Imagine that there are 1000 consumers For each consumer the

Imagine that there are 1,000 consumers. For each consumer, the willingness to pay for a widget is distributed uniformly over the interval [0,1] depending on the style of the widget. A retailer with a particular study of the good knows this distribution. Its costs are zero. Consumers do not know the style that the retailer has stocked and each incur a transport or search cost of T= 0.125. Once this cost is incurred it is sunk. At that point, a consumer in the retailer’s store will purchase the product so long as the consumer’s valuation is greater than or equal to the price charged by the retailer. A) Show that facing a random selection of customers, the retailer’s profit maximizing price is p= 0.5. B) Show that with T= 0.125, all consumers will come to shop expecting a price of 0.5. What would happen if T= 0.15?

Solution

Given that n=1000, willingness to pay for a widget is distributed uniformly over the interval [0,1]. So, first we generating the random variables of uniform distribution

(a)

x=runif(1000,0,1)

[1] 0.456014690 0.377264587 0.969876089 0.792716538 0.853262495 0.022928924
   [7] 0.831932161 0.226129862 0.817012086 0.806990244 0.014304288 0.291641118
[13] 0.170963428 0.649449725 0.980949104 0.661851608 0.914641665 0.708642185
[19] 0.874844518 0.891369556 0.043878797 0.089360078 0.289294298 0.233731775
[25] 0.695385135 0.544178264 0.202037520 0.482454420 0.006720301 0.576511641
[31] 0.436844131 0.994492590 0.324629167 0.216972207 0.715682755 0.145887394
[37] 0.083188537 0.372925829 0.442112637 0.664237099 0.185807051 0.991280041
[43] 0.350436643 0.243295024 0.940440122 0.144618715 0.305170211 0.386064510
[49] 0.732110926 0.777351499 0.551334258 0.882188730 0.965594030 0.220075823
[55] 0.801455447 0.354613904 0.245168564 0.067359129 0.109255190 0.636053742
[61] 0.292686479 0.209435234 0.647088725 0.291826057 0.420763935 0.966416646
[67] 0.346485871 0.450709354 0.976559004 0.114470432 0.778250357 0.151392655
[73] 0.214815714 0.323068148 0.571884325 0.528205239 0.075014475 0.436946618
[79] 0.322429217 0.046066032 0.933869036 0.707564753 0.898330194 0.646473803
[85] 0.499864510 0.159774271 0.765620926 0.298936432 0.162760028 0.099174869
[91] 0.463620499 0.469686955 0.272723342 0.786762496 0.272445919 0.537882201
[97] 0.938425679 0.982523483 0.190317844 0.688158745 0.063006530 0.856632924
[103] 0.426210659 0.542979247 0.677368106 0.748866566 0.301882507 0.508993309
[109] 0.628266460 0.812822498 0.834692991 0.207713743 0.613878531 0.546261253
[115] 0.134304534 0.324850196 0.674283592 0.684859506 0.420550406 0.700458606
[121] 0.097437164 0.517921448 0.125011664 0.487302281 0.015289473 0.140925844
[127] 0.897108517 0.506346202 0.887421436 0.682143022 0.380515867 0.439162727
[133] 0.851226174 0.818015810 0.695495429 0.554943012 0.353436350 0.122463164
[139] 0.481769152 0.500481972 0.544267242 0.080302949 0.539464455 0.127022652
[145] 0.135894132 0.753023186 0.756420672 0.573677481 0.504417075 0.907229845
[151] 0.334712515 0.593581417 0.543986678 0.582362873 0.680412332 0.874429647
[157] 0.944336886 0.090804628 0.335494239 0.138781146 0.785225947 0.878348227
[163] 0.776386620 0.487745636 0.986849789 0.495883350 0.396363315 0.525174325
[169] 0.309072755 0.695209009 0.077991909 0.892195799 0.117042499 0.696167247
[175] 0.846371114 0.558592697 0.625947913 0.958787289 0.636320916 0.047453406
[181] 0.547728610 0.209188428 0.852958606 0.093678342 0.334171944 0.432920240
[187] 0.109596015 0.240764947 0.499164366 0.771308030 0.658854320 0.664222218
