-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy patholigopoly_plots.html
979 lines (654 loc) · 22.5 KB
/
oligopoly_plots.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
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
<!DOCTYPE html>
<html lang="" xml:lang="">
<head>
<title>4.4: Monopolistic Competition & Oligopoly</title>
<meta charset="utf-8" />
<meta name="author" content="Ryan Safner Assistant Professor of Economics [email protected] ryansafner/microS20 microS20.classes.ryansafner.com " />
<script src="libs/header-attrs/header-attrs.js"></script>
<link href="libs/remark-css/hygge.css" rel="stylesheet" />
<link rel="stylesheet" href="custom.css" type="text/css" />
<link rel="stylesheet" href="custom-fonts.css" type="text/css" />
</head>
<body>
<textarea id="source">
class: center, middle, inverse, title-slide
# 4.4: Monopolistic Competition & Oligopoly
## ECON 306 · Microeconomic Analysis · Spring 2020
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:[email protected]"><i class="fa fa-paper-plane fa-fw"></i> [email protected]</a> <br> <a href="https://github.com/ryansafner/microS20"><i class="fa fa-github fa-fw"></i> ryansafner/microS20</a><br> <a href="https://microS20.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i> microS20.classes.ryansafner.com</a><br>
---
class: inverse, center, middle
# Imperfect Competition
---
# Imperfect Competition
.center[
]
---
# Imperfect Competition
.center[
]
---
# Imperfect Competition
.center[
]
---
# Imperfect Competition
.center[
]
---
class: inverse, center, middle
# Monopolistic Competition
---
# Monopolistic Competition
.pull-left[
- .hi[Monopolistic competition]: hybrid of monopoly and competition, where **each firm has _some_ market power**
1. Goods are .hi-purple[*imperfect* substitutes]
- consumers recognize non-price differences between sellers' goods
2. .hi-purple[Free Entry and exit] (no barriers)
3. Each firm is a .hi-purple[price-searcher]
- faces own downward-sloping demand
]
.pull-right[
.center[
]
]
---
# Monopolistic Competition Model: Short Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]
.pull-right[
- **Short Run**: Firm acts as a monopolist
]
---
# Monopolistic Competition Model: Short Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-2-1.png" width="504" />
]
.pull-right[
- **Short Run**: Firm acts as a monopolist:
- `\(q^*\)`: where `\(MR(q)=MC(q)\)`
]
---
# Monopolistic Competition Model: Short Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-3-1.png" width="504" />
]
.pull-right[
- **Short Run**: Firm acts as a monopolist:
- `\(q^*\)`: where `\(MR(q)=MC(q)\)`
- `\(p^*\)`: at market demand for `\(q^*\)`
]
---
# Monopolistic Competition Model: Short Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]
.pull-right[
- **Short Run**: Firm acts as a monopolist:
- `\(q^*\)`: where `\(MR(q)=MC(q)\)`
- `\(p^*\)`: at market demand for `\(q^*\)`
- Earns `\(\pi = [p^*-AC(q^*)]q^*\)`
]
---
# Monopolistic Competition Model: Long Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]
.pull-right[
- **Long Run**: market becomes competitive (*no barriers to entry!*)
- `\(\pi > 0\)` attracts entry into industry
- Demand for each firm's product will *decrease* (and become more *elastic*), until...
]
---
# Monopolistic Competition Model: Long Run
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-6-1.png" width="504" />
]
.pull-right[
- **Long Run**: market becomes competitive (*no barriers to entry!*)
- `\(\pi > 0\)` attracts entry into industry
- Demand for each firm's product will *decrease* (and become more *elastic*), until...
