Quasi-geometric discounting in cash-in-advance economy

Accepted Manuscript Quasi-geometric discounting in cash-in-advance economy Daiki Maeda

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S0304-4068(18)30116-2 https://doi.org/10.1016/j.jmateco.2018.09.004 MATECO 2272

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Journal of Mathematical Economics

Received date : 18 January 2018 Revised date : 23 July 2018 Accepted date : 19 September 2018 Please cite this article as: Maeda D., Quasi-geometric discounting in cash-in-advance economy. Journal of Mathematical Economics (2018), https://doi.org/10.1016/j.jmateco.2018.09.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Quasi-geometric discounting in cash-in-advance economy∗ Daiki Maeda† July 23, 2018

Abstract This paper develops a dynamic general equilibrium model in which households have a preference of quasi-geometric discounting, as in Krusell et al. (2002), and face a cash-inadvance constraint. Through this extension, we show that this economy engages in oversaving, although households have present bias, and the optimal growth rate of money when there is present bias is lower than one when there is not. Keywords: Quasi-geometric discounting, time-inconsistency, cash-in-advance JEL classification: E21, E40, E70

1

Introduction

Recently, human behavior with time-inconsistency has been observed in many economic experiments and empirical studies. Salois and Moss (2011) directly estimate the hyperbolic discounting parameter from asset market data. A representative study by Krusell et al. (2002) (hereafter KKS) introduces quasi-geometric discounting into the standard discrete-time dynamic general equilibrium model. KKS solves the household’s problem as a game between the current self and future self. They obtain the result in which three saving rates—for maximizing welfare, the central planner’s solution without commitment, and in a competitive equilibrium—are different. Furthermore, they find that the welfare in the competitive equilibrium is larger than that in the central planner’s solution. ∗

I would like to thank Koichi Futagami, Shinsuke Ikeda, Tatsuro Iwaisako, and the seminar participants at the 2017 Japanese Economic Association Spring Meeting at Ritsumeikan University, and the workshop participants at Kobe University for their helpful comments and suggestions on the previous version of this paper. † Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043, Japan E-mail addresses: [email protected]

1

In our paper, we extend KKS to a monetary economy. We assume that households face a cash-in-advance (CIA) constraint which requires them to hold money before they consume. Through this assumption, we find that the saving rate is excessive, although households have present bias. This is why the CIA constraint plays a role of commitment for making them save more in the model. Households with quasi-geometric discounting have an incentive to make themselves after a period save more. Therefore, they want to accumulate lesser money so as not to consume much in future. The CIA constraint can make this successful because the cash holding is determined in the last period and households cannot change it in the current period when households face it. Since money accumulated is too little in our model, households consume too little. Therefore, we generate a high savings rate even in a model with quasigeometric discounting. This result is quite different from that of KKS in which the saving rate in a competitive equilibrium is too small, relative to the optimal rate. Moreover, we also obtain the following result. The optimal growth rate of money when there is present bias is lower than the growth rate when there is no present bias. This is why there is no commitment device for labor supply by households, in contrast to the case of consumption. Then, households increase the present leisure, and labor supply of households will be too small. Since the decreasing growth rate of money increases labor supply under the CIA constraint as pointed out by previous researchers, for example Walsh (2010, Ch3), the lower growth rate of money is optimal for increasing labor supply. Next, we briefly mention related studies. Gong and Zhu (2009), Boulware et al. (2013), and Graham and Snower (2008, 2013) incorporate money and hyperbolic discounting into a dynamic general equilibrium model. Gong and Zhu (2009) and Boulware et al. (2013) use the money-inthe utility model, and Graham and Snower (2008, 2013) use the New Keynesian model with the stickiness of the wage. Other recent works study the model assuming hyperbolic discounting, but not the existence of money, including Hiraguchi (2014, 2016) and Ojima (2018). Finally, the rest of this paper is organized as follows. Section 2 provides our model. Section 3 studies the optimization of households and equilibrium. Section 4 analyzes welfare. Section 5 analyzes the monetary policy. Section 6 concludes the paper.