[193] 0.823465616 0.747643978 0.974822376 0.846975623 0.752215478 0.913379400
[199] 0.187203152 0.563833689 0.774753690 0.209311303 0.490098566 0.227315288
[205] 0.604412556 0.556904467 0.447308990 0.983113076 0.767661256 0.017312355
[211] 0.740833541 0.686494179 0.471141204 0.983274240 0.484460683 0.281155506
[217] 0.234303370 0.044476425 0.955790623 0.672671629 0.296217490 0.359515398
[223] 0.291258986 0.690726828 0.461576543 0.086944135 0.362735874 0.634914998
[229] 0.707006241 0.097598035 0.413846065 0.410473114 0.796340301 0.288612925
[235] 0.906165030 0.302112745 0.349447054 0.620804276 0.222359780 0.923585766
[241] 0.261950006 0.325415405 0.604086989 0.568045493 0.282502993 0.081810884
[247] 0.288557781 0.280188025 0.486282226 0.132909102 0.827992533 0.181816688
[253] 0.218185372 0.959422827 0.000101344 0.628375869 0.866203324 0.735546983
[259] 0.782017567 0.927029538 0.491529035 0.394997264 0.388546312 0.799912119
[265] 0.999222202 0.247152055 0.126633456 0.115612835 0.954224813 0.376913787
[271] 0.129936204 0.401256673 0.167908561 0.697794248 0.738608090 0.153460831
[277] 0.233393734 0.991471621 0.912794209 0.548654386 0.081670386 0.289927141
[283] 0.567766151 0.869245004 0.644819197 0.657365806 0.455195060 0.335408593
[289] 0.313972502 0.627411776 0.459653740 0.495689544 0.043614570 0.593280377
[295] 0.565166208 0.269247741 0.768869359 0.214734206 0.150378671 0.414836896
[301] 0.133496830 0.249089861 0.134304051 0.183870204 0.792430322 0.608659125
[307] 0.205434182 0.132001657 0.179511258 0.024279479 0.465359442 0.566809050
[313] 0.267533416 0.948204833 0.391679741 0.395911593 0.446994657 0.768332642
[319] 0.835940857 0.969151412 0.302665253 0.393006047 0.330340737 0.404839532
[325] 0.116136799 0.334524215 0.626634211 0.673014653 0.453760802 0.438126254
[331] 0.681783199 0.268487829 0.718301705 0.338277236 0.491871823 0.764549803
[337] 0.937493434 0.392566114 0.620963793 0.271738884 0.376526656 0.164380141
[343] 0.955802910 0.076460332 0.015470808 0.469103629 0.127484675 0.316763133
[349] 0.003234727 0.978725891 0.868210229 0.829505353 0.316002380 0.654037387
[355] 0.226034704 0.092363853 0.239825160 0.884692993 0.519932671 0.955432407
[361] 0.020103327 0.480349245 0.219676864 0.580721362 0.763150533 0.111723589
[367] 0.432606476 0.233814426 0.896905138 0.779284948 0.655762143 0.265846483
[373] 0.998594052 0.194376479 0.729989073 0.314499757 0.786705410 0.594111366
[379] 0.413532920 0.541096019 0.914167993 0.275260648 0.861704816 0.470226520
[385] 0.481021797 0.464061290 0.104119059 0.551640938 0.464588828 0.691944648
[391] 0.166709766 0.654022823 0.819956944 0.958055675 0.553506803 0.816894128
[397] 0.282678348 0.631308894 0.892736071 0.555559580 0.723906581 0.775725360
[403] 0.154018107 0.305249684 0.128935594 0.150918101 0.875427687 0.498218931
[409] 0.568666225 0.349541256 0.264640393 0.441857543 0.829591239 0.417456459
[415] 0.162024105 0.502971723 0.170649109 0.546134019 0.308763712 0.829648761
[421] 0.328447946 0.594306568 0.620603453 0.478654965 0.834866536 0.933587600
[427] 0.186180796 0.208686601 0.638701676 0.721376422 0.038973112 0.329279063
[433] 0.480985240 0.591423783 0.757031864 0.697647803 0.010400392 0.666297142
[439] 0.603005531 0.689365306 0.607362230 0.251026150 0.129639651 0.438477868
[445] 0.736980277 0.684277334 0.935868894 0.582510316 0.411530391 0.263117569
[451] 0.894363732 0.239972805 0.347722611 0.410395859 0.691398945 0.005984595
[457] 0.984082939 0.307698713 0.832515746 0.039797307 0.847054609 0.955068015
