-
Notifications
You must be signed in to change notification settings - Fork 0
/
ExponentialGrowth_Dessler.html
873 lines (769 loc) · 37.3 KB
/
ExponentialGrowth_Dessler.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<meta name="viewport" content="width=device-width, initial-scale=1.0" /><meta name="generator" content="Docutils 0.17.1: http://docutils.sourceforge.net/" />
<title>Exponential Growth — Climate Change Science and Engineering</title>
<!-- Loaded before other Sphinx assets -->
<link href="../_static/styles/theme.css?digest=1999514e3f237ded88cf" rel="stylesheet">
<link href="../_static/styles/pydata-sphinx-theme.css?digest=1999514e3f237ded88cf" rel="stylesheet">
<link rel="stylesheet"
href="../_static/vendor/fontawesome/5.13.0/css/all.min.css">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="../_static/vendor/fontawesome/5.13.0/webfonts/fa-solid-900.woff2">
<link rel="preload" as="font" type="font/woff2" crossorigin
href="../_static/vendor/fontawesome/5.13.0/webfonts/fa-brands-400.woff2">
<link rel="stylesheet" type="text/css" href="../_static/pygments.css" />
<link rel="stylesheet" href="../_static/styles/sphinx-book-theme.css?digest=5115cc725059bd94278eecd172e13a965bf8f5a9" type="text/css" />
<link rel="stylesheet" type="text/css" href="../_static/togglebutton.css" />
<link rel="stylesheet" type="text/css" href="../_static/copybutton.css" />
<link rel="stylesheet" type="text/css" href="../_static/mystnb.css" />
<link rel="stylesheet" type="text/css" href="../_static/sphinx-thebe.css" />
<link rel="stylesheet" type="text/css" href="../_static/design-style.b7bb847fb20b106c3d81b95245e65545.min.css" />
<!-- Pre-loaded scripts that we'll load fully later -->
<link rel="preload" as="script" href="../_static/scripts/pydata-sphinx-theme.js?digest=1999514e3f237ded88cf">
<script data-url_root="../" id="documentation_options" src="../_static/documentation_options.js"></script>
<script src="../_static/jquery.js"></script>
<script src="../_static/underscore.js"></script>
<script src="../_static/doctools.js"></script>
<script src="../_static/clipboard.min.js"></script>
<script src="../_static/copybutton.js"></script>
<script src="../_static/scripts/sphinx-book-theme.js?digest=9c920249402e914e316237a7dbc6769907cce411"></script>
<script>let toggleHintShow = 'Click to show';</script>
<script>let toggleHintHide = 'Click to hide';</script>
<script>let toggleOpenOnPrint = 'true';</script>
<script src="../_static/togglebutton.js"></script>
<script>var togglebuttonSelector = '.toggle, .admonition.dropdown, .tag_hide_input div.cell_input, .tag_hide-input div.cell_input, .tag_hide_output div.cell_output, .tag_hide-output div.cell_output, .tag_hide_cell.cell, .tag_hide-cell.cell';</script>
<script src="../_static/design-tabs.js"></script>
<script>const THEBE_JS_URL = "https://unpkg.com/thebe@0.8.2/lib/index.js"
const thebe_selector = ".thebe,.cell"
const thebe_selector_input = "pre"
const thebe_selector_output = ".output, .cell_output"
</script>
<script async="async" src="../_static/sphinx-thebe.js"></script>
<script>window.MathJax = {"options": {"processHtmlClass": "tex2jax_process|mathjax_process|math|output_area"}}</script>
<script defer="defer" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
<link rel="index" title="Index" href="../genindex.html" />
<link rel="search" title="Search" href="../search.html" />
<link rel="prev" title="Exponential Growth" href="../chapter10.html" />
<meta name="viewport" content="width=device-width, initial-scale=1" />
<meta name="docsearch:language" content="None">
<!-- Google Analytics -->
</head>
<body data-spy="scroll" data-target="#bd-toc-nav" data-offset="60">
<!-- Checkboxes to toggle the left sidebar -->
