ЭКОНОМЕТРИКА и математическая экономика / Солопахо А.В. Математика в экономике (ТамбовГТУ)

^_l_jfbgbjh\Zggmx fh^_ev Ijb wlhf \Z`gh qlh ihemq_ggZy fh^_ev g_ y\ey_lky ebg_cghc b l_f kZfuf g_ fh`_l [ulv j_r_gZ kbfie_dk f_lh^hf

H^gZdh k ihfhsvx bl_jZpbhggh]h rZ]h\h]h ijhp_kkZ ih^[hjZ ieZgZ fh`gh gZclb __ ijb[eb`_ggh_ j_r_gb_ NZdlbq_kdb ieZg ^ey dhlhjh]h ]ZjZglbjh\ZggZ lZ beb bgZy

ijb h]jZgbq_gbyo ihh[hjm^h\Zgbx

jZ[hq_fm \j_f_gb

fZl_jbZevguf j_kmjkZf

h[t_fm khhl\_lkl\_ggh \ ieZgh\uo b khihklZ\bfuo p_gZf

ih h[t_fm \ gZlmjZevghf \ujZ`_gbb

Ijbklmibf d ih^[hjm ^himklbfh]h ieZgZ RZ] Imklv gZijbf_j

Ih h[t_fm \ ieZgh\uo b khihklZ\bfuo p_gZf Z lZd `_ ih h[t_fm \ gZlmjZevghf \ujZ`_gbb ieZg y\ey_lky ^himklbfuf

GZoh^bf l_i_jv agZq_gb_ nmgdpbhgZeZ

<hafh`gh ihemq_ggh_ agZq_gb_ fh`gh kqblZlv ^himklbfuf H^gZdh fu \b^bf qlh mf_gvrb\ \uimkd \lhjh]h ijh^mdlZ b m\_ebqb\ \uimkd lj_lv_]h j_amevlZl \_jhylgh memqrblky

RZ] JZkkfhljbf ieZg

Ih h[t_fm \ ieZgh\uo b khihklZ\bfuo p_gZf Z lZd `_ ih h[t_fm \ gZlmjZevghf \ujZ`_gbb ieZg y\ey_lky ^himklbfuf

GZoh^bf l_i_jv agZq_gb_ nmgdpbhgZeZ

Ihemq_ggh_ agZq_gb_ agZqbl_evgh ij_\urZ_l ij_^u^msb_ GZijZ\e_gby baf_g_gby ieZgZ \_^msb_ d ^Zevg_cr_fm _]h memqr_gbx m`_ g_ klhev hq_\b^gu dZd gZ ij_^u^msbo rZ]Zo Ihwlhfm hklZgh\bfky gZ ^hklb]gmlhf j_amevlZl_

G_h[oh^bfh _s_ jZa kdZaZlv qlh j_r_gb_ jZkkfZljb\Z_fuo fh^_e_c hkh[_ggh \ kemqZ_ j_Zevguo aZ^Zq k [hevrbf dhebq_kl\hf i_j_f_gguo b h]jZgbq_gbc ij_^klZ\ey_lky \hafh`guf ebrv k bkihevah\Zgb_f W<F < wlhf kemqZ_ fu e_]dh k ihfhsvx ba\_klguo qbke_gguo f_lh^h\ fh]eb [u gZclb ijZdlbq_kdb lhqgh_ j_r_gb_ aZ^Zqb hilbfbaZpbb ieZgZ

L_fZ WE?F?GLU L?HJBB B=J

H^gZ ba oZjZdl_jguo q_jl \kydh]h h[s_kl\_ggh]h beb khpbZevgh wdhghfbq_kdh]h y\e_gby khklhbl \ fgh`_kl\_gghklb \ fgh]hklhjhgghklb bkoh^h\ wlh]h y\e_gby \ gZebqbb klhjhg gZ^_e_gguo jZaebqgufb bgl_j_kZfb beb ohly [u \ gZebqbb g_kdhevdbo jZaebqguo Zdlb\guo lhq_d aj_gby gZ y\e_gb_ b _]h bkoh^

LZd gZijbf_j ijboh^blky kqblZlvky k jZaebqb_f ebqguo dhee_dlb\guo b h[s_kl\_gguo bgl_j_kh\ < mkeh\byo ieZgh\hc kbkl_fu ohayckl\Z g_h[oh^bfh khq_lZlv hljZke_\u_ b j_]bhgZevgu_ bgl_j_ku Z lZd`_ ih^qZk ijhlb\hj_qZsb_ ^jm] ^jm]m bgl_j_ku hl^_evguo \_^hfkl\ Hkgh\gu_ ljm^ghklb \g_^j_gby b jZkijhkljZg_gby gh\h\\_^_gbc ij_^hij_^_eyxlky ijhlb\hj_qb_f f_`^m k_]h^gyrg_c \u]h^hc b

