative formulation implying the same de…nition as the one we use here is used by Buiter (1981, p. 773). He de…nes the pure rate of time preference as “the marginal rate of substitution between consumption” in two periods
“when equal amounts are consumed in both periods, minus one.” A derivation of the time preference rate for a two-period model is in appendix A.1 of Bossmann, Kleiber and Wälde (2007).
The OLG model goes back to Samuelson. For presentations in textbooks, see e.g. Blan-chard and Fischer (1989), Azariadis (1993) or de la Croix and Michel (2002). Applications of OLG models are more than numerous. For an example concerning bequests and wealth distributions, see Bossmann, Kleiber and Wälde (2007). See also Galor and Moav (2006) and Galor and Zeira (1993).
The presentation of the Lagrangian is inspired by Intriligator (1971, p. 28 - 30). Treat-ments of shadow prices are available in many other textbooks (Dixit, 1989, ch. 4; Intrili-gator, 1971, ch. 3.3). More extensive treatments of di¤erence equations and the implicit function theorem can be found in many introductory “mathematics for economists”books.
There is an interesting discussion on the empirical relevance of exponential discount-ing. An early analysis of the implications of non-exponential discounting is by Strotz (1955/56). An overview is provided by Frederick et al. (2002). An analysis using sto-chastic continuous time methods is by Gong et al. (2007).
Exercises chapter 2
Applied Intertemporal Optimization
Optimal consumption in two-period discrete time models
1. Optimal choice of household consumption Consider the following maximization problem,
cmaxt;ct+1
Ut= v (ct) + 1
1 + v (ct+1) (2.6.1)
subject to
wt+ (1 + r) 1wt+1= ct+ (1 + r) 1ct+1: Solve it by using the Lagrangian.
(a) What is the optimal consumption path?
(b) Under what conditions does consumption rise?
(c) Show that the …rst-order conditions can be written as u0(ct) =u0(ct+1) = [1 + r] : What does this equation tell you?
2. Solving by substitution
Consider the maximization problem of section2.2.1 and solve it by inserting. Solve the constraint for one of the control variables, insert this into the objective function and compute …rst-order conditions. Show that the same results as in (2.2.4) and (2.2.5) are obtained.
3. Capital market restrictions
Now consider the following budget constraint. This is a budget constraint that would be appropriate if you want to study the education decisions of households.
The parameter b amounts to schooling costs. Inheritance of this individual under consideration is n.
Ut = ln ct+ (1 ) ln ct+1 subject to
b + n + (1 + r) 1wt+1 = ct+ (1 + r) 1ct+1:
(a) What is the optimal consumption pro…le under no capital market restrictions?
(b) Assume loans for …nancing education are not available, hence savings need to be positive, st 0. What is the consumption pro…le in this case?
4. Optimal investment
Consider a monopolist investing in its technology. Technology is captured by mar-ginal costs ct. The chief accountant of the …rm has provided the manager of the
…rm with the following information,
= 1+ R 2; t= p (xt) xt ctxt It; ct+1 = ct f (I1) :
Assume you are the manager. What is the optimal investment sequence I1, I2? 5. A particular utility function
Consider the utility function U = ct+ ct+1; where 0 < < 1: Maximize U subject to an arbitrary budget constraint of your choice. Derive consumption in the …rst and second period. What is strange about this utility function?
6. Intertemporal elasticity of substitution
Consider the utility function U = c1t + c1t+1:
(a) What is the intertemporal elasticity of substitution?
(b) How can the de…nition in (2.2.8) of the elasticity of substitution be transformed into the maybe better known de…nition
ij = d ln (ci=cj) d ln ucj=uci
= pj=pi ci=cj
d (ci=cj) d (pj=pi)? What does ij stand for in words?
7. An implicit function
Consider the constraint x2 x1 x1 = b.
(a) Convince yourself that this implicitly de…nes a function x2 = h (x1) : Can the function h (x1)be made explicit?
(b) Convince yourself that this implicitly de…nes a function x1 = k (x2) : Can the function k (x2) be made explicit?
(c) Think of a constraint which does not de…ne an implicit function.
8. General equilibrium
Consider the Diamond model for a Cobb-Douglas production function of the form Yt = KtL1 and a logarithmic utility function u = ln cyt + ln cot+1.