- .hi-purple[Long run equilibrium]: firms earn `\(\pi=0\)` where `\(p=AC(q)\)`<sup>.red[1]</sup>
]
---
# Monopolistic Competition vs. Perfect Competition
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-7-1.png" width="504" />
]
.pull-right[
- **Perfect competition** `\((q_c, p_c)\)`
- `\(p_c = MC(q)\)`, .hi-purple[allocatively efficient]
- `\(q_c\)` where `\(P=MC(q)\)`
- Maximum .blue[consumer surplus]
- No **DWL**
]
---
# Monopolistic Competition vs. Perfect Competition
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]
.pull-right[
- **Monopolistic competition** `\((q_m, p_m)\)`
- `\(p_m = AC(q)\)`
- but not `\(AC_{min}\)`, .hi-purple[productive inefficiency]
- `\(q_m < q_c\)`, where `\(MR(q)=MC(q)\)`
- `\(p_m > MC(q)\)`, .hi-purple[allocative inefficiency]
- Less .blue[Consumer Surplus]
- **Deadweight loss**
]
---
# Monopolistic Competition vs. Perfect Competition
.pull-left[
<img src="oligopoly_plots_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]
.pull-right[
- Like a monopolist, produces less `\(q\)` at a higher `\(p\)` than competition
- But like perfect competition, still no `\(\pi\)` in the long run!
]
---
class: inverse, center, middle
# Oligopoly
---
# Oligopoly
.pull-left[
.center[
]
]
.pull-right[
- .hi[Oligopoly]: industry with a few large sellers with market power
- Other features can vary
- May sell similar or different goods
- May have barriers to entry
- Key: Firms make .hi-purple[strategic choices], interdependent on one another
- For modeling simplicity:
- .hi-purple[Duopoly]: a market with 2 sellers
]
---
# Oligopoly: Modelling
.pull-left[
.center[
]
]
.pull-right[
- Unlike PC or monopoly, no single "theory of oligopoly"
- Depends heavily on assumptions made about interactions and choice variables
- One certainty: oligopoly is a strategic interaction between few firms (i.e. ideal for game theory)
]
---
# Game Theory
.pull-left[
.center[
]
]
.pull-right[
- .hi[Game theory]: a set of tools that model strategic interactions ("games") between rational players, 3 elements:
1. **Players**
2. **Strategies** that players can choose from
3. **Payoffs** to each player that are *jointly-determined* from combination of all players' strategies
]
---
# Game Theory vs. Decision Theory Models I
.pull-left[
.center[
]
]
.pull-right[
- Traditional economic models are often called .hi[Decision theory]:
- .hi-purple[Optimization models] **ignore all other agents** and just focus on how can **you** maximize **your** objective within **your** constraints
- Consumers max utility; firms max profit, etc.
- **Outcome**: .hi-purple[optimum]: decision where *you* have no better alternatives
]
---
# Game Theory vs. Decision Theory Models I
.pull-left[
.center[
]
]
.pull-right[
- Traditional economic models are often called .hi[Decision theory]:
- .hi-purple[Equilibrium models] assume that there are **so many agents** that **no agent's decision can affect the outcome**
- Firms are price-takers or the *only* buyer or seller
- **Ignores all other agents' decisions**!
- **Outcome**: .hi-purple[equilibrium]: where *nobody* has no better alternatives
]
---
# Game Theory vs. Decision Theory Models III
.pull-left[
.center[
]
]
.pull-right[
- .hi[Game theory models] directly confront **strategic interactions** between players
- How each player would respond to a strategy chosen by other player(s)
- Lead to a stable outcome where everyone has considered and chosen their best responses
- **Outcome**: .hi-purple[Nash equilibrium]: where *nobody* has a better strategy *given the strategies everyone else is playing*
]
---
# Equilibrium in Oligopoly
.pull-left[
.center[
]
]
.pull-right[
- What does .hi-purple["equilibrium"] mean in an *oligopoly*?
