㸨㟁Ꮚ࣓ࢹᕤᏛ⛉
㧗ᑓᩘᏛ₇⩦ၥ㢟ࡋ࡚ࡢ࣑ࢡࣟ⤒῭Ꮫ
̿ᚤศࠊ೫ᚤศ㛵ࡍࡿ₇⩦ၥ㢟̿
ᔱ ୍ே
㸨
㸦㸰㸮㸯㸵ᖺ㸯㸯᭶㸰㸲᪥ཷ⌮㸧
㸯㸬ࡲ࠼ࡀࡁ ⩌㤿㧗ᑓ㟁Ꮚ࣓ࢹᕤᏛ⛉ࡽࡣẖᖺࡢࡼ࠺Ꮫ ࡢ⤒῭Ꮫ㒊⦅ධࡍࡿᏛ⏕ࡀ࠸ࡿࠋ⤒῭Ꮫ㒊ࡀ㧗ᑓ⏕ࢆཷ ࡅධࢀࡿ࡞せᅉࡋ࡚ࡣࠊ㧗ᑓ༞ᴗ⏕ࡣ୍ᐃࣞ࣋ࣝࡢᩘ ᏛࡢᏛຊࢆ᭷ࡍࡿࡇ࠶ࡿᛮࢃࢀࡿ㸬ࡲࡓࠊ༞ᴗ◊✲ ࡞࡛ࠊ⤒῭Ꮫ㒊㐍ᏛࡍࡿᏛ⏕ࡶᏛࡪ࠺ࡕ࣑ࢡ ࣟ⤒῭Ꮫࡢศ㔝ࡣ㧗ᑓ㸰ࠊ㸱ᖺ⏕ࡢᏛ⏕࡞ࡽࡤ⤒῭Ꮫࡢ ▱㆑ࡀ࠶ࡲࡾ࡞ࡃ࡚ࡶゎࡅࡿ⯆῝࠸ィ⟬ၥ㢟ࡀ࠶ࡿࡇ ࡀࢃࡗࡓ㸬ᮏㄽᩥ࡛ࡣࠊ㧗ᑓ㸰㸪㸱ᖺ⏕⛬ᗘࡢᚤศࠊ ೫ᚤศ㛵ࡍࡿຊࡀ࠶ࢀࡤゎࡃࡇࡢ࡛ࡁࡿ࣑ࢡࣟ⤒῭ Ꮫࡢศ㔝ࡢィ⟬ၥ㢟ࡀ࠶ࡿࡇࢆ⤂ࡍࡿ㸬 ලయⓗࡣⱱᮌ1)ࡼࡾࠊ࠸ࡃࡘࡢᆺⓗ࡞ၥ㢟ࢆ⤂ ࡍࡿࠋၥ㢟ᩥࡘ࠸࡚ࡣࢫ࣮࣌ࢫࡢ㛵ಀୖࠊཎᩥࡢពࢆ ᦆࡡ࡞࠸⛬ᗘᅽ⦰ࡋ࡚⤂ࡍࡿ㸬࡞࠾ࠊྠ᭩ࡣ㒊࡛ 㸯㸯㸳ၥࡀ㍕ࡗ࡚࠸ࡿࡀࠊ๓༙ࡢ㸵㸰ၥࡀ࣑ࢡࣟ⤒῭Ꮫ 㛵ࡍࡿၥ㢟࡛࠶ࡿ㸬ࡑࡢ࠺ࡕࠊ㸯㸲ၥ⛬ᗘࡀ㧗ᑓ㸰ᖺ⏕⛬ ᗘ࡛ゎࡅࡿ㸯ኚᩘࡢᚤศ㛵ࡍࡿၥ㢟࡛࠶ࡾࠊ㸯㸶ၥ⛬ᗘ ࡀ㧗ᑓ㸱ᖺ⏕⛬ᗘ࡛ゎࡅࡿ೫ᚤศ㛵ࡍࡿၥ㢟࡛࠶ࡾࠊࡑ ࡢࡢ㸲㸮ၥࡣᩘᏛⓗࡣึ➼ⓗ࡛࠶ࡿࡋ࡚ࡶ࠶ࡿ⛬ ᗘࡢ⤒῭Ꮫ㛵ࡍࡿ▱㆑ࠊᖖ㆑ࡀᚲせᛮࢃࢀࡿၥ㢟࡛࠶ ࡿ㸬⤒῭Ꮫ㛵ࡍࡿ࠶ࡿ⛬ᗘࡢ▱㆑ࡀᚲせ࡞ၥ㢟ࡣࢤ࣮ ࣒ࡢ⌮ㄽ㛵ࡍࡿၥ㢟ࡸࠊᾘ㈝⪅వࠊ⛯㔠࡞ࡢᴫᛕࡀ ࠶ࢀࡤゎࡃࡇࡢ࡛ࡁࡿၥ㢟ࡶྵࡲࢀࡿ㸬࡞࠾ࠊ➹⪅ࡣ㧗 ᑓᩘᏛ㛵ࡋ࡚ࡣࡦ࠾ࡾࡢᩍᤵ⤒㦂ࡀ࠶ࡿࡀࠊ⤒῭Ꮫ 㛵ࡋ࡚ࡣࢇᏛࢇࡔ⤒㦂ࡢ࡞࠸⪅࡛࠶ࡿ㸬 㸰㸬ᚤศࠊ೫ᚤศ࡛ゎࡅࡿၥ㢟 ࡇࡢ❶࡛ࡣⱱᮌࡢၥ㢟㞟ࡼࡾᚤศࠊ೫ᚤศࢆ࠺ࡇࡀ ࡛ࡁࢀࡤࠊ⤒῭Ꮫࡢ▱㆑ࡀࢇ࡞ࡃ࡚ࡶゎࡃࡇࡢ࡛ ࡁࡿၥ㢟ࡀ࠶ࡿࡇࢆලయⓗ⤂ࡍࡿ㸬࡞࠾㈝⏝㛵ᩘ㸰㸧 ࡞ࠊ⌮ゎࡍࡿࡢࡀẚ㍑ⓗᐜ᫆ᛮࢃࢀࡿ⤒῭Ꮫ⏝ㄒࡣࡑ ࡢࡲࡲ࠺㸬࠸ࡃࡘࡢ⤒῭Ꮫࡢᑓ㛛⏝ㄒࡘ࠸࡚ࡶࠊ▱ ࡽ࡞ࡃ࡚ࡶၥ㢟ࢆゎࡃ࠺࠼࡛ࡉᨭ㞀࡞ࡽ࡞࠸ᛮ ࢃࢀࡿࡶࡢࡣࡑࡢࡲࡲ࠺㸬ࡲࡓࠊၥ㢟␒ྕ࠾ࡼࡧ㸦ⱱᮌ ゎ㸧ࡣⱱᮌࡢၥ㢟㞟࠾ࡅࡿၥ㢟␒ྕ࠾ࡼࡧࡑࡢゎ⟅ࡢᢤ ⢋࡛࠶ࡿ㸬㸦ゎ㸧࠶ࡿࡢࡣ➹⪅ࡢゎ࡛࠶ࡿ㸬 㸰㸬㸯 ₶ࢆ᭱ࡍࡿၥ㢟 ၥ㢟㸯㸷➇தᕷሙ࠾࠸࡚ࠊ࠶ࡿ㈈ࢆ⏕⏘ࡋ࡚࠸ࡿ ᴗࡢ⥲㈝⏝㛵ᩘࡀࠊ[[[ [ ⏕⏘㔞 ࡛♧ࡉࢀ࡚࠸ ࡿࡍࡿ㸬㈈ࡢ౯᱁ࡀ㸴㸮࡛࠼ࡽࢀࡓࡁࠊࡇࡢᴗࡢ ₶ࡀ᭱࡞ࡿ⏕⏘㔞ࡣ࠸ࡃࡘ࡞ࡿ㸬 㸦ゎ㸧 ⏕⏘ࡋࡓศࡔࡅࢀࡿ⪃࠼ࡿࠋ₶㛵ᩘ I [ 㸱㸧 ࡣୖ㧗 [ ⥲㈝⏝㛵ᩘࡢᕪ࡛࠼ࡽࢀࡿ㸬ࡍ࡞ࢃࡕࠊ I [ [ [[[ ࡞ࡿI̓ [ [ [ ࡼ ࡾࠊI [ ࡢ [! ࠾ࡅࡿ್᭱ࢆ࠼ࡿ [ࡣ [ ồࡲ ࡿ㸬 㸦ⱱᮌゎ㸧౯᱁㸻㝈⏺㈝⏝4)ࡼࡾ [ ồࡲࡿ㸳㸧㸬 㸰㸬㸰 ⊂༨ᴗࡢ₶ࢆ᭱ࡍࡿၥ㢟 ၥ㢟㸱㸶 ⊂༨ᴗࡢ⏕⏘ࡍࡿ㈈ࡘ࠸࡚ࠊ㟂せ㛵ᩘ㝈 ⏺ ㈝ ⏝ 㛵 ᩘ ࡀ ḟ ࡢ ࡼ ࠺ ࠼ ࡽ ࢀ ࡚ ࠸ ࡿ 㸬 㟂 せ 㛵 ᩘ:D=10-2P, 㝈⏺㈝⏝㛵ᩘ:x-5 ࡇࡢࡁࠊ⊂༨౯᱁⏕⏘ 㔞ࡣ࠸ࡃࡽ࡞ࡿ㸬 㸦ゎ㸧 ⊂༨ᕷሙ࡛ࡣ㟂せ㹂⏕⏘㔞 [ ࡣ➼ࡋ࠸㸬₶㛵 ᩘࢆ I [ ࠊ㈝⏝㛵ᩘࢆ J [ ࡍࡿࠊI [ [ [J [ ࡞ࡿ㸬 I̓ [ [ [ [ ࡼࡾࠊ[ 3 ồࡲࡿ㸬 㸰㸬㸱 ₶ࡢࢆ᭱ࡍࡿၥ㢟㸦㸰ኚᩘ㸧 ၥ㢟㸴㸯 ࠶ࡿᐇᴗᐙࡣࠊ; ㈈ࢆ⏕⏘ࡍࡿᴗ㸯㈈ࢆ⏕ ⏘ࡍࡿᴗ㸯ࡢ㸰ࡢ୧᪉ࢆ⤒Ⴀࡋ࡚࠸ࡿ㸬 ࡇࡢࠊᐇᴗᐙ ࡀࠊ୧ᴗࡽࡢ₶ࢆ᭱ࡍࡿࡼ࠺⾜ືࡍࡿሙྜࠊ୧ ᴗࡢ⏕⏘㔞ࡣࡑࢀࡒࢀ࠸ࡃࡽ࡞ࡿ㸬 ᴗ㸯ࡢ㈝⏝ 㛵ᩘࡣ [ࠊᴗࡢ㈝⏝㛵ᩘࡣ \[\ࠊ; ㈈ࠊ< ㈈ࡢ ᕷሙ౯᱁ࡣࡑࢀࡒࢀࠊ㸳㸮㸪㸲㸮ࡍࡿ㸬 㸦ゎ㸧㸰ࡘࡢᴗࡢ₶㛵ᩘࡢ I [ f(x)=50x-(2x2+10)+40y-(y2+2xy+10)࡞ࡿ㸬"# "$ =50-4x-2y=0,%%"#"& =40-2x-2y=0 ࡼࡾ x=5, y=15 ồࡲࡿ㸬
㸰㸬㸲 㸰ࡘࡢ㸰ኚᩘ㛵ᩘࡢ⛸⥺ࡢⅬࢆồࡵࡿၥ㢟 ၥ㢟㸲㸲 ྠࡌ㈈ࢆ⏕⏘ࡍࡿᴗ㸯ࠊᴗ㸰ࡽ࡞ࡿ」༨ ᕷሙࡢ㟂せ㛵ᩘࡀࠊ' 3 3 ㈈ࡢ౯᱁ ࡛♧ࡉࢀ࡚࠸ࡿ ࡍࡿ㸬 ᴗ㸯ࡢ⥲㈝⏝㛵ᩘࡣ [ࠊᴗ㸰ࡢ⥲㈝⏝㛵ᩘ ࡣ \ ࡛࠶ࡿࡍࡿ㸬ࢡ࣮ࣝࣀ࣮ᆒ⾮࠾ࡅࡿ୧ᴗࡢ⏕ ⏘㔞ࡣࡑࢀࡒࢀ࠸ࡃࡽ࡞ࡿ㸬 㸦ゎ㸧 㸰ࡘࡢᴗࡢ₶㛵ᩘ f(x,y),g(x,y)ࡣ࠾ࡢ࠾ࡢࠊ f(x,y)=(20-x/2-y/2)x-2x, g(x,y)=(20-x/2-y/2)y-2y ࡞ࡿ㸬"# "$ =18-x-y/2=0 ,%%"#
"&=18-x/2-y=0 ࡼࡾ x=y=12 ồࡲࡿ㸬 ⿵㊊㸧 ࡘࡢ␗࡞ࡿ ኚᩘ㛵ᩘࢆ⪃࠼ࠊྛࠎࡢ㛵ᩘࡘ࠸ ࡚ࠊ᪉ࡢኚᩘࢆᡤࡋ࡚᭱࡞ࡿ᮲௳ࢆ೫ᚤศࡼ ࡾồࡵࡿ㸬ࡇࡢࡼ࠺࡞⪃࠼᪉ࢆࡍࡿࡇࡣ㏻ᖖࡢ㧗ᑓᩘᏛ ࡛ࡣ࠶ࡲࡾ࡞࠸ࡢ࡛ࡣ࡞࠸⪃࠼ࡽࢀࡿ㸬 㸰㸬㸳 ᦆ┈ศᒱⅬࢆồࡵࡿၥ㢟 ၥ㢟㸰㸮 ࠶ࡿᴗࡣ➇தᕷሙ࠾࠸࡚⏕⏘≀ࢆ㈍ ࡋ࡚࠸ࡲࡍ㸬ࡇࡢᴗࡢ⥲㈝⏝㛵ᩘࡀࠊI [ [[[ ࡛♧ࡉࢀ࡚࠸㸬ࡇࡢᴗࡢᦆ┈ศᒱⅬ࠾ࡅࡿ⏕⏘㔞ࡣ࠸ ࡃࡽ࡞ࡾࡿ㸬 㸦ⱱᮌゎ㸧ᖹᆒ㈝⏝ I [ [ I̓ [ 㝈⏺㈝⏝ࡼࡾࠊ [ [[ 㸬ࡇࢀࡼࡾࠊ[ 㸬
