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Keynes–Ramsey rule

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some time, but this does not mean that our whole income should be saved. The more we save the sooner we shall reach bliss, but the less enjoyment we shall have now, and we have to set the one against the other. Mr. Keynes has shown me that the rule governing the amount to be saved can be determined
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Mathematically, the Keynes–Ramsey rule is a necessary first-order condition for an
412: 382: 342: 331: 210: 48: 583: 20: 544:(1984). "Notes on the Keynes–Ramsey Rule". In Ingham, A.; Ulph, A. M. (eds.). 626: 44: 471: 579: 488:, p. 545): “Enough must therefore be saved to reach or approach 463: 132:{\displaystyle {\dot {c}}(t)=\sigma \cdot (r-\rho )\cdot c(t)} 571: 448:
Ramsey, F. P. (1928). "A Mathematical Theory of Saving".
427:(Second ed.). New York: McGraw-Hill. p. 91. 303: 261: 219: 177: 148: 68: 411: 315: 285: 243: 201: 163: 131: 624: 377: 548:. London: Palgrave Macmillan. pp. 93–104. 607: 509:Mathematical Optimization and Economic Theory 512:. Englewood Cliffs: Prentice-Hall. pp.  502: 338:, who provided an economic interpretation. 614: 600: 421:"Growth Models with Consumer Optimization" 334:, who derived it in 1928, and his mentor 391:. Cambridge: MIT Press. pp. 41–43. 59:, the Keynes–Ramsey rule may look like 625: 485: 447: 330:The Keynes–Ramsey rule is named after 16:Mathematical formula in macroeconomics 540: 35:choice. Usually it is expressed as a 566: 493:at once from these considerations.” 13: 534: 14: 659: 570: 31:condition for the optimality of 39:relating the rate of change of 496: 478: 441: 405: 371: 280: 268: 244:{\displaystyle \rho \in (0,1)} 238: 226: 196: 190: 158: 152: 126: 120: 111: 99: 87: 81: 1: 546:Demand, Equilibrium and Trade 364: 202:{\displaystyle {\dot {c}}(t)} 586:. You can help Knowledge by 316:{\displaystyle \sigma >0} 7: 648:Economics and finance stubs 352: 10: 664: 565: 388:Lectures on Macroeconomics 359:Ramsey–Cass–Koopmans model 345:problem, also known as an 325:elasticity of substitution 286:{\displaystyle r\in (0,1)} 57:Ramsey–Cass–Koopmans model 55:. If derived from a basic 53:elasticity of substitution 643:Mathematical optimization 209:its change over time (in 33:intertemporal consumption 504:Intriligator, Michael D. 638:Intertemporal economics 379:Blanchard, Olivier Jean 347:Euler–Lagrange equation 323:is the (intertemporal) 582:-related article is a 317: 287: 245: 203: 165: 133: 51:, and (intertemporal) 417:Sala-i-Martin, Xavier 318: 288: 246: 204: 166: 134: 37:differential equation 301: 259: 217: 175: 164:{\displaystyle c(t)} 146: 66: 336:John Maynard Keynes 171:is consumption and 313: 295:real interest rate 283: 241: 199: 161: 129: 25:Keynes–Ramsey rule 595: 594: 434:978-0-262-02553-9 187: 78: 655: 616: 609: 602: 574: 567: 559: 528: 527: 500: 494: 482: 476: 475: 458:(152): 543–559. 