- In competition or monopoly, a unique `\((q^*,p^*)\)` for whole market such that **no firms/consumers have incentives to change price**
]
---
# Equilibrium in Oligopoly
.pull-left[
.center[
]
]
.pull-right[
- Oligopoly: use game-theoretic .hi-purple[Nash Equilibrium]:
- no player wants to change their strategy **given all other player' strategies**
- each player is playing a **best response** against other players' strategies
]
---
# As a Prisoner's Dilemma I
.pull-left[
.center[
]
]
.pull-right[
- Suppose we have a **duopoly** between Apple and Google
- Each is planning to launch a new tablet, and choose to sell it at a **High Price** or a **Low Price**
]
---
# As a Prisoner's Dilemma I
.pull-left[
- Payoff matrix represents profits to each firm
- First number in each box goes to .hi-red[Row player (Apple)]
- Second number in each box goes to .hi-blue[Column player (Google)]
]
.pull-right[
.center[
]
]
---
# As a Prisoner's Dilemma II
.pull-left[
- From .hi-red[Apple]'s perspective:
- .hi-red[Low Price] is a .hi-purple[dominant strategy] for .hi-red[Apple]
]
.pull-right[
.center[
]
]
---
# As a Prisoner's Dilemma II
.pull-left[
- From Google's perspective:
- .hi-blue[Low Price] is a .hi-purple[dominant strategy] for .hi-blue[Google]
]
.pull-right[
.center[
]
]
---
# As a Prisoner's Dilemma II
.pull-left[
- .hi[Nash equilibrium]: (.hi-red[Low Price], .hi-blue[Low Price])
- neither player has an incentive to change price, *given the other's price*
]
.pull-right[
.center[
]
]
---
# As a Prisoner's Dilemma III
.pull-left[
- .hi[Nash equilibrium]: (.hi-red[Low Price], .hi-blue[Low Price])
- neither player has an incentive to change price, *given the other's price*
- A possible **Pareto improvement**: (.hi-red[High Price], .hi-blue[High Price])
- Both players are better off, nobody worse off!
- Is it a Nash Equilibrium?
]
.pull-right[
.center[
]
]
---
class: inverse, center, middle
# Cartels
---
# As a Prisoner's Dilemma IV
.pull-left[
- Google and Apple could .hi-purple[collude] with one another and agree to both raise prices
- .hi[Cartel]: group of sellers coordinate to raise prices to act like a collective monopoly and split the profits
]
.pull-right[
.center[
]
]
---
# Instability of Cartels
.pull-left[
- Cartels often **unstable**:
- Incentive for each member to cheat is too strong
- Entrants (non-cartel members) can threaten lower prices
- Difficult to monitor whether firms are upholding agreement
- Cartels are illegal, must be discrete
]
.pull-right[
.center[
]
]
---
# Attempts to Sustain Collusion I
.pull-left[
.center[
]
]
.pull-right[
Archer Daniels Midland (USA), Ajinomoto (Japan), Koywa Hakko Kogyo (Japan), Sewon American Inc (South Korea) held secret meetings to fix the price of lysine, a food additive to animal feed in the 1990s.
]
---
# Attempts to Sustain Collusion I
.pull-left[
.center[
]
]
.pull-right[
Archer Daniels Midland (USA), Ajinomoto (Japan), Koywa Hakko Kogyo (Japan), Sewon American Inc (South Korea) held secret meetings to fix the price of lysine, a food additive to animal feed in the 1990s.
An internal FBI informant brought the cartel down.
]
---
# Attempts to Sustain Collusion II
.pull-left[
.center[
]
]
.pull-right[
- 1950s market for turbines (for electric utility companies)
- A triopoly by market share:
- GE: 60%
- Westinghouse: 30%
- Allied-Chalmers: 10%
- Maintained this equilibrium with clever coordination
]
---
# Attempts to Sustain Collusion II
.pull-left[
.center[
]
]
.pull-right[
- Utility companies solicit bids to build turbines:
- If bid comes on day 1-17 on *lunar* calendar
- Westinghouse & A-C bid prohibitively high
- Ensures GE won
]
---
# Attempts to Sustain Collusion II
.pull-left[
.center[
]
]
.pull-right[
- Utility companies solicit bids to build turbines:
- If bid comes on day 18-25 on *lunar* calendar
- GE & A-C bid prohibitively high
- Ensures Westinghouse won
]
---
# Attempts to Sustain Collusion II
.pull-left[
.center[
]
]
.pull-right[
- Utility companies solicit bids to build turbines:
- If bid comes on day 26-28 on *lunar* calendar