㸦⿵㊊㸧I [ [ I̓ [ ࡢゎࢆ [ D ࡍࡿࠊ᭤⥺ \ I [
ୖࡢⅬ DI D ࠾ࡅࡿ᥋⥺ࡣཎⅬࢆ㏻ࡿࠋ3 I̓ D ࡍ ࡿࠊୖグࡢ᥋⥺ࡣ \ 3 [ ࡞ࡿ㸦ᅗ㸯㸧㸬ࡶࡋࡶࠊ⏕⏘ရ ࡢ౯᱁ 3 ࡀ 3!3 ࢆ‶ࡓࡏࡤࠊ┈ࡢฟࡿ⏕⏘㔞ࡢ⠊ᅖࡀᏑ ᅾࡍࡿ㸬ࡇࡢࡼ࠺ࠊ᭤⥺ࡢ᥋⥺࡛ཎⅬࢆ㏻ࡿࡶࡢࢆồࡵ ࡼ࠸࠺ᩘᏛࡢၥ㢟ࡀ⤒῭Ꮫⓗࡣពࢆࡶࡘ㸬࡞࠾ࠊ᧯ ᴗṆⅬࢆồࡵࡿၥ㢟ࡶᮏ㉁ⓗࡣྠࡌᆺࡢၥ㢟࡛࠶ࡿ㸬 ᅗ㸯 ᭤⥺ \ I [ ࠾ࡼࡧ┤⥺ \ 3 [ ࡢࢢࣛࣇ 㸰㸬㸴 㸰ኚᩘ㛵ᩘࡢ᮲௳ࡁ್᭱ࢆồࡵࡿၥ㢟 ၥ㢟㸮㸯 㸰㈈ࠊX,Y ࢆᾘ㈝ࡍࡿಶேࡢຠ⏝㛵ᩘࡀࠊ U=2XY ࡛♧ࡉࢀ࡚࠸ࡿ㸬 X ㈈ࡢ౯᱁ࡀ㸰ࠊY ㈈ࡢ౯᱁ ࡀ㸲ࠊᡤᚓࡀ㸯㸲㸲࡛࠶ࡿࡁࠊຠ⏝ࢆ᭱ࡋࡼ࠺ࡍ ࡿࡇࡢಶேࡣࠊX ㈈ࢆ࠸ࡃࡽᾘ㈝ࡍࡿ㸬 㸦ゎ㸯㸧 2x+4y=144 ࡢࡶ࡛ U(x,y)ࢆ᭱ࡍࡿ᮲௳ࢆ ồࡵࡿၥ㢟⪃࠼ࡿࡇࡀ࡛ࡁࡿ.U(x,y)=k=୍ᐃࡢ᭤⥺ ┤⥺ 2x+4y=144 ࡀ᥋ࡍࡿ᮲௳ࠋⅬ(x,y)࡛୧⪅ࡀ᥋ࡍࡿ ࡍࡿ U(x+'x,y+'y)=U(x,y)+(U/(x'x+(U/(y'y=U(x,y) ࡼࡾࠊ᥋⥺ࡢഴࡁࡣ'y/'x=-(U/(x/(U/(y=-2/4. ࡇࢀࡼࡾ (U/(x /2=(U/(y /4 ࢆᚓࡿ㸴㸧㸬ࡇࢀࡼࡾࠊx=36 ࢆᚓࡿ. ᅗ㸰 ┤⥺ ;< 8 [\ ୍ᐃࡢ᭤⥺ࡀ᥋ࡍࡿࡁ ࡢ≧ἣࢆ࠶ࡽࢃࡍࢢࣛࣇ 㸦ゎ㸰㸧㧗ᑓ➨㸱Ꮫᖺ⛬ᗘ࡛Ꮫࡪࣛࢢࣛࣥࢪࣗᩘἲࢆ ⏝࠸㛵ಀᘧࢆᑟࡃࡇࡶྍ⬟࡛࠶ࡿ I [\) 8 [\ ) [\ ࠾ࡁࠊ (I([ (I(\ ࡼࡾࠊ(8([)[ (8(\)\ ࢆᚓ ࡿ ࡇࢀࡽࠊ(U/(x /2=(U/(y /4 ࢆᚓࡿ 㸱㸬ࡲࡵ⪃ᐹ ࣑ࢡࣟ⤒῭Ꮫࡢศ㔝ࡢィ⟬ၥ㢟࠾࠸࡚ࠊࡈࡃࢃࡎࡢ ⤒῭㛵ࡍࡿ▱㆑ࡀ࠶ࢀࡤࠊ㧗ᑓ➨㸰Ꮫᖺࠊ➨㸱Ꮫᖺ⛬ᗘ ࡢᚤศࠊ೫ᚤศࡢຊࡼࡾゎࡃࡇࡢ࡛ࡁࡿၥ㢟ࡀከࡃ࠶ ࡿࡇࢆ♧ࡋࡓ㸬ᩘᏛࡀ⌮ᕤ⣔ࡢᑓ㛛ศ㔝࡛ࢃࢀࡿࡢࡳ ࡞ࡽࡎࠊ⤒῭࡞ࡢࡢศ㔝࡛ࡶᚲせ࡞ࡿࡇࢆ▱ࡿࡇ ࡣᩘᏛࢆᏛ⩦ࡍࡿࡇࡢ㔜せᛶࢆㄆ㆑ࡍࡿࡇࡘ ࡞ࡀࡿᮇᚅࡉࢀࡿ㸬ࡲࡓࠊ㧗ᑓ⏕ࡀ⤒῭Ꮫ㛵ࡍࡿィ⟬ ၥ㢟ࡩࢀࡿࡇࡣࠊᩘᏛࡢຊࢆ⏝࠸࡚♫┠ࢆ㛤ࡁࠊ ᩍ㣴ࢆ㧗ࡵࡿࡢᙺ❧ࡘࡇࡶᮇᚅࡉࢀࡿ㸬ᩘᏛࡢศ㔝௨ እࡢᩘᏛࡢၥ㢟ࢆᩘᏛࡢຊࢆ⏝࠸࡚ゎࡃࡇࡼࡾࠊ┿ࡢ ᩘᏛࡢຊࢆᾰ㣴ࡍࡿࡇࡀ࡛ࡁࡿࡢ࡛ࡣ࡞࠸ᮇᚅࡉ ࢀࡿ㸬 ࡞࠾ࠊ࣐ࢡࣟ⤒῭Ꮫ࠾࠸࡚ࡣࠊ㧗ᑓ➨㸯Ꮫᖺ⛬ᗘࡢ➼ ẚᩘิࡢ₇⩦ၥ㢟࡞ࡿ㔠㛵ࡍࡿၥ㢟ࡸྛ⏘ᴗ㛫ࡢ ᾘ㈝㔞ࡸ⏕⏘㔞ࡢ㛵ಀࡀ⾜ิࡢᙧ࡛⾲ࡉࢀࡿ⏘ᴗ㐃㛵⾲㸵㸧 ࡢၥ㢟࡞ࡀ࠶ࡿ㸬ᚋ⪅࡛ࡣ㏫⾜ิࢆồࡵࡿࡇࡼࡾࠊ ࠉࠉ7+(*810$̻.2+6(15(9,(:1R
⤒῭Ἴཬຠᯝࢆ⟬ฟࡍࡿࡇࡀ࡛ࡁࡿ㸬ࡇࢀࡣᩘᏛࡀ⤒῭ Ꮫ࠾࠸࡚ᙺ❧ࡘዲ࡛࠶ࡾࠊ㧗ᑓ⏕࡞ࡽ➨㸱Ꮫᖺ࠶ࡓ ࡾ࡛ࡶ༑ศ⌮ゎྍ⬟࡞ࡶࡢ࡛࠶ࡿ㸬 ㅰ㎡ ཧ⪃ᩥ⊩ ࠾ࡅࡿ㢟ࡢᘬ⏝ࢆᛌࡃチྍࡋ࡚ୗࡉࡗࡓࠊ ⱱᮌ႐ஂ㞝ඛ⏕୪ࡧ㐌หఫᏯ᪂⪺♫ᵝࡣᚰࡼࡾឤㅰ ⏦ࡋ࠶ࡆࡲࡍ㸬 ཧ⪃ᩥ⊩ὀグ 㸯㸧ⱱᮌ႐ஂ㞝ࠊࡽࡃࡽࡃ࣑ࢡ࣭࣐ࣟࢡࣟ⤒῭Ꮫධ㛛 ィ ⟬ၥ㢟⦅ࠊSS ࠊ㐌หఫᏯ᪂⪺♫ 㸰㸧࠶ࡿရ≀ࢆ㔞 [ ࡔࡅ⏕⏘ࡍࡿࡢᚲせ࡞㈝⏝ࡢࡇ 