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Global Dynamics and Optimal Policy in the Ak Model with Anticipated Future Consumption

  • Manuel A. Gomez EMAIL logo
Veröffentlicht/Copyright: 13. März 2024
The B.E. Journal of Theoretical Economics
Aus der Zeitschrift The B.E. Journal of Theoretical Economics Band 24 Heft 2

Abstract

This paper analyzes the Ak model with anticipated future consumption. In the model with internal anticipation individual’s utility depends on current consumption and a forward-looking reference level which is formed from individual’s own future consumption. The global dynamics of the economy is characterized by means of a qualitative phase diagram analysis. In the model with external anticipation the consumption reference level is formed from economy-wide average future consumption, which is taken as given by individuals and causes the competitive equilibrium to be inefficient. Characterizing the global dynamics of the economy is required to characterize an optimal fiscal policy that corrects the inefficiency brought about by this external effect.

Keywords: anticipated consumption; dynamics; phase diagram analysis; optimal fiscal policy
JEL Classification: O41; H21
Corresponding author: Manuel A. Gomez, Department of Economics, Facultad de Economía y Empresa, Universidade da Coruña, Campus de Elviña, A Coruna, 15071, Spain; and ECOBAS, Universidade da Coruña, C+D Group, Departamento de Economía, 15071 A Coruña, España, E-mail:

Funding source: MCIN/AEI/10.13039/501100011033

Award Identifier / Grant number: PID2021-127599NB-I00

Funding source: European Regional Development Fund A way of making Europe

Award Identifier / Grant number: PID2021-127599NB-I00

Acknowledgments

I gratefully acknowledge the helpful comments of an anonymous referee. Grant PID2021-127599NB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”.

Appendix A: The Condition sign det H u ( c , 1 ) = constant

To give an intuition on the meaning of condition P#6, let us note definition (14),

π ( c ) ≡ u A ( c , 1 ) u C ( c , 1 ) = u A ( C , A ) u C ( C , A ) ≡ d C d A u = c o n s t a n t ,

so −π(c) is the slope of the indifference curves of u depicted in the (A, C)–plane, and its absolute value is the marginal rate of substitution (MRS) of A for C. Differentiating π(c) we obtain

π ′ ( c ) = u C A ( c , 1 ) u C ( c , 1 ) − u C C ( c , 1 ) u A ( c , 1 ) u C ( c , 1 ) 2 = u C C ( c , 1 ) u A A ( c , 1 ) − u C A ( c , 1 ) 2 v u C ( c , 1 ) 2 ,

so that sign π ′ ( c ) = sign det H u ( c , 1 ) , where we have used that

u A ( c , 1 ) = − c u C A ( c , 1 ) + u A A ( c , 1 ) v , u C ( c , 1 ) = − c u C C ( c , 1 ) + u C A ( c , 1 ) v .

We have that

∂ ∂ C u A ( c , 1 ) u C ( c , 1 ) = ∂ c ∂ C d d c u A ( c , 1 ) u C ( c , 1 ) = 1 A π ′ ( c ) , ∂ ∂ A u A ( c , 1 ) u C ( c , 1 ) = ∂ c ∂ A d d c u A ( c , 1 ) u C ( c , 1 ) = − 1 A c π ′ ( c ) .

Hence, the convexity of the indifference curves can be studied from

d 2 C d A 2 = − ∂ ∂ C u A ( c , 1 ) u C ( c , 1 ) d C d A − ∂ ∂ A u A ( c , 1 ) u C ( c , 1 ) = 1 A π ′ ( c ) c + π ( c ) ,

where P#3 entails that c + π(c) > 0, so sign d 2 C / d A 2 = sign π ′ ( c ) .