<input type="checkbox" class="sidebar-toggle" name="__navigation" id="__navigation" aria-label="Toggle navigation sidebar">
<label class="overlay overlay-navbar" for="__navigation">
<div class="visually-hidden">Toggle navigation sidebar</div>
</label>
<!-- Checkboxes to toggle the in-page toc -->
<input type="checkbox" class="sidebar-toggle" name="__page-toc" id="__page-toc" aria-label="Toggle in-page Table of Contents">
<label class="overlay overlay-pagetoc" for="__page-toc">
<div class="visually-hidden">Toggle in-page Table of Contents</div>
</label>
<!-- Headers at the top -->
<div class="announcement header-item noprint"></div>
<div class="header header-item noprint"></div>
<div class="container-fluid" id="banner"></div>
<div class="container-xl">
<div class="row">
<!-- Sidebar -->
<div class="bd-sidebar noprint" id="site-navigation">
<div class="bd-sidebar__content">
<div class="bd-sidebar__top"><div class="navbar-brand-box">
<a class="navbar-brand text-wrap" href="../index.html">
<!-- `logo` is deprecated in Sphinx 4.0, so remove this when we stop supporting 3 -->
<img src="../_static/ce401.png" class="logo" alt="logo">
<h1 class="site-logo" id="site-title">Climate Change Science and Engineering</h1>
</a>
</div><form class="bd-search d-flex align-items-center" action="../search.html" method="get">
<i class="icon fas fa-search"></i>
<input type="search" class="form-control" name="q" id="search-input" placeholder="Search this book..." aria-label="Search this book..." autocomplete="off" >
</form><nav class="bd-links" id="bd-docs-nav" aria-label="Main">
<div class="bd-toc-item active">
<ul class="nav bd-sidenav bd-sidenav__home-link">
<li class="toctree-l1">
<a class="reference internal" href="../intro.html">
Computational Notebooks
</a>
</li>
</ul>
<ul class="current nav bd-sidenav">
<li class="toctree-l1 has-children">
<a class="reference internal" href="../chapter1.html">
An Introduction to the Climate Problem
</a>
<input class="toctree-checkbox" id="toctree-checkbox-1" name="toctree-checkbox-1" type="checkbox"/>
<label for="toctree-checkbox-1">
<i class="fas fa-chevron-down">
</i>
</label>
<ul>
<li class="toctree-l2">
<a class="reference internal" href="../chapter1/FairbanksTemperatures.html">
Comparing temperature distributions
</a>
</li>
</ul>
</li>
<li class="toctree-l1 has-children">
<a class="reference internal" href="../chapter2.html">
Is the Climate Changing?
</a>
<input class="toctree-checkbox" id="toctree-checkbox-2" name="toctree-checkbox-2" type="checkbox"/>
<label for="toctree-checkbox-2">
<i class="fas fa-chevron-down">
</i>
</label>
<ul>
<li class="toctree-l2">
<a class="reference internal" href="../chapter2/LaurentideIceSheet_and_SeaLevel.html">
Approximate the Decrease in Sea-level from the Laurentide Ice Sheet
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter2/plotBerkeleyEarth.html">
Plot global temperature anomalies from Berkeley Earth
</a>
</li>
</ul>
</li>
<li class="toctree-l1 has-children">
<a class="reference internal" href="../chapter3.html">
Radiation and Energy Balance
</a>
<input class="toctree-checkbox" id="toctree-checkbox-3" name="toctree-checkbox-3" type="checkbox"/>
<label for="toctree-checkbox-3">
<i class="fas fa-chevron-down">
</i>
</label>
<ul>
<li class="toctree-l2">
<a class="reference internal" href="../chapter3/PlancksLaw.html">
Calculate Planck curves as a function of wavelength and temperature
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter3/RadiationInSpectralBands.html">
Calculate Percentage of Radiation in Spectral Bands
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter3/PowerEmittedByHuman.html">
Calculate the Power Emitted by a Human
