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by: Dr. Janiya Bernier


Dr. Janiya Bernier

GPA 3.77


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Class Notes
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This 4 page Class Notes was uploaded by Dr. Janiya Bernier on Thursday October 22, 2015. The Class Notes belongs to ECON 161 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/226719/econ-161-university-of-california-berkeley in Economcs at University of California - Berkeley.




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Date Created: 10/22/15
Hamiltonians largely cribbed from Maurice Obstfeld s Guide for the Perplexed The Value Function An altemative way to think about maximizing utility in the Ramsey modela way that you will see a lot of in the futureinvolves a particular lnction called the Hamiltonian Recall that we had our intensive form utility function ie no U feiptuctalt Ie p 6391 alt 270 20 1 e I3 p n l eg And that we have our budget constraint 1me ampe quotg kz 0 tgtoo And that the household s intensive form asset holdings evolve according to alk 7 fki Ci quotgki Let s set up a value function rm 1 Vk0 nlt1 aXU nggxlggoe BtuctaltJ Bellman s Principle Because it is a value lnctionnadopting s to denote argument values that maximize the objectiveswe can write T Vk0 nngie auct dt 2 e ampuct alt e TVkT This is Bellman s principle of dynamic programming Now let s consider a discrete time analogue of our problem with time divided into periods of duration Ah U Ze pMuchAh 0 subject to ktAh k Ahfkt cg quotgk Then Bellman s principle tells us that mm maxuc Ah e We 1 And after a little manipulation WAh m l VkAhflte c ngltDe men 6 0 IQ XJLM0 2 mjVUc Ahfkc ngk M The Hamiltonian Let Ah approach zero and we have 0 9 0 Vkfk C 71 gk 3109 0 nagtltuc Vkfk c n gk Mk The terms inside the braces have a name the Hamiltonian William R Hamilton b 1805 Dublin d 1865 Dublin You form the Hamiltonian by taking the contemporanous piece of the function to be maximizedhere the utility lnctionnand adding to it the derivative of the value functionthe costate variabletimes the rateofchange of the state variable Let me talk a bit about state control and costate variables And let me write 7 V k Maximize the term in the braces by choosing consumption c and nd m Vk 1x 60 At each moment the consumer can decide to consume a little bit more at the price of an in nitesimal drop in the capital stock An in nitesimal unit of additional consumption yields the marginal payolT of the term on the left but it also generates an in nitesimal fall in the capital stockand the derivative of the value function tells you how costly in utility terms this fall in the capital stock is This equation has a straightforward economic interpretation Ifyou increase consumption today you gain utility But everything has an opportunity cost What s the opportunity cost of increasing consumption today It s equal to the effect of changing today s consumption on future opportunities times the value of those future opportunities Now let s go back to 0 9 0 Vkfk C quot gk 3Vk And let s differentiate the equation with respect to k taking c to be a function of k 0 ak VltIq 1 fk n g l3Vlq V kflqc n gk 0Himn9l5V9V399f9C ngk C And set V k 7 a9 dk 0f9ngB72EE 1 f39k quotg 3 7 Now this looks a lot like an asset pricing equation a dividend plus a capital gain equals a required rate of return The required rate of return is the adjusted rate of time discount beta The dividend is the added productive value of having an extra unit of capital on hand f ng The capital gain is the rate of change with time of the variable lambda These two conditionsan asset price condition that relates the state variable to the derivative with respect to the state variable of the value function and an asset accumulation condition that relates the opportunity cost of changing the state variable to the value of the state variableare going to come up again and again in intertemporal optimization But to return to our problem simply substitute in consumption for lambda and nd that ct 9 dt f kng 5 Ct 61 21 z fkngB z n p eg cl 0 0 The Euler equation So what s the point of the Hamiltonian methodology First it s used a lot in the literature You should see it now because you ll see i in more compleXit later Second it s a lot easier to say write down the Hamiltonian and solve then to run through the entire dynamicprogramming argument from rst principles all the time Third the Hamiltonian gives you interesting costate variablesthat have an interesting economic interpretation Phase Diagrams In addition to the equation for consumption growth we also have our capital accumulation equation dlddt c ngk and with the two of them we can start drawing phase diagrams Phase Diagrams dCdtO dkdtO Phase dynamics Saddle path What good are the other paths well suppose you have a terminal condition that you accumulate zero capital as of some point Look for where you are now and set c0 so you just hit the vertical aXis at that date Why is the dcdt0 line to the left of the Golden Rule maximum of the dlddt0 curve


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