If u A > 0, so both C and A are ‘goods’, the indifference curves are downward sloping and strictly convex (resp., concave) if π′(c) > 0 (resp., π′(c) < 0), and so, an increase in C, keeping A constant, causes the indifference curves to become steeper (flatter), and an increase in A, keeping C constant, causes the indifference curves to become flatter (steeper). If u A < 0, so A is a bad, the indifference curves are upward sloping and strictly concave (resp., convex) if π′(c) < 0 (resp., π′(c) > 0), and so, an increase in C, keeping A constant, causes the indifference curves to become flatter (steeper), and an increase in A, keeping C constant, causes the indifference curves to become steeper (flatter). Thus, the higher is the value of C, the higher is the increase in C needed to compensate a given increase in A to keep utility constant. Hence, the more meaningful situation seems to be that π′(c) > 0 if u A > 0 and π′(c) < 0 if u A < 0. The assumption P#6 that sign det H u ( c , 1 ) = sign π ′ ( c ) is constant for all c simply states that the effect of an increase of C or A on the MRS goes always in the same direction so that the indifference curves are either strictly concave or strictly convex.

To illustrate the former discussion, let us consider the CES utility function

u ( C , A ) = 1 1 − ϵ C ϕ + γ A ϕ 1 + γ ( 1 − ϵ ) / ϕ , 0 < ϕ < 1 , ϵ > 1 − ϕ , ϵ ≠ 1 , γ > − 1 ,

which tends to the additive specification

u ( C , A ) = 1 1 − ϵ C + γ A 1 + γ 1 − ϵ

as ϕ → 1, and to the multiplicative specification

u ( C , A ) = ( C A γ ) ( 1 − ϵ ) / ( 1 + γ ) 1 − ϵ

as ϕ → 0. This specification implies that

u C ( C , A ) = C ϕ − 1 1 + γ C ϕ + γ A ϕ 1 + γ ( 1 − ϵ − ϕ ) / ϕ , u A ( C , A ) = γ A ϕ − 1 1 + γ C ϕ + γ A ϕ 1 + γ ( 1 − ϵ − ϕ ) / ϕ , π ( c ) = u A ( c , 1 ) u C ( c , 1 ) = γ c 1 − ϕ ,

and so, sign π ′ ( c ) = sign ( γ ) = sign ( u A ) . Hence, in the altruistic specification γ > 0 we have that π′(c) > 0, the utility function is strictly concave and the indifference curves are strictly decreasing and strictly convex. In the jealousy specification γ < 0 we have that π′(c) < 0 and the indifference curves are strictly increasing and strictly concave.

Appendix B: Dynamics of the Centrally Planned Economy

From (6) we have that

(B.1) λ = u C ( C , A ) / ( 1 + ρ q ) ,

(B.2) μ = q u C ( C , A ) / ( 1 + ρ q ) .

Using that q ̇ / q = μ ̇ / μ − λ ̇ / λ , taking into account (7) and (8), together with (B.2) and the homogeneity of degree −v of u C and u A , we get Eq. (11).

Log-differentiating (B.1), after simplification we get

C ̇ C = u C ( C , A ) C u C C ( C , A ) λ ̇ λ − A u C A ( C , A ) u C ( C , A ) A ̇ A + ρ q ̇ 1 + ρ q .

Now, using (3) and (7) and the homogeneity of degree −v of u C , we can obtain that

(B.3) C ̇ C = σ ( c ) B − v A ̇ A − ρ q ̇ 1 + ρ q − β + A ̇ A ,

where σ(c) is defined in (13). From c ̇ / c = C ̇ / C − A ̇ / A , using (B.3) and (3), we immediately get Eq. (10). Finally, Eq. (12) is obtained using that a ̇ / a = A ̇ / A − K ̇ / K , taking into account (3) and (5).

Appendix C: Dynamics of the Market Economy with Government

The agent maximizes her utility (1) subject to the budget constraint (27), taking as given C ̃ and the initial condition on capital. The Hamiltonian of this problem is

H = u ( C , A ) + λ D ( 1 − τ Y ) B K − ( 1 + τ C ) C + S ,

where λ D is the shadow value of capital. The first-order conditions for an interior optimum, which are also sufficient, are

(C.1) ∂ H ∂ C = 0   ⇒   u C ( C , A ) = ( 1 + τ C ) λ D ,

(C.2) ∂ H ∂ K = β λ D − λ ̇ D   ⇒   ( 1 − τ Y ) B = β − λ ̇ D / λ D ,

together with the initial condition, K(0) = K 0, and the usual transversality condition lim t→∞e−βt λ D K = 0. From (27) and (28), we get the resources’ constraint

(C.3) K ̇ = B K − C .