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter3/BodiesEmittingInARoom.html">
Bodies Emitting in a Room
</a>
</li>
</ul>
</li>
<li class="toctree-l1 has-children">
<a class="reference internal" href="../chapter4.html">
A Simple Climate Model
</a>
<input class="toctree-checkbox" id="toctree-checkbox-4" name="toctree-checkbox-4" type="checkbox"/>
<label for="toctree-checkbox-4">
<i class="fas fa-chevron-down">
</i>
</label>
<ul>
<li class="toctree-l2">
<a class="reference internal" href="../chapter4/SolarConstant.html">
Calculation of the Solar Constant
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter4/Insolation.html">
Insolation
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter4/InfraredAbsorptionByTheAtmosphere.html">
Infrared Absorption by the Atmosphere
</a>
</li>
<li class="toctree-l2">
<a class="reference internal" href="../chapter4/EquilibriumTemperatureOfEarth.html">
Calculation of the Equilibrium Temperature of Earth
</a>
</li>
</ul>
</li>
<li class="toctree-l1 current active has-children">
<a class="reference internal" href="../chapter10.html">
Exponential Growth
</a>
<input checked="" class="toctree-checkbox" id="toctree-checkbox-5" name="toctree-checkbox-5" type="checkbox"/>
<label for="toctree-checkbox-5">
<i class="fas fa-chevron-down">
</i>
</label>
<ul class="current">
<li class="toctree-l2 current active">
<a class="current reference internal" href="#">
Exponential Growth
</a>
</li>
</ul>
</li>
</ul>
</div>
</nav></div>
<div class="bd-sidebar__bottom">
<!-- To handle the deprecated key -->
<div class="navbar_extra_footer">
Powered by <a href="https://jupyterbook.org">Jupyter Book</a>
</div>
</div>
</div>
<div id="rtd-footer-container"></div>
</div>
<!-- A tiny helper pixel to detect if we've scrolled -->
<div class="sbt-scroll-pixel-helper"></div>
<!-- Main content -->
<div class="col py-0 content-container">
<div class="header-article row sticky-top noprint">
<div class="col py-1 d-flex header-article-main">
<div class="header-article__left">
<label for="__navigation"
class="headerbtn"
data-toggle="tooltip"
data-placement="right"
title="Toggle navigation"
>
<span class="headerbtn__icon-container">
<i class="fas fa-bars"></i>
</span>
</label>
</div>
<div class="header-article__right">
<div class="menu-dropdown menu-dropdown-launch-buttons">
<button class="headerbtn menu-dropdown__trigger"
aria-label="Launch interactive content">
<i class="fas fa-rocket"></i>
</button>
<div class="menu-dropdown__content">
<ul>
<li>
<a href="https://mybinder.org/v2/gh/vonw/CE401/main?urlpath=tree/chapter10/ExponentialGrowth_Dessler.ipynb"
class="headerbtn"
data-toggle="tooltip"
data-placement="left"
title="Launch on Binder"
>
<span class="headerbtn__icon-container">
<img src="../_static/images/logo_binder.svg">
</span>
<span class="headerbtn__text-container">Binder</span>
</a>
</li>
</ul>
</div>
</div>
<button onclick="toggleFullScreen()"
class="headerbtn"
data-toggle="tooltip"
data-placement="bottom"
title="Fullscreen mode"
>
<span class="headerbtn__icon-container">
<i class="fas fa-expand"></i>
</span>
</button>
<div class="menu-dropdown menu-dropdown-repository-buttons">
<button class="headerbtn menu-dropdown__trigger"
aria-label="Source repositories">
<i class="fab fa-github"></i>
</button>
<div class="menu-dropdown__content">
<ul>
<li>
<a href="https://github.com/vonw/CE401"
class="headerbtn"
data-toggle="tooltip"
data-placement="left"
title="Source repository"
>
<span class="headerbtn__icon-container">
<i class="fab fa-github"></i>
</span>
<span class="headerbtn__text-container">repository</span>
</a>
</li>
<li>
<a href="https://github.com/vonw/CE401/issues/new?title=Issue%20on%20page%20%2Fchapter10/ExponentialGrowth_Dessler.html&body=Your%20issue%20content%20here."