2

2

The model

In this section, we explain the goods market, the capital and labor markets, money, and households in this economy. First, we explain the goods market. In this economy, a good exists that is produced by inputting capital and labor. The production function is a Cobb-Douglas type function, as in KKS. Because we assume that capital is fully depreciated, the resource constraint is as follows: α ¯1−α Ak¯t−1 lt = c¯t + k¯t ,

(1)

where k¯t is the capital accumulated by period t, ¯lt is the labor supply, c¯t is consumption, these variables are aggregate values in this economy, A > 0 is the productivity parameter, and α ∈ (0, 1) is the capital share. We also assume that the goods market is perfectly competitive. Second, we explain the capital and labor markets. These markets are perfectly competitive, implying marginal-product pricing of the capital and labor inputs: α−1 ¯1−α rt = αAk¯t−1 lt α ¯−α wt = (1 − α)Ak¯t−1 lt .

(2) (3)

Third, we explain money. Money is supplied by the government, which increases money at a constant growth rate, θ. Therefore, we obtain the dynamics of the real money stock as follows:

m ¯t =

1+θ m ¯ t−1 , 1 + πt

(4)

where m ¯ t is the real money stock and πt is the inflation rate. The government transfers all of the money that it issues in each period to households:

τt =

θ m ¯ t−1 , 1 + πt

(5)

where τt is the real transfer to households. At the end of this section, we explain households. There is one unit of a household and its size does not change. We assume that each household has one unit of time, and it is divided

3

into leisure and labor. We also assume a utility function with consumption and leisure:

u(ct , lt ) = ln ct + µ ln(1 − lt ),

(6)

where ct is consumption, lt is labor supply, and µ ≥ 0 is the parameter of the preference for leisure. In this economy, there are two assets: capital and money. Therefore, the household’s real budget constraint is:

kt + mt = rt kt−1 + wt lt +

mt−1 + τt − ct , 1 + πt

(7)

where kt is capital and mt is money held by a household. We suppose that households face the CIA constraint as follows:

ct ≤

mt−1 + τt . 1 + πt

(8)

Households have the preference of quasi-geometric discounting. Therefore, their time utility is:

Ut = u(ct , lt ) + β

∞ ∑

δ s−t u(cs , ls ), (β > 0, 0 < δ < 1)

(9)

s=t+1

where β is the parameter of present bias and δ is the discount factor. Similar to KKS, we assume that households cannot commit their future actions, which is discussed in detail in the next section.

3

Optimization of households and Equilibrium

In this section, we discuss optimization of households and the market equilibrium. We use dynamic programming to obtain the solution. Before we solve the household’s problem, we explain notations. Subscripts for all of the variables are omitted, such as l. “ ′ ” has been added to the superscript of the stock variable, which is accumulated in the next period, such as k ′ . From this point, we discuss households’ optimization problem. Households choose their ¯ ¯l), wage w(k, ¯ ¯l), the behavior, c, l, k ′ , and m′ , by assuming that the rental price of capital r(k, ¯ m, ¯ m), inflation rate π(k, ¯ ¯l), the dynamics of aggregate capital holdings k¯′ = G(k, ¯ the dynamics 4

of money m ¯t =

1+θ ¯ t−1 , 1+πt m

¯ m) and aggregate labor supply ¯l = J(k, ¯ depend on the aggregate

¯ labor supply ¯l, and the government’s policy θ, m capital k, ¯ are given. We also assume that the ¯ m, ¯ m, household behaves as given them future decision rules k ′ = g(k, m, k, ¯ ¯l), m′ = h(k, m, k, ¯ ¯l), ¯ m, and l = j(k, m, k, ¯ ¯l), and that the CIA constraint is binding. Here, we seek an equilibrium in which G, g, h, J, and j are time-invariant. The current self’s problem is: [ ( ) ] m ′ ′ ¯′ ′ ¯ V0 (k, m, k, m) ¯ = max ln + τ + µ ln(1 − l) + βδV (k , m , k , m ¯ ) l,k′ ,m′ 1+π

(10)

¯ ¯l)k + w(k, ¯ ¯l)l, s.t. k ′ + m′ = r(k,

(11)

¯ m) where V (k, m, k, ¯ is the value function after a period. The current self believes that the future self commits to adopting the decision rules g, h, and j after a period. Therefore, V is defined by: ¯ m) V (k, m, k, ¯ = = ln

(

∞ ∑

δ s−t u(cs , ls )

s=t

m +τ 1+π

)

¯ m, ¯ m, ¯ m, + µ ln(1 − j(k, m, k, ¯ ¯l)) + δV (g(k, m, k, ¯ ¯l), h(k, m, k, ¯ ¯l), k¯′ , m ¯ ′ ). (12)

The definition of equilibrium is as follows. Definition 1. A recursive competitive equilibrium for this economy consists of decision rules ¯ m, ¯ m, ¯ m, ¯ m), g(k, m, k, ¯ ¯l), h(k, m, k, ¯ ¯l), and j(k, m, k, ¯ ¯l), a value function V (k, m, k, ¯ pricing func¯ ¯l), w(k, ¯ ¯l) and π(k, ¯ m, ¯ m), tions r(k, ¯ ¯l), law of motion for aggregate capital k¯′ = G(k, ¯ the government’s policy m ¯′ =

1+θ ¯ 1+π m,

¯ m) and aggregate labor supply ¯l = J(k, ¯ such that:

¯ m), ¯ m), 1. given V (k, m, k, ¯ G(k, ¯ m ¯′ =

1+θ ¯ 1+π m,

¯ m), ¯ m, ¯ m, and J(k, ¯ the rules g(k, m, k, ¯ ¯l), h(k, m, k, ¯ ¯l),

¯ m, and j(k, m, k, ¯ ¯l) solve the optimization problem (10); ¯ m, ¯ m, ¯ m, ¯ m), 2. given g(k, m, k, ¯ ¯l), h(k, m, k, ¯ ¯l), j(k, m, k, ¯ ¯l), G(k, ¯ m ¯′ =

1+θ ¯ 1+π m,

¯ m), and J(k, ¯

¯ m) the function V (k, m, k, ¯ satisfies (12); ¯ ¯l) = αAk¯α−1 ¯l1−α , w(k, ¯ ¯l) = (1 − α)Ak¯α ¯l−α ; and 3. r(k, ¯ m, ¯ m, ¯ m), ¯ m, ¯ m, ¯ m, ¯ m, ¯ m). 4. g(k, ¯ k, ¯ ¯l) = G(k, ¯ h(k, ¯ k, ¯ ¯l) = m ¯ ′ and j(k, ¯ k, ¯ ¯l) = J(k, ¯ From this definition, we obtain the following proposition. 5

Proposition 1. For our economy, the recursive competitive equilibrium is given by: ¯ m) ¯ 1. V (k, m, k, ¯ = B + ln(m + θm) ¯ − ln m ¯ + D ln k¯ + E ln(k + F k) where D =

α(1−δ)−(1−αδ)(µ+δ) ,E (1−αδ)(1−δ)

=

¯ = sAk¯α J(k, ¯ m) 2. G(k) ¯ 1−α where s = ¯ m) J(k, ¯ = ¯l∗ =

µ+δ 1−δ ,

F =

βδ(1−αδ)+µ{[1−δ(1−β)]−αβδ} (1 αβδ(1−δ)

αβδ(δ+µ) βδ+µ[1−δ(1−β)] (this

+ θ) − 1

is the saving rate) and

1−α . 1−α+ 1+θ µ(1−s) βδ

Proof. See Appendix A. The remarkable point in this result is the saving rate s, which differs from the saving rate αβδ ( 1−δ(1−β) ) in KKS. The difference is obtained by combining quasi-geometric discounting and the

CIA constraint.

