From a chronological viewpoint, the starting point for modern growth theory is Ramsey's article on the optimal social savings behaviour entitled A Mathematical Theory of Saving. Since Ramsey applied variational calculus* to answer the question how an infinitely lived representative agent should allocate his resource over time, his work--regarded by Keynes as "one of the most remarkable contributions to mathematical economics ever made"(obituary notice of Ramsey)--was "terribly difficult reading for an economist" (Keynes, ibidem). In case you have troubles following Ramsey's arguments, I highly recommend reading "The Ramsey Exercise" over at the History of Economic Thought website. Unfortunately, the site fails to mention that in Ramsey's paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result.

Ramsey's growth model was refined by Cass (1965) and Koopmans (1965). Here is a short sketch (borrowing heavily on Robert Lucas' Lectures on Economic Growth): Consider a closed economy with competitive markets, with identical rational agents and a constant returns technology. At date t there are L(t) persons, the exogeneously given rate of growth of L(t) is n. Real, per capita consumption is a stream c(t) t ≥ 0, of units of a single good. Preferences over consumption streams are given bywhere the discount rate ρ and the coefficient of relativ risk aversion θ are both positive. The elasticity of substitution for this utility funtion is the constant σ = 1/θ. Hence, this form is either called the

The ressource allocation problem faced by this economy is simply to choose a time path c(t) for per capita consumption. Given a path c(t) and an initial capital stock K(0), the technology (2) then implies a time path K(t) for capital. One way to think about this allocation problem is to think of choosing c(t) at each date, given the values of K(t), A(t), and L(t) that have been attained by that date. Evidently, it will not be optimal to choose c(t) to maximize current-period utility, [1/(1 - θ)][c(t) - 1]^(1-θ)L(t), for the choice that achieves this is to set net investment d(K(t)/dt equal to zero: One needs to set some

*AFAIK Evans (1924) was the first to use variational calculus to address an economic issue.

related items: Growth Theory: Stylized Facts

Ramsey's growth model was refined by Cass (1965) and Koopmans (1965). Here is a short sketch (borrowing heavily on Robert Lucas' Lectures on Economic Growth): Consider a closed economy with competitive markets, with identical rational agents and a constant returns technology. At date t there are L(t) persons, the exogeneously given rate of growth of L(t) is n. Real, per capita consumption is a stream c(t) t ≥ 0, of units of a single good. Preferences over consumption streams are given bywhere the discount rate ρ and the coefficient of relativ risk aversion θ are both positive. The elasticity of substitution for this utility funtion is the constant σ = 1/θ. Hence, this form is either called the

*constant-relative-risk-aversion*utility (CRRA) function or the*constant intertemporal elasticity of substitution*(CIES) utility funtion. Production per capita of the one good is divided into consumption c(t) and capital accumulation. If we let K(t) denote the total stock of capital, then total output is L(t)c(t) + dK(t)/dt. Production is assumed to depend on the level of capital and labor inputs and on the level A(t) of the "technology" according towhere 0 <β< 1 and where the exogenously given rate of technological growth is μ>0.The ressource allocation problem faced by this economy is simply to choose a time path c(t) for per capita consumption. Given a path c(t) and an initial capital stock K(0), the technology (2) then implies a time path K(t) for capital. One way to think about this allocation problem is to think of choosing c(t) at each date, given the values of K(t), A(t), and L(t) that have been attained by that date. Evidently, it will not be optimal to choose c(t) to maximize current-period utility, [1/(1 - θ)][c(t) - 1]^(1-θ)L(t), for the choice that achieves this is to set net investment d(K(t)/dt equal to zero: One needs to set some