class="headerbtn"
data-toggle="tooltip"
data-placement="left"
title="Open an issue"
>
<span class="headerbtn__icon-container">
<i class="fas fa-lightbulb"></i>
</span>
<span class="headerbtn__text-container">open issue</span>
</a>
</li>
</ul>
</div>
</div>
<div class="menu-dropdown menu-dropdown-download-buttons">
<button class="headerbtn menu-dropdown__trigger"
aria-label="Download this page">
<i class="fas fa-download"></i>
</button>
<div class="menu-dropdown__content">
<ul>
<li>
<a href="../_sources/chapter10/ExponentialGrowth_Dessler.ipynb"
class="headerbtn"
data-toggle="tooltip"
data-placement="left"
title="Download source file"
>
<span class="headerbtn__icon-container">
<i class="fas fa-file"></i>
</span>
<span class="headerbtn__text-container">.ipynb</span>
</a>
</li>
<li>
<button onclick="printPdf(this)"
class="headerbtn"
data-toggle="tooltip"
data-placement="left"
title="Print to PDF"
>
<span class="headerbtn__icon-container">
<i class="fas fa-file-pdf"></i>
</span>
<span class="headerbtn__text-container">.pdf</span>
</button>
</li>
</ul>
</div>
</div>
<label for="__page-toc"
class="headerbtn headerbtn-page-toc"
>
<span class="headerbtn__icon-container">
<i class="fas fa-list"></i>
</span>
</label>
</div>
</div>
<!-- Table of contents -->
<div class="col-md-3 bd-toc show noprint">
<div class="tocsection onthispage pt-5 pb-3">
<i class="fas fa-list"></i> Contents
</div>
<nav id="bd-toc-nav" aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#examples-from-introduction-to-modern-climate-change-by-andrew-dessler">
Examples from
<em>
Introduction to Modern Climate Change
</em>
by Andrew Dessler
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#equation-10-1-and-table-10-1">
Equation 10.1 and Table 10.1
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-rule-of-72">
The Rule of 72
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#discounting">
Discounting
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-utility-of-discounting-is-that-it-allows-one-to-compare-costs-and-benefits-that-occur-at-different-times">
The utility of discounting is that it allows one to compare costs and benefits that occur at different times.
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-10-1-what-would-you-do-the-tv-example">
Example 10.1: What would you do? The TV example!
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-choice-of-discount-rate-makes-a-profound-difference-in-what-you-choose-to-do">
The choice of discount rate makes a profound difference in what you choose to do.
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#cost-benefit-analysis">
Cost-Benefit Analysis
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#so-how-do-we-determine-the-correct-discount-rate-to-use">
So how do we determine the correct discount rate to use?
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-choice-of-discount-rate-makes-a-huge-difference-in-climate-problems">
The choice of discount rate makes a huge difference in climate problems.
</a>
</li>
</ul>
</li>
</ul>
</nav>
</div>
</div>
<div class="article row">
<div class="col pl-md-3 pl-lg-5 content-container">
<!-- Table of contents that is only displayed when printing the page -->
<div id="jb-print-docs-body" class="onlyprint">
<h1>Exponential Growth</h1>
<!-- Table of contents -->
<div id="print-main-content">
<div id="jb-print-toc">
<div>
<h2> Contents </h2>
</div>
<nav aria-label="Page">
<ul class="visible nav section-nav flex-column">
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#examples-from-introduction-to-modern-climate-change-by-andrew-dessler">
Examples from
<em>
Introduction to Modern Climate Change
</em>
by Andrew Dessler
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#equation-10-1-and-table-10-1">
Equation 10.1 and Table 10.1
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-rule-of-72">
The Rule of 72
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#discounting">
Discounting
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-utility-of-discounting-is-that-it-allows-one-to-compare-costs-and-benefits-that-occur-at-different-times">
The utility of discounting is that it allows one to compare costs and benefits that occur at different times.
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#example-10-1-what-would-you-do-the-tv-example">
Example 10.1: What would you do? The TV example!
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-choice-of-discount-rate-makes-a-profound-difference-in-what-you-choose-to-do">
The choice of discount rate makes a profound difference in what you choose to do.
</a>
</li>
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#cost-benefit-analysis">
Cost-Benefit Analysis
</a>
</li>
</ul>
</li>
<li class="toc-h2 nav-item toc-entry">
<a class="reference internal nav-link" href="#so-how-do-we-determine-the-correct-discount-rate-to-use">
So how do we determine the correct discount rate to use?