4

Welfare Properties

¯ m) ¯ as the welfare function from section 4.2 of KKS We define the value function V0 (k, m, k, because households maximize this function in a competitive equilibrium. By maximizing the value function as in Appendix B, we obtain the saving rate, sop , and the labor supply, lop , which maximizes the value function as follows: αβδ (1 − αδ)[1 − δ(1 − β)] + αβδ 1−α ¯lop = . 1 − α + µ(1 − sop )

sop =

(13) (14)

We compare the saving rate of the competitive equilibrium and the optimal saving rate, and obtain the following proposition: Proposition 2. The saving rate is excessive when β < (>)1 if µ < (>) 1−αδ α . Proof. Since sop − s =

αβδ 2 (1−δ)(1−β)[αµ−(1−αδ)] {(1−αδ)[1−δ(1−β)]+αβδ}{βδ+µ[1−δ(1−β)]} ,

we find s > sop when β < (>)1

if µ < (>) 1−αδ α . From this proposition, we find that over-saving occurs when households have a present bias (β < 1) and a weak preference for leisure. This is why the CIA constraint plays a role of commitment in the model. Households with quasi-geometric discounting have two incentives which are consuming more in the current period and delaying saving in the future period. 6

The desire of delaying saving in the future period is realized by accumulating less money. By the CIA constraint, the cash holding is determined in the last period and households cannot change it in the current period. Therefore accumulating less money causes small consumption. In some cases, the incentive, which is accumulating less money, is stronger than that of the overconsumption induced by present bias. Then, we generate high saving rate even in a model with quasi-geometric discounting. To understand this outcome in detail, we consider the case in which labor supply is not determined endogenously, µ = 0. In this case, every household at any time cannot decide on the present consumption and decides only to invest capital and money. This implies that households choose the amount of consumption in the next period and in subsequent periods. This decision is equivalent to maximizing only the second term of (9). Therefore, in this case, the saving rate is s = αδ, which is the same as when β = 1 and is larger than the case without the CIA constraint. Although households at any time succeed to make themselves save more in the future, they are forced not to consume more, which is why over-saving occurs. The saving rate is excessive when 0 < µ <

1−αδ α

for almost the same reason.

A household’s decision making is affected by present bias because the household cannot commit future labor supply. However, the effect of β is limited when µ is small. In this case, labor supply and income do not decrease as much. Therefore, the saving rate is excessive.

5

Monetary policy

In this section, we discuss monetary policy, that is controlling the growth rate of money, θ, by ¯ m) the government. We define the value function V0 (k, m, k, ¯ as the welfare function as in section 4. Therefore, the government maximizes the following equation as given households’ response, ¯ and the accumulated money, m, the accumulated capital, k, ¯ in past time: ¯ m) V0 (k, ¯ = ln(1 − s)Ak¯α (¯l∗ )1−α + µ ln(1 − ¯l∗ ) + βδV (k¯′ , m ¯ ′ ).

7

(15)

From Proposition 1, we find that θ affects only labor supply. Therefore, the optimal monetary policy is maximizing the following function:

v(θ) = [1−δ(1 − β)]µ ln(1 + θ) { } [ ] 1−α 1+θ − [1 − δ(1 − β)]µ + [1 − δ(1 − β) + αβδ] ln 1 − α + µ(1 − s) . 1 − αδ βδ (16) From this equation, we obtain the optimal monetary policy as follows: θ∗ = βδ

1 − sop − 1. 1−s

Since the second term of (16), [1 − δ(1 − β)]µ +

1−α 1−αδ [(1 − δ(1 − β)) + αβδ],

(17)

is positive, we find

that this function is concave. Therefore, θ¯ which satisfies v ′ (θ) = 0 is optimal. When there is no present bias, β = 1, the optimal growth rate of money is θ∗ |β=1 = δ − 1. Using this equation and (17), we obtain: θ∗ − θ∗ |β=1 = (β − 1)

βδ(1 − αδ)[1 − αδ(1 − δ)] + µ{(1 − δ)(1 − αδ)[1 − δ(1 − β)] + βδ[1 − αδ{1 − α(1 − δ)}]} . {1 − δ(1 − β)[1 − α(1 − δ)]}{µ[1 − δ(1 − β) + αβδ] + βδ(1 − α)} (18)

This implies that the optimal growth rate of money when there is present bias, β < 1, is smaller than that when there is no present bias. This is why there is no commitment device for labor supply by households. If β < 1, households prefer the present leisure more than a future one. When there is no commitment device for labor supply as in this model, current households cannot stop increasing leisure households in the future. Therefore, households supply smaller labor when they have present bias than when they do not have it. As a result, if β is small, the labor supply may be too small. From Proposition 1, the decreasing growth rate of money increases labor supply. This effect is the same as that pointed out by Walsh (2010, Ch.3) Therefore, the lower growth rate of money is optimal.