[463] 0.519728732 0.338208369 0.515017924 0.303276403 0.474127236 0.480251013
[469] 0.315041670 0.310383151 0.718761340 0.628974175 0.816472427 0.679656544
[475] 0.273254942 0.507028946 0.588256444 0.127227664 0.975938269 0.134291416
[481] 0.625245243 0.715715052 0.711118049 0.154666932 0.521508798 0.377684841
[487] 0.537205308 0.152020830 0.532615447 0.465771348 0.725221018 0.973876633
[493] 0.578021833 0.102194783 0.832184871 0.352575325 0.159285532 0.969319535
[499] 0.897227706 0.839029613 0.154200372 0.639638071 0.786162010 0.109996374
[505] 0.093452483 0.817921812 0.003947767 0.810453171 0.749291247 0.920532965
[511] 0.504377148 0.057174805 0.899816722 0.823018391 0.317578045 0.289705265
[517] 0.739313256 0.599661723 0.003432884 0.862423849 0.223020781 0.550987423
[523] 0.490071480 0.857910281 0.099361687 0.284472311 0.388981156 0.560161466
[529] 0.096988001 0.010198082 0.094068245 0.942742181 0.434822623 0.911529133
[535] 0.759175523 0.131127558 0.913441786 0.470034204 0.358444107 0.364041571
[541] 0.310426387 0.951440288 0.209238516 0.491648076 0.828928960 0.199710259
[547] 0.143237418 0.943864075 0.819103787 0.994096906 0.649292854 0.661925519
[553] 0.313893165 0.631225670 0.523456858 0.208755109 0.743949130 0.271031378
[559] 0.862520068 0.267327814 0.923014251 0.044182619 0.086229393 0.385039424
[565] 0.475900497 0.756979928 0.757424311 0.054893886 0.769732967 0.758382536
[571] 0.282954321 0.383669596 0.384194959 0.516150264 0.981948273 0.410528223
[577] 0.751147010 0.386393209 0.783458724 0.058782616 0.427499315 0.391298520
[583] 0.931776701 0.556793871 0.120572432 0.707962551 0.462226641 0.474767615
[589] 0.629507714 0.606827530 0.334634477 0.434235582 0.070544063 0.882443544
[595] 0.082204119 0.667417050 0.442296623 0.592619126 0.924746159 0.282088651
[601] 0.383615842 0.236533270 0.869970504 0.208162494 0.425854363 0.380281711
[607] 0.477206018 0.072167866 0.857954875 0.245514621 0.645281228 0.451620808
[613] 0.139806294 0.115306781 0.096205555 0.256335655 0.831125081 0.608189689
[619] 0.866267245 0.510264975 0.282855421 0.004223587 0.643981041 0.186199050
[625] 0.381990915 0.768104834 0.812902989 0.367536535 0.527308929 0.273442268
[631] 0.053050711 0.745008680 0.839887692 0.144418685 0.926190011 0.428756422
[637] 0.991833667 0.924514581 0.890513874 0.510225368 0.107084739 0.615642092
[643] 0.005026421 0.008752150 0.603581012 0.431711325 0.629028223 0.175652734
[649] 0.501570310 0.214949786 0.923657804 0.268559732 0.931650449 0.723272851
[655] 0.679978470 0.024338673 0.468577795 0.748782691 0.075614018 0.397286430
[661] 0.193887698 0.006936170 0.194229350 0.693586107 0.286546644 0.844922758
[667] 0.068704801 0.412780717 0.620740432 0.432977412 0.803606532 0.226635758
[673] 0.230176982 0.242935993 0.831149479 0.375128252 0.151039314 0.032120243
[679] 0.187756766 0.445850281 0.900050319 0.504257069 0.122215969 0.926908098
[685] 0.246762938 0.052370280 0.859242800 0.922434842 0.946510519 0.955421069
[691] 0.842434506 0.184011241 0.017509430 0.723062478 0.839897690 0.913261308
[697] 0.962646526 0.143115642 0.357171688 0.715602629 0.272174830 0.384010024
[703] 0.824607254 0.239523810 0.814679147 0.333760003 0.487270405 0.588821623
[709] 0.518436000 0.679966103 0.999071825 0.927369152 0.201890565 0.997368379
[715] 0.051162064 0.168492585 0.354512884 0.190412811 0.218096917 0.356653462
[721] 0.914154787 0.461214456 0.516486750 0.071678607 0.955463334 0.993863089
[727] 0.439455344 0.015145283 0.509487258 0.847208447 0.880935599 0.893728087
[733] 0.672986012 0.001613999 0.254960291 0.465463046 0.620218204 0.501511684