</a>
<ul class="nav section-nav flex-column">
<li class="toc-h3 nav-item toc-entry">
<a class="reference internal nav-link" href="#the-choice-of-discount-rate-makes-a-huge-difference-in-climate-problems">
The choice of discount rate makes a huge difference in climate problems.
</a>
</li>
</ul>
</li>
</ul>
</nav>
</div>
</div>
</div>
<main id="main-content" role="main">
<div>
<section class="tex2jax_ignore mathjax_ignore" id="exponential-growth">
<h1>Exponential Growth<a class="headerlink" href="#exponential-growth" title="Permalink to this headline">#</a></h1>
<section id="examples-from-introduction-to-modern-climate-change-by-andrew-dessler">
<h2>Examples from <em>Introduction to Modern Climate Change</em> by Andrew Dessler<a class="headerlink" href="#examples-from-introduction-to-modern-climate-change-by-andrew-dessler" title="Permalink to this headline">#</a></h2>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
</pre></div>
</div>
</div>
</div>
<section id="equation-10-1-and-table-10-1">
<h3>Equation 10.1 and Table 10.1<a class="headerlink" href="#equation-10-1-and-table-10-1" title="Permalink to this headline">#</a></h3>
<div class="math notranslate nohighlight">
\[ F = P (1 + rate/100)^n \]</div>
<p>where F is the future values, P is the initial value, rate is the percentage rate of increase (interest), and n is the number of years.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">P</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">rate</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">101</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">P</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">rate</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'Wow! That $100 investment really grew! It is $'</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">F</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span> <span class="o">+</span> <span class="s1">' after 101 years!'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>Wow! That $100 investment really grew! It is $1515867.36 after 101 years!
</pre></div>
</div>
</div>
</div>
</section>
<section id="the-rule-of-72">
<h3>The Rule of 72<a class="headerlink" href="#the-rule-of-72" title="Permalink to this headline">#</a></h3>
<p>To determine the doubling time, one can recognize that F/P is 2. So, then one can determine n if the rate is known.</p>
<div class="math notranslate nohighlight">
\[ \frac{F}{P} = 2 = (1 + r)^n \]</div>
<p>where r is the fractional rate of increase (not the percentage)</p>
<p>Take the natural log of both sides of the equation.</p>
<div class="math notranslate nohighlight">
\[ ln(2) = ln[(1 + r)^n] = n * ln(1 + r) \]</div>
<p>So,</p>
<div class="math notranslate nohighlight">
\[ n = \frac{ln(2)}{ln(1 + r)} \]</div>
<div class="math notranslate nohighlight">
\[ n = \frac{ln(2)}{r} \frac{r}{ln(1 + r)} \]</div>
<p>If r is small (e.g., 8% = 0.08), then</p>
<div class="math notranslate nohighlight">
\[ \frac{r}{ln(1 + r)} = \frac{0.08}{ln(1 + 0.08)} = 1.03949 \]</div>
<p>and</p>
<div class="math notranslate nohighlight">
\[ n \approx \frac{0.72}{r} \approx \frac{72}{100 * r} \]</div>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">r</span> <span class="o">=</span> <span class="mf">0.08</span>
<span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">2</span><span class="p">)</span> <span class="o">*</span> <span class="n">r</span> <span class="o">/</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">r</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output text_plain highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>0.7205174673600471
</pre></div>
</div>
</div>
</div>
<p>So, if the rate of interest is constant at 7.2%, it will take 10 years to double your money.</p>
</section>
</section>
<hr class="docutils" />
<section id="discounting">
<h2>Discounting<a class="headerlink" href="#discounting" title="Permalink to this headline">#</a></h2>
<p>Discounting simply solves our original equation for P.</p>
<div class="math notranslate nohighlight">
\[ P = \frac{F}{(1 + rate/100)^n} \]</div>
<p>The value today of a future expense or benefit is known as the <em>present value</em>. The process of calculating the present value is referred to as <em>discounting</em>. <em>“Money today is worth more to you than money in the future.”</em></p>
<section id="the-utility-of-discounting-is-that-it-allows-one-to-compare-costs-and-benefits-that-occur-at-different-times">
<h3>The utility of discounting is that it allows one to compare costs and benefits that occur at different times.<a class="headerlink" href="#the-utility-of-discounting-is-that-it-allows-one-to-compare-costs-and-benefits-that-occur-at-different-times" title="Permalink to this headline">#</a></h3>
<hr class="docutils" />
<p>How much would we need to invest to have $25,000 in 15 years at a 5% interest rate.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">=</span> <span class="mi">25000</span>
<span class="n">rate</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">15</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">F</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">rate</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'We would need to invest $</span><span class="si">{</span><span class="n">P</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> now (at 5%) to have $25000 in 15 years.'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s1">'$</span><span class="si">{</span><span class="n">P</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1"> is the present value.'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'This calculation is called discounting.'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The rate of 5</span><span class="si">% i</span><span class="s1">s called the discount rate, because it quantifies the rate at which money loses value in the future.'</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'In other words, money loses 5</span><span class="si">% o</span><span class="s1">f its value each year.'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>We would need to invest $12025.43 now (at 5%) to have $25000 in 15 years.