㸱㸧⤒῭Ꮫ࡛ࡣ₶㛵ᩘࡣ* ; ࠸࠺グྕࡀࢃࢀࡿࡇ ࡀከ࠸ࡀࠊࡇࡇ࡛ࡣ⤒῭ᏛࢆᏛࢇࡔࡇࡀ࡞࠸㧗ᑓ⏕ࢆᛕ 㢌⨨࠸࡚ I [ ࡋࡓ㸬 㸲㸧㈝⏝㛵ᩘࡢᑟ㛵ᩘࡢࡇ㸬 㸳㸧ཧ⪃ᩥ⊩㸯㸧ࡣබົဨヨ㦂➼ࡢᑐ⟇⏝ࡢၥ㢟㞟࠸࠺ ⨨࡙ࡅࡶ࠶ࡾࠊ▷㛫࡛ၥ㢟ࢆゎࡃࡓࡵࡢࠊ⤒῭ᏛࢆᏛ ࢇࡔ⤒㦂ࡢ࠶ࡿேྥࡅ≉Ṧゎἲࡀ♧ࡉࢀ࡚࠸ࡿሙྜࡶ࠶ ࡿ㸬 㸴㸧ࡇࡢ㛵ಀᘧࡀཧ⪃ᩥ⊩ ࡛ࡣࠊ⤒῭ᏛࡢほⅬࡽࠊゎ ἲࡢฟⓎⅬࡢ㸯ࡘࡋ࡚ᣲࡆࡽࢀ࡚࠸ࡿ㸬 㸵㸧ࢳࣕࣥࠊ࢙࢘ࣥࣛࢺࠊ⌧௦⤒῭ᏛࡢᩘᏛⓗᇶ♏㸦ୖ㸧ࠊ SSࠊࢩ࣮࢚࣮ࣆ࣮ฟ∧
Exercises of Mathematics for Kosen Students in Micro Economics
-exercises in differential and partial
differential-Kazuto Oshima
We show that there are several calculation problems in micro economics that can be used as exercises in differential and partial differential for Kosen students. It is expected that to solve mathematical exercises in micro economics will help Kosen students to be concerned with society and develop their culture.