[m^msbf wnn_dlhf < wlhf kfuke_ fh`gh kdZaZlv qlh ex[h_ khpbZevgh wdhghfbq_kdh_ y\e_gb_ gZ^_e_gh q_jlZfb dhgnebdlZ KblmZpby gZau\Z_lky dhgnebdlghc _keb \ g_c mqZkl\mxl klhjhgu bgl_j_ku dhlhjuo iheghklvx beb qZklbqgh ijhlb\hiheh`gu

L_hjby b]j hlghkblky d fZl_fZlbq_kdhfm h[_ki_q_gbx fh^_ebjh\Zgby y\e_gbc \ hkgh\ghf khpbZevgh wdhghfbq_kdhc oZjZdl_jZ

Hkgh\gu_ ihgylby b h[sZy deZkkbnbdZpby b]j

<kydZy fZl_fZlbq_kdZy fh^_ev khpbZevgh wdhghfbq_kdh]h y\e_gby beb dhgnebdlZ

^he`gZ \ujZ`Zlv ijbkmsb_ _fm q_jlu l _ ih djZcg_c f_j_ hljZ`Zlv ke_^mxsb_ _]h dhfihg_glu

Z aZbgl_j_kh\Zggu_ klhjhgu [ \hafh`gu_ ^_ckl\by dZ`^hc ba klhjhg \ bgl_j_ku klhjhg

>Z^bf nhjfZevgh_ hibkZgb_ mdZaZgguo dhfihg_gl dhgnebdlZ \\h^y ijb wlhf ijbgylmx l_jfbgheh]bx b h[hagZq_gby

AZbgl_j_kh\Zggu_ klhjhgu [m^ml ^Ze__ gZau\Zlvky b]jhdZfb beb ebpZfb Z fgh`_kl\h \k_o b]jhdh\ [m^_l h[hagZqZlvky q_j_a , >Ze__ fu h]jZgbqbfky jZkkfhlj_gb_f kemqZy dh]^Z fgh`_kl\h , dhg_qgh G_ gZjmrZy h[sghklb fh`gh ijbgbfZlv qlh , ^i`

Ex[h_ \hafh`gh_ ^ey b]jhdZ L • , ^_ckl\b_ gZau\Z_lky _]h kljZl_]b_c

Fgh`_kl\h \k_o kljZl_]bc b]jhdZ L h[hagZqbf q_j_a ;L < mkeh\byo dhgnebdlZ dZ`^uc b]jhd L • , \u[bjZ_l g_dhlhjmx k\hx kljZl_]bx [L • ;L \ j_amevlZl_ q_]h kdeZ^u\Z_lky gZ[hj kljZl_]bc o [ [ [Q gZau\Z_fuc kblmZpb_c Fgh`_kl\h \k_o kblmZpbc h[hagZqbf =

AZbgl_j_kh\Zgghklv b]jhdh\ \ujZ`Z_lky \ lhf qlh dZ`^hfm b]jhdm L • , \ lhc beb bghc kblmZpbb [ • = ijbibku\Z_lky qbkeh \ujZ`Zxs__ kl_i_gv m^h\e_l\hj_gby _]h bgl_j_kh\ \ wlhc kblmZpbb Wlh qbkeh gZau\Z_lky \ub]jur_f b]jhdZ L \ kblmZpbb o b h[hagZqZ_lky q_j_a GL o <_ebqbgZ GL o gZau\Z_lky nmgdpb_c \ub]jurZ b]jhdZ L b

h[uqgh y\ey_lky ^_ckl\bl_evguf qbkehf

< wlbo mkeh\byo ijhl_dZgb_ dhgnebdlZ khklhbl \ \u[hj_ dZ`^uf b]jhdhf L • , _]h kljZl_]bb [L • ;L b \ ihemq_gbb bf \ keh`b\r_cky kblmZpbb o [ [ [Q ba g_dhlhjh]h bklhqgbdZ \ub]jurZ GL o LZdbf h[jZahf fgh`_kl\h b]jhdh\ bo kljZl_]bc b bo nmgdpbc \ub]jurZ iheghklvx hibku\Zxl g_dhlhjmx b]jm = qlh nhjfZevgh aZibku\Zxl lZd