In what follows we will use that C ̃ = C in a symmetric equilibrium. Log-differentiating (C.1), using (C.2), we get the Euler equation

C ̇ C = − u C ( C , A ) C u C C ( C , A ) ( 1 − τ Y ) B − τ ̇ C 1 + τ C + u C A ( C , A ) u C ( C , A ) A ̇ − β .

Denoting c ≡ C/A and a ≡ A/K, and using the homogeneity of degree −v of u C , this equation can be rewritten as

(C.4) C ̇ C = σ ( c ) ( 1 − τ Y ) B − τ ̇ C 1 + τ C − v u C ( c , 1 ) + c u C C ( c , 1 ) u C ( c , 1 ) A ̇ A − β = σ ( c ) ( 1 − τ Y ) B − τ ̇ C 1 + τ C − v A ̇ A − β + A ̇ A ,

where σ(c) is defined in (13).

From c ̇ / c = C ̇ / C − A ̇ / A , using (C.4) and (3), we get (29), and from a ̇ / a = A ̇ / A − K ̇ / K , using (C.3) and (3), we get (30).

References

Abel, A. B. 1990. “Asset Prices Under Habit Formation and Catching Up with the Joneses.” The American Economic Review 80 (2): 38–42.10.3386/w3279Suche in Google Scholar

Bambi, M., and F. Gozzi. 2020. “Internal Habits Formation and Optimality.” Journal of Mathematical Economics 91: 165–72. https://doi.org/10.1016/j.jmateco.2020.09.008.Suche in Google Scholar

Baucells, M., and S. Bellezza. 2016. “Temporal Profiles of Instant Utility During Anticipation, Event, and Recall.” Management Science 63 (3): 729–48. https://doi.org/10.1287/mnsc.2015.2362.Suche in Google Scholar

Becker, G. S., and K. M. Murphy. 1988. “A Theory of Rational Addiction.” Journal of Political Economy 96 (4): 675–700. https://doi.org/10.1086/261558.Suche in Google Scholar

Caplin, A., and J. Leahy. 2001. “Psychological Expected Utility Theory and Anticipatory Feelings.” Quarterly Journal of Economics 116 (1): 55–79. https://doi.org/10.1162/003355301556347.Suche in Google Scholar

Carroll, C. D., J. Overland, and D. N. Weil. 1997. “Comparison Utility in a Growth Model.” Journal of Economic Growth 2 (4): 339–67. https://doi.org/10.1023/a:1009740920294 10.3386/w6138Suche in Google Scholar

Carroll, C. D., J. Overland, and D. N. Weil. 2000. “Saving and Growth with Habit Formation.” The American Economic Review 90 (3): 341–55. https://doi.org/10.1257/aer.90.3.341.Suche in Google Scholar

Chatterjee, B., R. Escobar-Posada, and G. Monteiro. 2023. “Anticipation in Leisure – Effects on Labor-Leisure Choice.” International Journal of Economic Theory 19 (2): 384–412. https://doi.org/10.1111/ijet.12358.Suche in Google Scholar

Christiano, L. J., M. Eichenbaum, and C. L. Evans. 2005. “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy.” Journal of Political Economy 113 (1): 1–45. https://doi.org/10.1086/426038.Suche in Google Scholar

Chun, H. H., K. Diehl, and D. J. MacInnis. 2017. “Savoring an Upcoming Experience Affects Ongoing and Remembered Consumption Enjoyment.” Journal of Marketing 81 (3): 96–110. https://doi.org/10.1509/jm.15.0267.Suche in Google Scholar

Constantinides, G. M. 1990. “Habit Formation: A Resolution of the Equity Premium Puzzle.” Journal of Political Economy 98 (3): 519–43. https://doi.org/10.1086/261693.Suche in Google Scholar

Faria, J. R., and P. McAdam. 2013. “Anticipation of Future Consumption: A Monetary Perspective.” Journal of Money, Credit, and Banking 45 (2–3): 423–47. https://doi.org/10.1111/jmcb.12008.Suche in Google Scholar

Faria, J. R., and P. McAdam. 2018. “The Green Golden Rule: Habit and Anticipation of Future Consumption.” Economics Letters 172: 131–3. https://doi.org/10.1016/j.econlet.2018.08.026.Suche in Google Scholar