$12025.43 is the present value.
This calculation is called discounting.
The rate of 5% is called the discount rate, because it quantifies the rate at which money loses value in the future.
In other words, money loses 5% of its value each year.
</pre></div>
</div>
</div>
</div>
</section>
<hr class="docutils" />
<section id="example-10-1-what-would-you-do-the-tv-example">
<h3>Example 10.1: What would you do? The TV example!<a class="headerlink" href="#example-10-1-what-would-you-do-the-tv-example" title="Permalink to this headline">#</a></h3>
<p>Discounting can be used to help make financial decisions.</p>
<p>Let’s say you’re buying a new TV and you’re given two payment options:</p>
<ol class="simple">
<li><p>Pay $1000. now.</p></li>
<li><p>Get it with no money down and pay $1100. in 1 year.</p></li>
</ol>
<p>Which option is best for you?</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># No discounting.</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'You pay $1000. now, so it costs you $1000. in dollars today. So, the present value is $1000.'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>You pay $1000. now, so it costs you $1000. in dollars today. So, the present value is $1000.
</pre></div>
</div>
</div>
</div>
<p>But what is the present value of $1100. in one year.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="c1"># Discount option at 5% in one year</span>
<span class="n">P</span> <span class="o">=</span> <span class="mi">1000</span>
<span class="n">discountRate</span> <span class="o">=</span> <span class="mi">5</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">F</span> <span class="o">=</span> <span class="n">P</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">discountRate</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The value of the money after 1 year AT A DISCOUNT RATE OF 5</span><span class="si">% i</span><span class="s1">s $'</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">F</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The value of the money after 1 year AT A DISCOUNT RATE OF 5% is $1050.00
</pre></div>
</div>
</div>
</div>
<p>The option that costs you the least amount of money is the one with the lowest present value.</p>
<p>Therefore, you choose to pay $1000. now (because the present value of $1100. in one year from now, discounted at 5%, is $1050.)</p>
<p>But what is the discount rate is higher, say 15%?</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">F</span> <span class="o">=</span> <span class="mi">1100</span>
<span class="n">discountRate</span> <span class="o">=</span> <span class="mi">15</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">F</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">discountRate</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The value of the money after 1 year AT A DISCOUNT RATE OF 15</span><span class="si">% i</span><span class="s1">s $'</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">P</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The value of the money after 1 year AT A DISCOUNT RATE OF 15% is $956.52
</pre></div>
</div>
</div>
</div>
<p>Now the option that costs you the least amount of money is the present value using the 15% discount rate.</p>
<p>Therefore, you choose to pay $1100. next because it will only cost you $956 in today’s dollars.</p>
</section>
<section id="the-choice-of-discount-rate-makes-a-profound-difference-in-what-you-choose-to-do">
<h3>The choice of discount rate makes a profound difference in what you choose to do.<a class="headerlink" href="#the-choice-of-discount-rate-makes-a-profound-difference-in-what-you-choose-to-do" title="Permalink to this headline">#</a></h3>
</section>
<hr class="docutils" />
<section id="cost-benefit-analysis">
<h3>Cost-Benefit Analysis<a class="headerlink" href="#cost-benefit-analysis" title="Permalink to this headline">#</a></h3>