G O O G O O

Qlh[u khdjZlblv mihlj_[e_gb_ bg^_dkh\ fgh`_kl\Z kljZl_]bc b]jhdh\ b ZglZ]hgbklbq_kdhc b]j_ [m^ml h[uqgh h[hagZqZlvky q_j_a ; b < Z nmgdpby \ub]jurZ GL

q_j_a + >ey Zglh]hgbklbq_kdbo b]j k ^\mfy b]jhdZfb \\h^blky ke_^mxs hij_^_e_gb_

Hij_^_e_gb_ Imklv dZ`^hc kljZl_]bb ]h b]jhdZ \aZbfhh^gha gZqgh ihklZ\e_gZ \ khhl\_lkl\b_ kljhdZ g_dhlhjhc fZljbpu : Z dZ`^hc kljZl_]bb ]h b]jhdZ \aZbfhh^ghagZqgh ihklZ\e_g \ khhl\_lkl\b_ klhe[_p wlhc fZljbpu FZljbpm : gZau\Zxl ieZl_`ghc beb fZljbp_c b]ju _keb _z we_f_gl DLM jZ\_g \ub]jurm ]h b]jhdZ l _ ijhb]jurm ]h ijb \u[hj_ f b]jhdhf L c kljZl_]bb Z f M c

Ihwlhfm ZglZ]hgbklbq_kdZy b]jZ iheghklvx hibku\Z_lky _^bgkl\_gghc fZljbp_c b \ khhl\_lkl\bb k wlbf gZau\Z_lky fZljbqghc

Ijbf_j R\_cgh_ ij_^ijbylb_ ieZgbjm_l d fZkkh\hfm \uimkdm gh\mx fh^_ev h^_`^u Kijhk gZ wlm fh^_ev g_ fh`_l [ulv lhqgh hij_^_e_g H^gZdh fh`gh

ij_^iheh`blv qlh _]h \_ebqbgZ oZjZdl_jbam_lky lj_fy \hafh`gufb khklhygbyfb, ,,

,,, K mq_lhf wlbo khklhygbc ZgZebabjmxlky ljb \hafh`guo \ZjbZglZ \uimkdZ ^Zgghc fh^_eb : ; < DZ`^uc ba wlbo \ZjbZglh\ lj_[m_l k\hbo aZljZl b h[_ki_qb\Z_l \ dhg_qghf kq_l_ jZaebqguc wnn_dl Ijb[uev luk j dhlhjmx ihemqZ_l ij_^ijbylb_ ijb ^Zgghf h[t_f_ \uimkdZ fh^_eb b khhl\_lkl\mxs_f khklhygbb kijhkZ hij_^_ey_lky fZljbp_c

<u[hj \ lhf beb bghf kfuke_ gZbemqr_]h \ZjbZglZ \uimkdZ fh^_e_c fh`gh ihgbfZlv dZd fZljbqgmx b]jm

Ihgylb_ hilbfZevghklb kljZl_]bb

L_hjby b]j dZd fZl_fZlbq_kdZy ^bkpbiebgZ \ __ kh\j_f_gghf khklhygbb aZgbfZ_lky ghjfZlb\guf bamq_gb_f b]j l _ kqblZ_l k\h_c aZ^Zq_c mklZgh\blv dZdh_ ih\_^_gb_ b]jhdh\ ke_^m_l kqblZlv hilbfZevguf jZamfguf p_e_khh[jZaguf

HilbfZevghklv kljZl_]bb hq_\b^gh fh`gh ihgbfZlv ih jZaghfm Lh _klv ihgylby hilbfZevghklb \ l_hjbb b]j b hilbfZevgh]h j_r_gby b]ju g_ y\eyxlky h^ghagZqgufb Zijbhjgufb Z[khexlgufb <f_kl_ k l_f wlb ihgylby y\eyxlky h[t_dlb\gufb l _ dZ`^uc \ZjbZgl hilbfZevghklb ih^^Z_lky lhqghfm hibkZgbx ijb ihfhsb g_^\mkfuke_gguo fZl_fZlbq_kdbo nhjfmebjh\hd L_f kZfuf jZaebqgu_ kh^_j`Zl_evgu_ ij_^klZ\e_gby h[ hilbfZevghklb fh]ml ijb\h^blv d hlebqZxsbfky fZl_fZlbq_kdbf fh^_eyf

Hkgh\gufb kh^_j`Zl_evgufb q_jlZfb hilbfZevghklb \ ijbf_g_gbb d bkoh^m beb d fgh`_kl\m bkoh^h\ dhgnebdlZ fh`gh kqblZlv bglmblb\gu_ ij_^klZ\e_gby h \u]h^ghklb b kijZ\_^eb\hklb