8

6

Conclusion

This paper studies a general equilibrium model in which households have a preference of quasigeometric discounting and face a CIA constraint. This paper makes two contributions. First, we show that over-saving can occur although households have present bias. This is why the CIA constraint plays a role of commitment in the model. Households with quasi-geometric discounting have an incentive to make themselves after a period save more. Therefore, they want to accumulate lesser money so as not to consume much in future. The CIA constraint can make this successful because the cash holding is determined in the last period and households cannot change it in the current period when households face it. Since money accumulated is too little in our model, households consume too little. Therefore, we generate high savings rate even in a model with quasi-geometric discounting. Second, we obtain the following result. The optimal growth rate of money when there is present bias is smaller than the growth rate when there is no present bias. This is why there is no commitment device for labor supply by households, in contrast to the case of consumption. Then, households increase the present leisure more, and labor supply of households will be too small. Because the decreasing growth rate of money increases labor supply, the lower growth rate of money is optimal for increasing labor supply. However, these results imply that the government can improve the welfare of households more by taxing capital. In our setting, monetary policy cannot affect the saving rate. If the government taxes capital income or accumulation, it can improve welfare by decreasing the saving rate when over-saving occurs. Although this extension is beyond the scope of the present paper, it is worth exploring in the future.

9

References [1] Boulware, K., David, R., Robert R., Ume. E., 2013. Time inconsistency and the long-run effects of inflation. Economics Letters. 120 267-270. [2] Gong, L., Zhu, S., 2009. Hyperbolic Discounting in a Stochastic Monetary Growth Model. Mimeo. [3] Graham, L., Snower, D. J., 2008. Hyperbolic Discounting and the Phillips Curve. Journal of Money, Credit and Banking. 40(2-3), 428-448. [4] Graham, L., Snower, D. J., 2013. Hyperbolic Discounting and Positive Optimal Inflation. Macroeconomic Dynamics 17(3),591-620. [5] Hiraguchi R., 2014. On the neoclassical growth model with non-constant discounting. Economics Letter. 125, 175-178. [6] Hiraguchi R., 2016. On a two-sector endogenous growth model with quasi-geometric discounting. Journal of Mathematical Economics. 65, 26-35. [7] Krusell, P., Kuruscu, B., Smith, A., 2000. Tax-Policy with Quasi-Geometric Discounting. International Economic Journal 14-3. [8] Krusell, P., Kuruscu, B., Smith, A., 2002. Equilibrium welfare and government policy with quasi-geometric discounting. Journal of Economic Theory 105, 42-72. [9] Ojima, T., 2018. General Equilibrium Dynamics with Naive and Sophisticated Hyperbolic Consumers in an Overlapping Generations Economy. Economica. 85, 281-304. [10] Salois, M. J., Moss, C. B., 2011. A direct test of hyperbolic discounting using market asset data. Economics Letters. 112, 290-292. [11] Walsh, C., E., 2010 Monetary Theory and Policy-Third Edition. Ch3 The MIT Press.

10

Appendix A

Proof of Proposition1

¯ m) We solve this problem by the Guess and Verify method. The value function V (k, m, k, ¯ is guessed as follows: ¯ m) ¯ V (k, m, k, ¯ = B + ln(m + θm) ¯ − ln m ¯ + D ln k¯ + E ln(k + F k),

(A1)

where B, D, E, and F are constant. Since we assume that (8) is binding, we obtain the first-order conditions with respect to k ′ , m′ , and l as follows: βδE −λ=0 + F k¯′ βδ −λ=0 m′ + θm ¯′ µ − + λw = 0, 1−l k′

(A2) (A3) (A4)

where λ is the Lagrangian multiplier of (11). Next, we discuss the market equilibrium. First, we discuss the asset markets. From (A2) and (A3), we have: k ′ = E(m′ + θm ¯ ′ ) − F k¯′ .