Fuhrer, J. C. 2000. “Habit Formation in Consumption and its Implications for Monetary-Policy Models.” The American Economic Review 90 (3): 367–90. https://doi.org/10.1257/aer.90.3.367.Suche in Google Scholar

Gilovich, T., A. Kumar, and L. Jampol. 2015. “A Wonderful Life: Experiential Consumption and the Pursuit of Happiness.” Journal of Consumer Psychology 25 (1): 152–65. https://doi.org/10.1016/j.jcps.2014.08.004.Suche in Google Scholar

Gómez, M. A. 2008. “Convergence Speed in the Ak Endogenous Growth Model with Habit Formation.” Economics Letters 100 (1): 16–21. https://doi.org/10.1016/j.econlet.2007.10.022.Suche in Google Scholar

Gómez, M. A. 2021. “On the Closed-Form Solution of an Endogenous Growth Model with Anticipated Consumption.” Journal of Mathematical Economics 95: 102471. https://doi.org/10.1016/j.jmateco.2021.102471.Suche in Google Scholar

Gómez, M. A. 2022. “The Good, the Bad and the Worse: Current, Past and Future Consumption Externalities and Equilibrium Efficiency.” Journal of Economics 137: 195–228. https://doi.org/10.1007/s00712-022-00788-6.Suche in Google Scholar

Gómez, M. A., and G. Monteiro. 2022. “Anticipated Future Consumption in an Endogenous Growth Model.” Macroeconomic Dynamics 26 (5): 1202–38. https://doi.org/10.1017/s136510052000053x.Suche in Google Scholar

Guo, L. 2020. “Anticipatory Consumptions.” Management Science 66 (8): 3717–34. https://doi.org/10.1287/mnsc.2019.3367.Suche in Google Scholar

Hardisty, D. J., and E. U. Weber. 2020. “Impatience and Savoring vs. Dread: Asymmetries in Anticipation Explain Consumer Time Preferences for Positive vs. Negative Events.” Journal of Consumer Psychology 30 (4): 598–613. https://doi.org/10.1002/jcpy.1169.Suche in Google Scholar

Harris, C. R. 2012. “Feelings of Dread and Intertemporal Choice.” Journal of Behavioral Decision Making 25 (1): 13–28. https://doi.org/10.1002/bdm.709.Suche in Google Scholar

He, Y., J. S. Dyer, and J. C. Butler. 2019. “A Decision-Making Model with Utility from Anticipation and Disappointment.” Journal of Multi-Criteria Decision Analysis 26 (1–2): 35–50. https://doi.org/10.1002/mcda.1657.Suche in Google Scholar

Kőszegi, B. 2006. “Emotional Agency.” Quarterly Journal of Economics 121 (1): 121–55. https://doi.org/10.1162/003355306776083554.Suche in Google Scholar

Köszegi, B., and M. Rabin. 2006. “A Model of Reference-Dependent Preferences.” Quarterly Journal of Economics 121 (4): 1133–65. https://doi.org/10.1162/qjec.121.4.1133.Suche in Google Scholar

Köszegi, B., and M. Rabin. 2007. “Reference-Dependent Risk Attitudes.” The American Economic Review 97 (4): 1047–73. https://doi.org/10.1257/aer.97.4.1047.Suche in Google Scholar

Kumar, A., M. A. Killingsworth, and T. Gilovich. 2014. “Waiting for Merlot: Anticipatory Consumption of Experiential and Material Purchases.” Psychological Science 25 (10): 1924–31. https://doi.org/10.1177/0956797614546556.Suche in Google Scholar

Kuznitz, A., S. Kandel, and V. Fos. 2008. “A Portfolio Choice Model with Utility from Anticipation of Future Consumption and Stock Market Mean Reversion.” European Economic Review 52 (8): 1338–52. https://doi.org/10.1016/j.euroecorev.2008.07.002.Suche in Google Scholar

Leitmann, G. 1981. “The Calculus of Variations and Optimal Control.” In Mathematical Concepts and Methods in Science and Engineering, Vol. 24. New York: Springer Science+Business Media.10.1007/978-1-4899-0333-4Suche in Google Scholar

Loewenstein, G. 1987. “Anticipation and the Valuation of Delayed Consumption.” Economic Journal 97 (387): 666–84. https://doi.org/10.2307/2232929.Suche in Google Scholar