Fh`gh gZijbf_j hilbfZevghc kblmZpb_c kqblZlv lZdmx \ dhlhjhc h^gh\j_f_ggh

^hklb]Zxl k\hbo fZdkbfmfh\ nmgdpbb \ub]jurZ dZ`^h]h ba b]jhdh\ Mkeh\b_ hilbfZevghklb \ wlhf kfuke_ ^ey kblmZpbb o ^ey b]ju nhjfZevgh fh`gh aZibkZlv dZd

GL o GL o ^ey ex[uo L • , [ • =

<u]h^ghklv lZdhc kblmZpbb ykgZ

G_ljm^gh \b^_lv h^gZdh qlh kms_kl\h\Zgb_ \ [_kdhZebpbhgghc b]j_ kblmZpbc hilbfZevguo \ lhevdh qlh hibkZgghf kfuke_ y\ey_lky kjZ\gbl_evgh j_^dbf bkdexq_gb_f dZd b ex[h_ kh\iZ^_gb_ fZdkbfmfh\ g_kdhevdbo nmgdpbc < kmsghklb j_Zebam_fhklv wlh]h ijbgpbiZ hilbfZevghklb khhl\_lkl\m_l keZ[hklb dhgnebdlguo q_jl fh^_ebjm_fh]h y\e_gby [ebahklb p_e_c _]h mqZklgbdh\

?kl_kl\_ggh ihwlhfm bkdZlv ^jm]b_ ij_^klZ\e_gby h[ hilbfZevghklb [ulv fh`_l g_ klhev [_kkihjgu_ gh aZlh [he__ qZklh j_Zebam_fu_

H^ghc ba gZb[he__ ieh^hl\hjguo nhjf j_ZebaZpbb ij_^klZ\e_gbc h[ hilbfZevghklb fh`gh kqblZlv ihgylb_ jZ\gh\_kby khklhys__ \ ke_^mxs_f

Hij_^_e_gb_ KblmZpby gZau\Z_lky jZ\gh\_kghc _keb gb h^bg ba b]jhdh\ g_ aZbgl_j_kh\Zg \ lhf qlh[u __ gZjmrblv l _ hldehgblvky hl g__

?keb b]jZ = y\ey_lky ZglZ]hgbklbq_kdhc lh jZ\gh\_kgZy kblmZpby o bf__l \b^ o

m b nhjfZevgh ^ey jZ\gh\_kghc kblmZpbb fh`gh aZibkZlv

G o m d G o m d G o m

ijb ex[uo [ • ; \ • < M`_ ba lh]h qlh \ bkihevam_lky g_ kljh]h_ g_jZ\_gkl\h

f_gvr_ beb jZ\gh d ke_^m_l qlh jZ\gh\_kguo kblmZpbc fh`_l [ulv g_kdhevdh Hij_^_e_gb_ < kemqZ_ ZglZ]hgbklbq_kdhc b]ju kblmZpby jZ\gh\_kby gZau\Z_lky

k_^eh\hc lhqdhc HdZau\Z_lky qlh nmgdpby \ub]jurZ b]ju \h \k_o __ k_^eh\uo lhqdZo ijbgbfZ_l h^gh b lh `_ agZq_gb_ dhlhjh_ gZau\Z_lky p_ghc b]ju beb agZq_gb_f b]ju

Bf_ggh kblmZpbb jZ\gh\_kby fh]ml [ulv ij_^f_lhf mklhcqb\uo ^h]h\hjh\ f_`^m b]jhdZfb <kydZy ihiuldZ aZnbdkbjh\Zlv \ ^h]h\hj_ g_jZ\gh\_kgmx kblmZpbx [m^_l hagZqZlv qlh ohly [u m h^ghc ba ^h]h\Zjb\Zxsboky klhjhg gZc^_lky lZdZy kljZl_]by qlh \u[hj __ \f_klh ij_^mkfhlj_gghc ^h]h\hjhf m\_ebqbl \ub]jur wlh]h b]jhdZ L_f kZfuf \hagbdZxl fhlb\u d gZjmr_gbx ^h]h\hjZ

H[uqgh bf_ggh jZ\gh\_kgu_ kljZl_]bb kqblZxl hilbfZevgufb b gZau\Zxl bo j_r_gb_f b]ju