(A5)

This equation substitutes the budget constraint (11), and we obtain: m′ =

In the equilibrium, c¯ =

1+θ ¯ 1+π m

1 (rk + wl + F k¯′ − Eθm ¯ ′ ). 1+E

(A6)

because (8) is binding. The right-hand side of this equation equals

¯ the right-hand side of equation (4). Therefore, c¯ = m ¯ ′ . By the assumption, m = m ¯ and k = k. From the above, goods market clearing condition (1), rental price of capital (2), and wage (3), we obtain: m ¯′ =

1+F Ak¯α ¯l1−α . 1 + F + E(1 + θ)

11

(A7)

Moreover, from (A7) and (4), we obtain: 1+F Ak¯α ¯l1−α 1 = 1+π [1 + F + E(1 + θ)](1 + θ) m ¯

(A8)

Next, we discuss the investment for capital. Since k¯′ = Ak¯α ¯l1−α − c¯ = Ak¯α ¯l1−α − m ¯ ′ from (1) and (8), we obtain: ¯ m) k¯′ = G(k, ¯ = sAk¯α ¯l1−α

(A9)

E(1 + θ) . 1 + F + E(1 + θ)

(A10)

where s ≡

Next, we discuss the labor market. From (A3) and (A4), we have:

1−l =

µ (m′ + θm ¯ ′ ). βδw

(A11)

From (A6), (A7), and (A9), we obtain: 1 m + θm ¯ = 1+E ′

′

{

} F E(1 + θ) + θ(1 + F ) ¯α ¯1−α rk + wl + Ak l . 1 + F + E(1 + θ)

(A12)

From this equation and (A11), we obtain: { [ ]} 1 F E(1 + θ) + θ(1 + F ) ¯α ¯1−α wl = βδ(1 + E)w − µ rk + Ak l . βδ(1 + E) + µ 1 + F + E(1 + θ)

(A13)

We substitute (2) and (3) into this equation and obtain: ¯l∗ =

βδ(1 − α)[1 + F + E(1 + θ)] . βδ(1 − α)[1 + F + E(1 + θ)] + µ(1 + θ)(1 + F )

(A14)

Moreover, (A13) and (A14) substitute (A12), and we obtain: βδr m + θm ¯ = βδ(1 + E) + µ ′

′

{

[

] } [(1 + E)βδ + µ](1 + F )(1 + θ) k+ − 1 k¯ αβδ[1 + F + E(1 + θ)]

(A15)

By the definition of equilibrium, policy functions satisfy (12). Therefore, the following equation

12

is satisfied: ¯ B + ln(m + θm) ¯ − ln m ¯ + D ln k¯ + E ln(k + F k) = δ[B + D ln s + E ln E] + (1 − δ) ln(1 − s) + [1 + δ(D + E)] ln A + µ[ln µ − ln(1 − α)] + δ(1 + E) ln βδ + (µ + δ + δE) ln α − ln(1 + θ) + {αµ + (1 − α)[1 + µ + δ(D + E)]} ln ¯l + ln(m + θm) ¯ − ln m ¯ + [α − δ − (1 − α)(µ + δE) + αδD − αµ] ln k¯ { [ ] } [(1 + E)βδ + µ](1 + F )(1 + θ) + (µ + δ + δE) ln k + − 1 k¯ . αβδ[1 + F + E(1 + θ)]

(A16)

From this equation, we obtain:

E = µ + δ + δE → E =

µ+δ 1−δ

(A17)

D = α − δ − (1 − α)(µ + δE) + αδD − αµ → D =

α(1 − δ) − (1 − αδ)(µ + δ) (1 − αδ)(1 − δ)

[(1 + E)βδ + µ](1 + F )(1 + θ) −1 αβδ[(1 + F ) + E(1 + θ)] βδ[1 − αδ] + µ{[1 − δ(1 − β)] − αβδ} (1 + θ) − 1. →F = αβδ(1 − δ)