Loewenstein, G., and J. Elster, eds. 1992. Choice Over Time. New York: Rusell Sage Foundation.Suche in Google Scholar

Monteiro, G., and S. J. Turnovsky. 2016. “Anticipated Consumption and its Impact on Capital Accumulation and Growth: ‘Forward-Looking’ vs. ‘Backward-Looking’ Consumption Reference.” International Journal of Economic Theory 12 (3): 203–32. https://doi.org/10.1111/ijet.12091.Suche in Google Scholar

Mukherjee, A., R. Smith, and S. Burton. 2023. “The Effect of Positive Anticipatory Utility on Product Pre-Order Evaluations and Choices.” Journal of the Academy of Marketing Science 51 (3): 551–69, https://doi.org/10.1007/s11747-021-00810-1.Suche in Google Scholar

Nowlis, S. M., N. Mandel, and D. B. McCabe. 2004. “The Effect of a Delay Between Choice and Consumption on Consumption Enjoyment.” Journal of Consumer Research 31 (3): 502–10. https://doi.org/10.1086/425085.Suche in Google Scholar

Smets, F., and R. Wouters. 2003. “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area.” Journal of the European Economic Association 1 (5): 1123–75. https://doi.org/10.1162/154247603770383415.Suche in Google Scholar

Van Boven, L., and L. Ashworth. 2007. “Looking Forward, Looking Back: Anticipation is More Evocative than Retrospection.” Journal of Experimental Psychology: General 136 (2): 289–300. https://doi.org/10.1037/0096-3445.136.2.289.Suche in Google Scholar

Wang, X. T. X., P. Wang, J. Lu, J. Zhou, G. Li, and S. Garelik. 2022. “Episodic Future Thinking and Anticipatory Emotions: Effects on Delay Discounting and Preventive Behaviors During Covid-19.” Applied Psychology: Health and Well-Being 14 (3): 842–61. https://doi.org/10.1111/aphw.12350.Suche in Google Scholar
Received: 2023-07-28
Accepted: 2024-02-16
Published Online: 2024-03-13

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Global Dynamics and Optimal Policy in the Ak Model with Anticipated Future Consumption
  4. Offsetting Distortion Effects of Head Starts on Incentives in Tullock Contests
  5. Collusive Price Leadership Among Firms with Different Discount Factors
  6. Motivating Loyal Bureaucrats in Sequential Agency
  7. Disclosure of Product Information After Price Competition
  8. Uncertain Commitment Power in a Durable Good Monopoly
  9. Optimal Trade Policy in a Ricardian Model with Labor-Market Search-and-Matching Frictions
  10. Consumer-Benefiting Transport Costs: The Role of Product Innovation in a Vertical Structure
  11. Information Disclosure by Informed Intermediary in Double Auction
  12. Notes
  13. Strategic Partial Inattention in Oligopoly
  14. The Role of Informative Advertising in Aligning Preferences Over Product Design
  15. To Bequeath, or Not to Bequeath? On Labour Income Risk and Top Wealth Concentration
https://doi.org/10.1515/bejte-2023-0080
Schlagwörter für diesen Artikel
anticipated consumption; dynamics; phase diagram analysis; optimal fiscal policy

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Global Dynamics and Optimal Policy in the Ak Model with Anticipated Future Consumption
  4. Offsetting Distortion Effects of Head Starts on Incentives in Tullock Contests
  5. Collusive Price Leadership Among Firms with Different Discount Factors
  6. Motivating Loyal Bureaucrats in Sequential Agency
  7. Disclosure of Product Information After Price Competition
  8. Uncertain Commitment Power in a Durable Good Monopoly
  9. Optimal Trade Policy in a Ricardian Model with Labor-Market Search-and-Matching Frictions
  10. Consumer-Benefiting Transport Costs: The Role of Product Innovation in a Vertical Structure
  11. Information Disclosure by Informed Intermediary in Double Auction
  12. Notes
  13. Strategic Partial Inattention in Oligopoly
  14. The Role of Informative Advertising in Aligning Preferences Over Product Design
  15. To Bequeath, or Not to Bequeath? On Labour Income Risk and Top Wealth Concentration
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