*value*or*price*on increments to capital. A central construct in the study of optimal allocations, allocations that maximize utility (1) subject to the technology (2), is the current-value Hamiltonian H defined by which is just the sum of current-period utility and (from (2)) the rate of increase of capital, the latter valued at the (shadow) "price" λ(t). An optimal allocation must maximize the expression H at each date t, provided the price λ(t) is correctly chosen. The first-order condition for maximizing H with respect to c iswhich is to say that goods must be so allocated at each date as to be equally valuable, on the margin, used either as consumption or as investment. It is known that the price λ(t) must satisfy at each date t if the solution c(t) is to yield an optimal path. Now if (4) is used to express c(t) as a funtion of λ(t), and this function is substituted in place of c(t) in (2) and (5), these two equations are a pair of first-order differential equations in K(t) and its "price" λ(t). Solving this system, there will be a one-parameter family of paths (K(t), λ(t))), satisfying the given initial condition on K(0). The*unique*member of this family that satisfies the transversality condition:is the optimal path! The next step is to construct from (2), (4), and (5) the system's*balanced growth path*: the particular solution (K(t), λ(t),c(t)) such that the rates of growth of each of these variables is constant. To cut a long story short: The interesting finding is that along a balanced path, the rate of growth of per capita magnitudes (κ, same for capital and consumption)is simply proportional to the given rate of technical change, μ, where the constant of proportionality is the inverse of labor's share, 1 - β. The rate of time preference ρ and the degree of risk aversion θ have no bearing on this long-run growth rate. Low time preference ρ and low risk aversion θ induce a high savings rate s, and high savings is, in turn, associated with relatively high output*levels*on a balanced path. A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster!*AFAIK Evans (1924) was the first to use variational calculus to address an economic issue.

related items: Growth Theory: Stylized Facts

Mahalanobis - am 2005-09-13 09:54 - Rubrik: EconoSchool

Mike Linksvayer (guest) meinte am 13. Sep, 21:10:

more wealth, same growth?

I don't follow the math, (so?) the following seems paradoxical:"A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster!"

In the long run how can a society be wealthier without growing faster? Output is higher in the initial state, so the same rate of growth results in more absolute wealth?

Mahalanobis antwortete am 14. Sep, 01:09:

Balanced Growth Path

We are looking at the **steady state**, so transitional dynamics don't matter. As the economy develops and approaches the steady state many forces act upon the savings rate. On one hand, agents will save less when they are poor due to the form of the utility function (declining marginal utility, consumption smoothing), on the other hand, the return on saving is higher when the capital stock is low - which actually should lead to a higher savings rate in poor economies. How the saving rate (and therefore output growth) develops over time for different countries depends on quite a lot of paramters.

However, the savings rate will be constant in the long-run (but not necessarily the same for different countries) and so will be output growth.

Maybe you have heard of the Solow model (you will find lots of information on the internet) where the savings rate is exogeneously given. Here it makes more sense to ask what happens if the savings rate is incrased by x%. The answer is that output growth would temporarily increase but after a while output growth would be the same as before. So an incrase in the savings rate only has a level effect.

Daniel Lam (guest) antwortete am 14. Sep, 04:08:

Risk Aversion?

Since there is no uncertainty in this model, I'm not sure that it adds anything to interpret theta as the coefficient of relative risk aversion. Its sole effect is that of (the reciprocal of) the elasticity of intertemporal substitution.
Mahalanobis antwortete am 14. Sep, 05:45:

Elasticity of Marginal Utility

Lucas notes in a footnote* that "the coefficient of risk aversion is sometimes called the intertemporal elasticity of substitution. Since all the models considered in this chapter are deterministic, this latter terminology might be more suitable."When thinking about the curvature of the utility function (u(c) → ln(c) as θ → 1 and u(c) → c -1 as θ → 0) I immediately categorize in risk averse, risk neutral, and risk loving. Maybe that's why I also tend to denote θ the coefficient of (relative) risk aversion. I do not spend more time thinking about the form of the utility function. Btw, it has never happend to me that I met a student who could correctly explain the Elasticity of Substitution.