451:Economic Journal 445: 439: 438: 413:Barro, Robert J. 409: 403: 402: 383:Fischer, Stanley 375: 322: 320: 319: 314: 292: 290: 289: 284: 250: 248: 247: 242: 208: 206: 205: 200: 189: 188: 180: 170: 168: 167: 162: 138: 136: 135: 130: 80: 79: 71: 663: 662: 658: 657: 656: 654: 653: 652: 633:Economic growth 623: 622: 621: 620: 563: 556: 537: 535:Further reading 532: 531: 524: 501: 497: 483: 479: 464:10.2307/2224098 446: 442: 435: 425:Economic Growth 410: 406: 399: 376: 372: 367: 355: 343:optimal control 332:Frank P. Ramsey 302: 299: 298: 260: 257: 256: 218: 215: 214: 211:Newton notation 179: 178: 176: 173: 172: 147: 144: 143: 70: 69: 67: 64: 63: 49:time preference 17: 12: 11: 5: 661: 651: 650: 645: 640: 635: 619: 618: 611: 604: 596: 593: 592: 575: 561: 560: 554: 536: 533: 530: 529: 522: 495: 477: 440: 433: 404: 397: 369: 368: 366: 363: 362: 361: 354: 351: 312: 309: 306: 282: 279: 276: 273: 270: 267: 264: 240: 237: 234: 231: 228: 225: 222: 198: 195: 192: 186: 183: 160: 157: 154: 151: 140: 139: 128: 125: 122: 119: 116: 113: 110: 107: 104: 101: 98: 95: 92: 89: 86: 83: 77: 74: 45:interest rates 21:macroeconomics 15: 9: 6: 4: 3: 2: 660: 649: 646: 644: 641: 639: 636: 634: 631: 630: 628: 617: 612: 610: 605: 603: 598: 597: 591: 589: 585: 581: 576: 573: 569: 568: 564: 557: 555:0-333-33184-2 551: 547: 543: 539: 538: 525: 523:0-13-561753-7 519: 515: 511: 510: 505: 499: 491: 487: 481: 473: 469: 465: 461: 457: 453: 452: 444: 436: 430: 426: 422: 418: 414: 408: 400: 398:0-262-02283-4 394: 390: 389: 384: 380: 374: 370: 360: 357: 356: 350: 348: 344: 339: 337: 333: 328: 326: 310: 307: 304: 296: 277: 274: 271: 265: 262: 254: 253:discount rate 235: 232: 229: 223: 220: 212: 193: 184: 181: 155: 149: 123: 117: 114: 108: 105: 102: 96: 93: 90: 84: 75: 72: 62: 61: 60: 58: 54: 50: 46: 42: 38: 34: 30: 26: 22: 588:expanding it 577: 562: 545: 508: 498: 486:Ramsey (1928 480: 455: 449: 443: 424: 407: 387: 373: 340: 329: 141: 24: 18: 41:consumption 627:Categories 365:References 580:economics 542:Bliss, C. 305:σ 266:∈ 224:∈ 221:ρ 185:˙ 115:⋅ 109:ρ 106:− 97:⋅ 94:σ 76:˙ 29:necessary 506:(1971). 419:(2004). 385:(1989). 353:See also 514:308–311 472:2224098 293:is the 251:is the 552:  520:  470:  431:  395:  297:, and 142:where 23:, the 578:This 490:bliss 468:JSTOR 43:with 27:is a 584:stub 550:ISBN 518:ISBN 484:See 429:ISBN 393:ISBN 308:> 460:doi 213:), 19:In 629:: 516:. 466:. 456:38 454:. 423:. 415:; 381:; 349:. 327:. 255:, 47:, 615:e 608:t 601:v 590:. 558:. 526:. 474:. 462:: 437:. 401:. 311:0 281:) 278:1 275:, 272:0 269:( 263:r 239:) 236:1 233:, 230:0 227:( 197:) 194:t 191:( 182:c 159:) 156:t 153:( 150:c 127:) 124:t 121:( 118:c 112:) 103:r 100:( 91:= 88:) 85:t 82:( 73:c

Index

macroeconomics
necessary
intertemporal consumption
differential equation
consumption
interest rates
time preference
elasticity of substitution
Ramsey–Cass–Koopmans model
Newton notation
discount rate
real interest rate
elasticity of substitution
Frank P. Ramsey
John Maynard Keynes
optimal control
Euler–Lagrange equation
Ramsey–Cass–Koopmans model
Blanchard, Olivier Jean
Fischer, Stanley
Lectures on Macroeconomics
ISBN
0-262-02283-4
Barro, Robert J.
Sala-i-Martin, Xavier
"Growth Models with Consumer Optimization"
ISBN
978-0-262-02553-9
Economic Journal
doi

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