[739] 0.135270093 0.596288985 0.765170902 0.412345233 0.814319271 0.850672180
[745] 0.954611589 0.460554996 0.123486499 0.566178435 0.105274921 0.353608650
[751] 0.793904127 0.896762931 0.901340026 0.298062382 0.732428424 0.669973892
[757] 0.348017400 0.127952424 0.901905190 0.091084878 0.366565739 0.104714119
[763] 0.085826507 0.706428397 0.388813067 0.880243422 0.240607179 0.115360542
[769] 0.320412849 0.126596824 0.629399960 0.192107022 0.696446448 0.173928346
[775] 0.300668138 0.882847705 0.432280857 0.865565139 0.991491257 0.731582637
[781] 0.815063890 0.390947497 0.632155522 0.544244739 0.041557485 0.271666405
[787] 0.573489276 0.282380947 0.870451281 0.018637272 0.017193133 0.055306112
[793] 0.691767592 0.202948314 0.074815559 0.248638300 0.035206627 0.858431174
[799] 0.148718925 0.418077512 0.891005121 0.320607182 0.162690239 0.665697171
[805] 0.184327374 0.923514942 0.987793045 0.363696011 0.162119316 0.859569228
[811] 0.140984077 0.201337705 0.839036733 0.761296694 0.642697772 0.154233583
[817] 0.955440049 0.027986004 0.883504780 0.998659762 0.143035913 0.096865078
[823] 0.678099489 0.657693527 0.382967677 0.346476638 0.615950975 0.849321384
[829] 0.767815803 0.333439662 0.996618066 0.573088007 0.217183533 0.364242429
[835] 0.932898709 0.376785543 0.545240080 0.306562237 0.006244211 0.617554812
[841] 0.315534246 0.945811513 0.415480287 0.030672572 0.714658091 0.664982644
[847] 0.525627550 0.792910202 0.133949209 0.409141587 0.955970225 0.611104671
[853] 0.475096155 0.464747230 0.467583645 0.847956885 0.570669891 0.310682415
[859] 0.163925772 0.893712073 0.167773787 0.644708186 0.604719736 0.218309560
[865] 0.776700907 0.081961660 0.495513093 0.101114808 0.476273650 0.394960369
[871] 0.384058868 0.191210473 0.922020009 0.680499485 0.204441564 0.797908518
[877] 0.877591976 0.163431418 0.713295399 0.183441540 0.114369424 0.151876793
[883] 0.288551545 0.473933880 0.936861812 0.817365777 0.466070015 0.492420256
[889] 0.282226897 0.930504278 0.445338726 0.716203830 0.149268794 0.725011319
[895] 0.515038839 0.054441410 0.246962999 0.009517025 0.244724636 0.822959490
[901] 0.753886830 0.547213650 0.224077433 0.972238222 0.740223524 0.768542438
[907] 0.074040548 0.368006382 0.509589464 0.698355164 0.390293045 0.059122349
[913] 0.307848943 0.870482175 0.895266580 0.122722715 0.457180649 0.927320528
[919] 0.079527115 0.203891597 0.332909486 0.514024490 0.851870720 0.250824761
[925] 0.539617472 0.618245117 0.364576300 0.456044517 0.869975438 0.046643047
[931] 0.672257289 0.528975357 0.931893472 0.320757414 0.982627708 0.413875485
[937] 0.891812389 0.803963009 0.188817104 0.222846496 0.477391089 0.099220486
[943] 0.365515998 0.822975883 0.557462685 0.289026990 0.371307965 0.774700636
[949] 0.285576581 0.248804803 0.468133600 0.745249395 0.867559256 0.960137309
[955] 0.047106965 0.395107792 0.874502115 0.283467547 0.949233127 0.758133483
[961] 0.026794938 0.287685327 0.299802973 0.124524133 0.692048660 0.820411809
[967] 0.155615000 0.704355842 0.004607145 0.421463715 0.088859503 0.796308352
[973] 0.210012669 0.551392959 0.074530150 0.631403730 0.654742725 0.209346641
[979] 0.841655196 0.603283111 0.592958029 0.373567467 0.032088855 0.577804767
[985] 0.709109305 0.214113422 0.803228684 0.278104092 0.206070874 0.301122506
[991] 0.815469258 0.584756653 0.817215012 0.661182737 0.626208931 0.050829495
[997] 0.435513700 0.723054568 0.343677806 0.830591328

T=0.125

So, the retailer profit for each consumer is

T=0.125
y=mean(x*T)

.5046824

(b) when we use T=0.15, there is not change in profit maximizing price.