(a) Derive the di¤erence equation for Kt: (b) Draw a phase diagram.
(c) What are the steady state consumption level and capital stock?
9. Sums
(a) Proof the statement of the second lemma in ch.2.5.1,
T
i=1iai = 1
1 a a1 aT
1 a T aT +1 : The idea is identical to the …rst proof in ch.2.5.1.
(b) Show that
k 1
s=0ck 1 s4 s= ck4 k c4 :
Both parameters obey 0 < c4 < 1 and 0 < v < 1: Hint: Rewrite the sum as ck 14 k 1s=0( =c4)sand observe that the …rst lemma in ch.2.5.1holds for a which are larger or smaller than 1.
10. Di¤erence equations
Consider the following linear di¤erence equation system yt+1 = a yt+ b; a < 0 < b;
(a) What is the …xpoint of this equation?
(b) Is this point stable?
(c) Draw a phase diagram.
Multi-period models
This chapter looks at decision processes where the time horizon is longer than two periods.
In most cases, the planning horizon will be in…nity. In such a context, Bellman’s optimality principle is very useful. Is it, however, not the only way to solve maximization problems with in…nite time horizon? For comparison purposes, we therefore start with the Lagrange approach, as in the last section. Bellman’s principle will be introduced afterwards when intuition for the problem and relationships will have been increased.
3.1 Intertemporal utility maximization
3.1.1 The setup
The objective function is given by the utility function of an individual,
Ut= 1=t tu (c ) ; (3.1.1)
where again as in (2.2.12)
(1 + ) 1; > 0 (3.1.2)
is the discount factor and is the positive time preference rate. We know this utility function already from the de…nition of the time preference rate, see (2.2.12). The util-ity function is to be maximized subject to a budget constraint. The di¤erence to the formulation in the last section is that consumption does not have to be determined for two periods only but for in…nitely many. Hence, the individual does not choose one or two consumption levels but an entire path of consumption. This path will be denoted by fc g : As t; fc g is a short form of fct; ct+1; :::g : Note that the utility function is a generalization of the one used above in (2.1.1), but is assumed to be additively separable.
The corresponding two period utility function was used in exercise set 1, cf. equation (2.6.1).
The budget constraint can be expressed in the intertemporal version by
1=t(1 + r) ( t)e = at+ 1=t(1 + r) ( t)w ; (3.1.3) 45
where e = p c : It states that the present value of expenditure equals current wealth at plus the present value of labour income w . Labour income w and the interest rate r are exogenously given to the household, its wealth level at is given by history. The only quantity that is left to be determined is therefore the path fc g : Maximizing (3.1.1) subject to (3.1.3) is a standard Lagrange problem.
3.1.2 Solving by the Lagrangian
The Lagrangian reads L = 1=t
tu (c ) + h
1=t(1 + r) ( t)e at 1=t(1 + r) ( t)w i
; where is the Lagrange multiplier. First-order conditions are
Lc = tu0(c ) + [1 + r] ( t)p = 0; t <1; (3.1.4)
L = 0; (3.1.5)
where the latter is, as in the OLG case, the budget constraint. Again, we have as many conditions as variables to be determined: there are in…nitely many conditions in (3.1.4), one for each c and one condition for in (3.1.5).
Do these …rst-order conditions tell us something? Take the …rst-order condition for period and for period + 1. They read
tu0(c ) = [1 + r] ( t)p ;
+1 tu0(c +1) = [1 + r] ( +1 t)p +1: Dividing them gives
1 u0(c )
u0(c +1) = (1 + r) p
p +1 () u0(c )
u0(c +1) = p
(1 + r) 1p +1: (3.1.6) Rearranging allows us to see an intuitive interpretation: Comparing the instantaneous gain in utility u0(c )with the future gain, discounted at the time preference rate, u0(c +1) ; must yield the same ratio as the price p that has to be paid today relative to the price that has to be paid in the future, also appropriately discounted to its present value price (1 + r) 1p +1: This interpretation is identical to the two-period interpretation in (2.2.6) in ch. 2.2.2. If we normalize prices to unity, (3.1.6) is just the expression we obtained in the solution for the two-period maximization problem in (2.6.1).