<p>Imagine our choice is between spending $100 billion today or $1 trillion in 100 years. Compare these two using a discount rate of 3%.</p>
<div class="cell docutils container">
<div class="cell_input docutils container">
<div class="highlight-ipython3 notranslate"><div class="highlight"><pre><span></span><span class="n">rate</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">F</span> <span class="o">=</span> <span class="mf">1e12</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">F</span> <span class="o">/</span> <span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="n">rate</span><span class="o">/</span><span class="mi">100</span><span class="p">)</span><span class="o">**</span><span class="n">n</span>
<span class="nb">print</span><span class="p">(</span><span class="s1">'The value of the money after 100 years AT A DISCOUNT RATE OF 3</span><span class="si">% i</span><span class="s1">s $'</span> <span class="o">+</span> <span class="sa">f</span><span class="s1">'</span><span class="si">{</span><span class="n">P</span><span class="si">:</span><span class="s1">.2f</span><span class="si">}</span><span class="s1">'</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="cell_output docutils container">
<div class="output stream highlight-myst-ansi notranslate"><div class="highlight"><pre><span></span>The value of the money after 100 years AT A DISCOUNT RATE OF 3% is $52032839850.21
</pre></div>
</div>
</div>
</div>
<p>Note that this equals only $52 billion, so we would prefer to pay $1 trillion in 100 years than pay $100 billion today</p>
</section>
</section>
<hr class="docutils" />
<section id="so-how-do-we-determine-the-correct-discount-rate-to-use">
<h2>So how do we determine the correct discount rate to use?<a class="headerlink" href="#so-how-do-we-determine-the-correct-discount-rate-to-use" title="Permalink to this headline">#</a></h2>
<p>The discount rate is a combination of two different judgements:</p>
<ol class="simple">
<li><p><em>Time discounting</em>, which is the preference to consume now rather than later.
Positive time discounting means that goods and services now are worth more than they are later.</p></li>
<li><p><em>Growth discounting</em> is the fact that a dollar means more to poor people than it does to rich people.
Would you pick up a $1 bill if you saw it on the street? What about a $20 bill?</p></li>
</ol>
<p>The discount rate used in present-value calculations is determined by combining the time and growth discount rates.</p>
<p>BUT economists argue about which rate to use: 1% to 4%.</p>
<section id="the-choice-of-discount-rate-makes-a-huge-difference-in-climate-problems">
<h3>The choice of discount rate makes a huge difference in climate problems.<a class="headerlink" href="#the-choice-of-discount-rate-makes-a-huge-difference-in-climate-problems" title="Permalink to this headline">#</a></h3>
</section>
</section>
</section>
<script type="text/x-thebe-config">
{
requestKernel: true,
binderOptions: {
repo: "binder-examples/jupyter-stacks-datascience",
ref: "master",
},
codeMirrorConfig: {
theme: "abcdef",
mode: "python"
},
kernelOptions: {
kernelName: "python3",
path: "./chapter10"
},
predefinedOutput: true
}
</script>
<script>kernelName = 'python3'</script>
</div>
</main>
<footer class="footer-article noprint">
<!-- Previous / next buttons -->
<div class='prev-next-area'>
<a class='left-prev' id="prev-link" href="../chapter10.html" title="previous page">
<i class="fas fa-angle-left"></i>
<div class="prev-next-info">
<p class="prev-next-subtitle">previous</p>
<p class="prev-next-title">Exponential Growth</p>
</div>
</a>
</div>
</footer>
</div>
</div>
<div class="footer-content row">
<footer class="col footer"><p>
By Von P. Walden, Washington State University<br/>
© Copyright 2022.<br/>
</p>
</footer>
</div>
</div>
</div>
</div>
<!-- Scripts loaded after <body> so the DOM is not blocked -->
<script src="../_static/scripts/pydata-sphinx-theme.js?digest=1999514e3f237ded88cf"></script>
</body>
</html>