(A18)

F =

(A19)

Finally, these equations substitute (A10) and (A14), and we obtain:

s=

αβδ(µ + δ) βδ + µ[1 − δ(1 − β)]

(A20)

(1 − α)βδ[βδ + µ(1 − δ(1 − β))] (1 − α)βδ[βδ + µ(1 − δ(1 − β))] + µ(1 + θ){βδ[1 − αδ] + µ[(1 − δ(1 − β)) − αβδ]} 1−α = . (A21) 1 − α + 1+θ βδ µ(1 − s)

¯l∗ = J(k, ¯ m) ¯ =

We find that these equations are consistent with the equations in Proposition 1.

13

B

Derivation of (13) and (14)

¯ m) By definition, the future value function V (k, m, k, ¯ must satisfy the following equation: ¯ m) V (k, m, k, ¯ =

∞ ∑

δ

s−t

u(cs , ls ) =

s=t

∞ ∑

δ

s−t

ln cs + µ

s=t

∞ ∑ s=t

δ s−t ln(1 − ls ).

(B1)

In this paper, the saving rate and labor supply are constant at any time. Therefore, the second term of this equation can be written as follows:

µ

∞ ∑ s=t

δ s−t ln(1 − ls ) =

µ ln(1 − l). 1−δ

(B2)

Since ct = (1 − s)Aktα lt1−α , the first term is written as follows: ∞ ∑ s=t

α

α

δ s−t ln cs = ln(1 − s)Ak α l1−α + δ ln(1 − s)Ak ′ l1−α + δ 2 ln(1 − s)Ak ′′ l1−α + · · · =

1−α 1 1 ′ 2 ′′ ln(1 − s) + ln l + ln A + α(ln | k + δ ln k + {z δ ln k + · ·}·). 1−δ 1−δ 1−δ ≡K

(B3)

I define the final term as K, which can be expressed as: K = ln k + δ ln sAk α l1−α + δ 2 ln sAk ′α l1−α + δ 3 ln sAk ′′α l1−α + · · · =

1−α δ δ ′ 2 ′′ ln s + δ ln l + ln A + ln k + αδ(ln | k + δ ln k + {z δ ln k + · ·}·). 1−δ 1−δ 1−δ

(B4)

=K

Therefore,

1 K= 1 − αδ

{

} δ [ln s + (1 − α) ln l + ln A] + ln k . 1−δ

(B5)

This equation substitutes (B3), and we obtain: ∞ ∑ s=t

δ s−t ln cs =

1−α 1 αδ α 1 ln(1 − s) + ln l + ln A + ln l + ln k. 1−δ (1 − δ)(1 − αδ) (1 − δ)(1 − αδ) (1 − δ)(1 − αδ) 1 − αδ (B6)

14

From (B2) and this, ¯ m) V (k, m, k, ¯ =

1 1 αδ α 1−α ln(1 − s) + ln A + ln s + ln k + ln l 1−δ (1 − δ)(1 − αδ) (1 − δ)(1 − αδ) 1 − αδ (1 − δ)(1 − αδ) {z } | ∑ ∞ s−t s=t δ

+

µ

ln cs

µ ln(1 − l) . |1∑− δ {z } ∞ s−t s=t δ

(B7)

ln(1−ls )

¯ m) Since V0 (k, m, k, ¯ = ln c + µ ln(1 − l) + βδV (k ′ , m′ , k¯′ , m ¯ ′ ), we substitute (B7) for this equation and obtain: ¯ m) V0 (k, m, k, ¯ 1 − δ(1 − β) 1 − δ(1 − β) αβδ αβδ ln(1 − s) + ln A + ln s + ln k 1−δ (1 − δ)(1 − αδ) (1 − δ)(1 − αδ) 1 − αδ (1 − α){(1 − αδ)[1 − δ(1 − β)] + αβδ} µ[1 − δ(1 − β)] + ln l + ln(1 − l). (B8) (1 − δ)(1 − αδ) 1−δ =

s and l, which maximize this equation, are (13) and (14).

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