- GE & Westinghouse bid prohibitively high
- Ensures Allied-Chalmers won
]
---
# Attempts to Sustain Collusion II
.pull-left[
.center[
]
]
.pull-right[
- Utility companies released their bids randomly, not according to lunar calendar
- Ensures the 60%-30%-10% distribution
- Cheating by one of the 3 firms easily monitored by other 2
- Nobody thought about the lunar calendar, until antitrust authorities caught on
]
---
# Attempts to Sustain Collusion III
.pull-left[
.center[
]
]
.pull-right[
- FCC Spectrum License auctions 1996-1997
- Firm seeking a license in particular location (and willing to fight for it) signals to other firms via ending its bid in the **telephone area code** digits
- e.g. $50,100,**202** for Washington DC (area code 202)
- Other firms let it win (in exchange for tacit agreement to do the same)
]
---
# Government-Sanctioned Cartels I
.pull-left[
- Like monopolies, some cartels exist because they are *supported* by governments or regulators, possibly by rent-seeking
- National Recovery Administration (1933-1935)
- cartelized most industries to artificially raise prices of goods
- found unconstitutional in *Schechter Poultry Corp. v. United States* (1935)
]
.pull-right[
.center[
]
]
---
# Government-Sanctioned Cartels II
.center[
]
---
# Government-Sanctioned Cartels II
.left-column[
.center[
]
]
.right-column[
"[B]ecause of their inability to maintain their cartels [prior to the ICC], railroads were big supporters of the [Interstate Commerce Act] because the newly-formed ICC could coordinate cartel prices...Using the new law as authority, the railroads revamped their freight classification, raised rates, eliminated passes and fare reductions, and revised less than carload rates on all types of goods, including groceries."
.source[Kolko, Gabriel, 1963, *The Triumph of Conservatism: A Reinterpretation of American History, 1900-1916*]
]
---
# Government-Sanctioned Cartels III
.pull-left[
.center[
Source: [NPR Planet Money](https://www.npr.org/sections/money/2015/06/24/417187957/episode-478-the-raisin-outlaw)
]
]
.pull-right[
> "Marvin Horne was known as the raisin outlaw. His crime: Selling 100% of his raisin crop, against the wishes of the Raisin Administrative Committee, a group of farmers that regulates the national raisin supply.
> He took the case all the way to the Supreme Court, which issued its final ruling this week."
]
---
# Government-Sanctioned Cartels IV
.pull-left[
.center[
]
]
.pull-right[
.center[
]
]
---
# Cartels: In Fiction I
.center[
<iframe width="560" height="315" src="https://www.youtube.com/embed/hGo5bxWy21g" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]
---
# Cartels: In Fiction II
.center[
<iframe width="560" height="315" src="https://www.youtube.com/embed/aT7TxMaZ4eM" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
]
---
# Comparing Industries
| Industry | Firms | Entry | Price (LR Eq.) | Output | Profits (LR) | Cons. Surplus | DWL |
|----------|-------|-------|--------|--------|--------------|---------------|-----|
| **Perfect competition** | Very many | Free | Lowest `\((MC)\)` | Highest | 0 | Highest | None |
| **Monopolistic competition** | Many | Free | Higher `\((p>MC)\)` | Lower | 0 | Lower | Some |
| **Oligopoly** (non-cooperative) | Few | Barriers? | Higher | Lower | Some | Lower | Some |
| **Monopoly**<sup>.red[1]</sup> (or cartel) | 1 | Barriers | Highest | Lowest | Highest | Lowest | Largest |
.footnote[<sup>.red[1]</sup> *Without* price-discrimination. Price-discrimination will increase output, increase profits, decrease consumer surplus, decrease deadweight loss]
</textarea>
<style data-target="print-only">@media screen {.remark-slide-container{display:block;}.remark-slide-scaler{box-shadow:none;}}</style>
<script src="https://remarkjs.com/downloads/remark-latest.min.js"></script>
<script src="macros.js"></script>
<script src="https://platform.twitter.com/widgets.js"></script>
<script>var slideshow = remark.create({
"highlightStyle": "ocean",
"highlightLines": true,