Ijbgpbi hilbfZevghklb \ [_kdhZebpbhgghc b]j_ khklhysbc \ hkms_kl\e_gbb b]jhdZfb __ kblmZpbc jZ\gh\_kby y\ey_lky [he__ keZ[uf b qZs_ j_Zebam_fuf q_f ijbgpbi \ujZ`Z_fuc khhlghr_gb_f H^gZdh jZ\gh\_kgu_ kblmZpbb hij_^_ey_fu_ khhlghr_gb_f lZd `_ kms_kl\mxl g_ ^ey \kydhc b]ju

HdZau\Z_lky ba wlhc kblmZpbb lZd`_ fh`gh ihiulZlvky gZclb \uoh^ Ijb hlkmlkl\bb \ b]j_ kblmZpbc jZ\gh\_kby ijb bkihevah\Zgbb b]jhdZfb _^bgkl\_gghc kljZl_]bb ba bf_xsboky m dZ`^h]h ba gbo fgh`_kl\Z _kl_kl\_ggh ihklZ\blv \hijhk h jZkrbj_gbb ihgylby kljZl_]bb LZdhf qlh[u kj_^b kblmZpbc khklZ\e_gguo ba gh\uo h[h[s_gguo kljZl_]bc \k_ `_ fh]eb [u gZclbkv jZ\gh\_kgu_

Hij_^_e_gb_ We_f_glu fgh`_kl\Z ; b < gZau\Zxl qbklufb kljZl_]byfb b]jhdh\ Hij_^_e_gb_ <_dlhj dZ`^Zy ba dhfihg_gl dhlhjh]h ihdZau\Z_l hlghkbl_evgmx

qZklhlm bkihevah\Zgby b]jhdhf khhl\_lkl\mxs_c qbklhc kljZl_]bb gZau\Z_lky kf_rZgghc kljZl_]b_c ^Zggh]h b]jhdZ

Ba ^Zggh]h hij_^_e_gby g_ihkj_^kl\_ggh ke_^m_l qlh kmffZ dhfihg_gl mdZaZggh]h

\_dlhjZ jZ\gZ _^bgbp_ Z kZfb dhfihg_glu g_ hljbpZl_evgu Ykgh lZd `_ qlh qbklmx kljZl_]bx fh`gh jZkkfZljb\Zlv dZd qZklguc kemqZc kf_rZgghc

M^h[ghc bgl_jij_lZpb_c kf_rZgghc kljZl_]bb y\ey_lky __ ij_^klZ\e_gb_ dZd kemqZcgh]h \u[hjZ b]jhdhf k\hbo qbkluo kljZl_]bc \ khhl\_lkl\bb k \_jhylghklyfb aZ^Z\Z_fufb hlghkbl_evgufb qZklhlZfb \kihfgbf deZkkbq_kdh_ hij_^_e_gb_

\_jhylghklb kh[ulby Ijbq_f \u[hju b]jhdh\ g_aZ\bkbfu Z \ub]jur \ kblmZpbb \ kf_rZgguo kljZl_]byo hij_^_ey_lky dZd fZl_fZlbq_kdh_ h`b^Zgb_ kemqZcgh]h \ub]jurZ

J_r_gb_ fZljbqghc b]ju \ qbkluo kljZl_]byo

fZdkbfbghf Z khhl\_lkl\mxsZy _fm kljZl_]by kljhdZ fZdkbfbgghc

:gZeh]bqgh \lhjhc b]jhd lZd`_ bkoh^y ba gZbom^r_]h bkoh^Z ^he`_g jZkkm`^Zlv

Ijbf_j >ey b]jh\hc fZljbpu ba ijbf_jZ gZclb k_^eh\u_ lhqdb _keb hgb

_klv GZoh`^_gb_ gb`g_c b \_jog_c p_g b]ju jZ\gh dZd b ijh\_jdZ kms_kl\h\Zgby k_^eh\uo lhq_d b bo gZoh`^_gb_ ^ey fZljbqguo b]j m^h[gh ijh\h^blv ih ke_^mxs_c ko_f_

<b^bf qlh fZdkbfbg bf__l f_klh \ c kljhd_ l _ ijb i_j\hc kljZl_]bb ]h b]jhdZ FbgbfZdk \ i_j\hf klhe[p_ l _ ijb c kljZl_]bb ]h b]jhdZ Ijb wlhf hgb kh\iZ^Zxl lh _klv bf__lky k_^eh\Zy lhqdZ b wlh kblmZpby [

J_r_gb_ fZljbqghc b]ju \ kf_rZgguo kljZl_]byo

B]jZ aZ^ZggZy g_dhlhjhc fZljbp_c fh`_l g_ bf_lv k_^eh\hc lhqdb