*after denoting θ (actually σ - he uses a different notation) the coefficient of (relative) risk aversion

kimcil (guest) antwortete am 7. Apr, 12:17:

I am preparing a research paper and collecting information on this topic. Your post is one of the better that I have read. Thank you for putting this information into one post.alfamart official partner merchandise fifa piala dunia brazil 2014

Unit Link Terbaik di Indonesia Commonwealth Life Investra Link

begedir antwortete am 4. May, 07:07:

I think there should be an official website for people who want to lodge a protest with the United States government. And to my mind, the strongest protest would be a promise never to come here. Instead of burning our flag or booing our athletes belajar grammar bahasa inggris , ingin hamil , kata kata untuk ibu
Paul N (guest) meinte am 15. Sep, 22:27:

When's Part II being released? I learn more on this blog than I do watching PBS.
asda (guest) antwortete am 26. Feb, 09:22:

I am interested in this subject matter and would like to explore out some more information as my colleague need information on this topicwebsite

radek (guest) meinte am 21. Sep, 02:13:

The fact that the same coefficient captures both elasticity of substitution and risk aversion causes some problems in growth-style models with uncertainty (like capital asset pricing models and Ramsey-based RBC models). And just intuitively, there is no reason to think that a person who is say risk averse, MUST regard consumption in different periods in a particular way. Some of the newer behavioral economics has sought to disentangle the two. I can't remember any papers specifically off the top of my head, but I believe Thaler has some stuff on it.BTW, even though you say you never met a student who could explain elasticity of substitution, you link to the wrong thing. That page is about substitution between capital and labor. Here we're talking about the substitution between consumption today and consumption tomorrow.

maskodok antwortete am 15. Jan, 01:32:

Interesting topic for a blog. I have been searching the Internet for fun and came upon your website. Fabulous post. Thanks a ton for sharing your knowledge! It is great to see that some people still put in an effort into managing their websites. I'll be sure to check back again real soon.Mobil Sedan COrolla,Idrpoker.com agen Texas poker Online Indonesia Terpercaya, Mobil Sedan COrolla, Cipto Junaedy

kurasakisan meinte am 30. Sep, 22:26:

It's my first visit and yet, totally great for first coming. Thank you for prividing useful content. Don't forget to visit some of my profiles like Eterna Profiles - Scoop It - Pocket Knowledge Profile - Cyberlaw Harvard - Hong Kong Protest via cnn
kurasakisan meinte am 30. Sep, 22:27:

It's my first visit and yet, totally great for first coming. Thank you for prividing useful content. Don't forget to visit some of my profiles like Eterna Profiles - Scoop It - Pocket Knowledge Profile - Cyberlaw Harvard - Hong Kong Protest via cnn
jacktoro meinte am 15. Oct, 00:54:

Interesting topic for a blog. I have been searching the Internet for fun and came upon your website. Fabulous post. Thanks a ton for sharing your knowledge! It is great to see that some people still put in an effort into managing their websites. I'll be sure to check back again real soon.Daftar Harga Laptop Terbaru 2014 semoga dengan Daftar Harga Laptop Acer 2014 kamu bisa mencari laptop sesuai budget kamu dan jangan lupa lihat juga Daftar Harga Laptop Lenovo 2014 Terupdate
mena99260mena99260 meinte am 21. Mar, 09:46:

Davidjohn

Der Schwerpunkt unseres Angebots liegt in der Konzeption, Gestaltung und Entwicklung von Websites sowie im Online Marketing. SEO-Optimierung, Social Media Marketing (SMM) oder Responsive Webdesign sind dabei keine Fremdwörter für uns, sondern Teil unserer täglichen Arbeit. Suchmaschinenoptimierung
AsharSeo meinte am 22. Mar, 07:15:

Auf über 300 m2 Fläche nden wir für Ihre Kunstwerke, Bilder und Fotos den passenden Rahmen, mit oder ohne Passepartout. Unsere top ausgestatte Rahmenwerkstatt ermöglicht uns schnell und in höchster Qualität auf jeden Kundenwunsch zu reagieren. Unsere über 20 Jahre Erfahrung sind ein großer Pluspunkt für unsere Kunden..Galerieschienen
mena99260mena99260 meinte am 25. Mar, 10:37:

Asiya

"A thrifty society will, in the long run, be wealthier than an impatient one, but it will not grow faster!"http://www.ibbycongress2014.org/?option=com_k2&view=itemlist&task=user&id=21612
mena99260mena99260 meinte am 26. Mar, 12:51:

Davidjohn

Bei uns finden Sie sämtliches Zubehör zum Aufhängen von Bildern. Vom einfachsten Bilderhaken bis hin zur Bildaufhängung mit Diebstahlschutz. Wir als Bilderrahmenprofi haben hier alle Produkte zur Bildaufhängung in einem Shop zusammengestellt. Zackenaufhänger
mena99260mena99260 meinte am 29. Mar, 09:07:

Davidjohn

I definitely enjoying every little bit of it. It is a great website and nice share. I want to thank you. Good job! You guys Extra resources do a great blog, and have some great contents. Keep up the good work.
mena99260mena99260 meinte am 29. Mar, 10:46:

Davidjohn

I am preparing a research paper and collecting information on this topic. Your post is one of the better that click for source I have read. Thank you for putting this information into one post.
mena99260mena99260 meinte am 5. Apr, 10:03:

Davidjohn

I definitely enjoying every little bit of it. It is a great website and nice share. I want to thank you. Good job! You guys do a great blog, and have some great contents. Keep up the good work. goji pro
mena99260mena99260 meinte am 7. Apr, 13:48:

Davidjohn

will, in the long run, be wealthier than an impatient one, but it will not grow faster! Ejaculation Guru Ebook
mena99260mena99260 meinte am 9. Apr, 08:34:

Davidjohn

Ixperia IT Schulung und Beratung An 18 Standorten in Deutschland bieten wir anspruchsvollen Führungskräften und Mitarbeitern erfolgreich Seminare zu den Themen IT Service Management, ITIL, Projektmanagement und Prince2 in exzellenter Atmosphäre. Sowohl fachlich als auch für die Durchführung erhalten wir von unseren Teilnehmern dafür die besten Beurteilungen mit 100% Erfolgsgarantie. ITIL Foundation
AsharSeo meinte am 11. Apr, 12:44:

Good examples are the parking lights and brake lights. Cynics would say that that is yet more proof of dishonest practices. Quite the opposite. It's proof of good quality.paleo diet recipes
jackman30 meinte am 12. Apr, 15:56:

jackman30

paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result. 15 seater minibus hire
jackman30 meinte am 13. Apr, 20:37:

jackman30

Ramsey's paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result. https://rebelmouse.com/
jackman30 meinte am 14. Apr, 16:04:

amazing selling machine

constant-relative-risk-aversion utility (CRRA) function or the constant intertemporal elasticity amazing selling machine
jackman30 meinte am 15. Apr, 15:52:

jackman30

per capita consumption is a stream c(t) t ≥ 0, of units of a single good. Preferences over consumption streams are given by http://www.twitter.com/newspaper_cat/
jackman30 meinte am 16. Apr, 07:21:

jackman30

problem is specified, i.e. they can also be seen as a stand-alone marginalist result. loft conversions
jackman30 meinte am 16. Apr, 12:59:

jackman30

paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result. Calgary online marketing agency
jackman30 meinte am 16. Apr, 17:44:

jackman30

Ramsey's paper the Euler equations are introduced before the optimization problem is specified, i.e. they can also be seen as a stand-alone marginalist result. easy video suite
jackman30 meinte am 17. Apr, 12:04:

jackman30

Real, per capita consumption is a stream c(t) t ≥ 0, of units of a single good. Preferences over consumption streams are given by double glazed sash windows london
AsharSeo meinte am 18. Apr, 12:53:

My husband furthermore i believed an individual cooperate part distribute of associated among writers that got hellos or her system truly. ...best wp themes