"countIncrementalSlides": false,
"ratio": "16:9",
"slideNumberFormat": "<div class=\"progress-bar-container\">\n <div class=\"progress-bar\" style=\"width: calc(%current% / %total% * 100%);\">\n </div>\n</div>"
});
if (window.HTMLWidgets) slideshow.on('afterShowSlide', function (slide) {
window.dispatchEvent(new Event('resize'));
});
(function(d) {
var s = d.createElement("style"), r = d.querySelector(".remark-slide-scaler");
if (!r) return;
s.type = "text/css"; s.innerHTML = "@page {size: " + r.style.width + " " + r.style.height +"; }";
d.head.appendChild(s);
})(document);
(function(d) {
var el = d.getElementsByClassName("remark-slides-area");
if (!el) return;
var slide, slides = slideshow.getSlides(), els = el[0].children;
for (var i = 1; i < slides.length; i++) {
slide = slides[i];
if (slide.properties.continued === "true" || slide.properties.count === "false") {
els[i - 1].className += ' has-continuation';
}
}
var s = d.createElement("style");
s.type = "text/css"; s.innerHTML = "@media print { .has-continuation { display: none; } }";
d.head.appendChild(s);
})(document);
// delete the temporary CSS (for displaying all slides initially) when the user
// starts to view slides
(function() {
var deleted = false;
slideshow.on('beforeShowSlide', function(slide) {
if (deleted) return;
var sheets = document.styleSheets, node;
for (var i = 0; i < sheets.length; i++) {
node = sheets[i].ownerNode;
if (node.dataset["target"] !== "print-only") continue;
node.parentNode.removeChild(node);
}
deleted = true;
});
})();
(function() {
"use strict"
// Replace <script> tags in slides area to make them executable
var scripts = document.querySelectorAll(
'.remark-slides-area .remark-slide-container script'
);
if (!scripts.length) return;
for (var i = 0; i < scripts.length; i++) {
var s = document.createElement('script');
var code = document.createTextNode(scripts[i].textContent);
s.appendChild(code);
var scriptAttrs = scripts[i].attributes;
for (var j = 0; j < scriptAttrs.length; j++) {
s.setAttribute(scriptAttrs[j].name, scriptAttrs[j].value);
}
scripts[i].parentElement.replaceChild(s, scripts[i]);
}
})();
(function() {
var links = document.getElementsByTagName('a');
for (var i = 0; i < links.length; i++) {
if (/^(https?:)?\/\//.test(links[i].getAttribute('href'))) {
links[i].target = '_blank';
}
}
})();
// adds .remark-code-has-line-highlighted class to <pre> parent elements
// of code chunks containing highlighted lines with class .remark-code-line-highlighted
(function(d) {
const hlines = d.querySelectorAll('.remark-code-line-highlighted');
const preParents = [];
const findPreParent = function(line, p = 0) {
if (p > 1) return null; // traverse up no further than grandparent
const el = line.parentElement;
return el.tagName === "PRE" ? el : findPreParent(el, ++p);
};
for (let line of hlines) {
let pre = findPreParent(line);
if (pre && !preParents.includes(pre)) preParents.push(pre);
}
preParents.forEach(p => p.classList.add("remark-code-has-line-highlighted"));
})(document);</script>
<script>
slideshow._releaseMath = function(el) {
var i, text, code, codes = el.getElementsByTagName('code');
for (i = 0; i < codes.length;) {
code = codes[i];
if (code.parentNode.tagName !== 'PRE' && code.childElementCount === 0) {
text = code.textContent;
if (/^\\\((.|\s)+\\\)$/.test(text) || /^\\\[(.|\s)+\\\]$/.test(text) ||
/^\$\$(.|\s)+\$\$$/.test(text) ||
/^\\begin\{([^}]+)\}(.|\s)+\\end\{[^}]+\}$/.test(text)) {
code.outerHTML = code.innerHTML; // remove <code></code>
continue;
}
}
i++;
}
};
slideshow._releaseMath(document);
</script>
<!-- dynamically load mathjax for compatibility with self-contained -->
<script>
(function () {
var script = document.createElement('script');
script.type = 'text/javascript';
script.src = 'https://cdn.bootcss.com/mathjax/2.7.1/MathJax.js?config=TeX-MML-AM_HTMLorMML';
if (location.protocol !== 'file:' && /^https?:/.test(script.src))
script.src = script.src.replace(/^https?:/, '');
document.getElementsByTagName('head')[0].appendChild(script);
})();
</script>
</body>
</html>