Mathematics for Economics

Program Studi Magister AgribisnisFakultas Pertanian Universitas Riau

Matematika Ekonomi

Objectives of mathematics for Objectives of mathematics for economists economists To understand mathematical economics

problems by being able to state the unknowns, the data and the conditions

To plan solutions to these problems by finding a connection between the data and the unknown

To carry out your plans for solving mathematical economics problems

To examine the solutions to mathematical economics problems for general insights into current and future problems

Tujuan matematika untuk ekonomi Untuk memahami masalah matematika

ekonomi dengan mampu negara diketahui, data dan kondisi.

Untuk merencanakan solusi untuk masalah ini dengan mencari hubungan antara data dan yang tidak diketahui.

Untuk melaksanakan rencana Anda untuk memecahkan masalah matematika ekonomi.

Untuk menguji solusi untuk masalah matematika ekonomi untuk wawasan umum menjadi masalah saat ini dan masa depan

Endogenous & Exogenous Endogenous & Exogenous Variables, constants, Variables, constants, parametersparameters = TR – TC (identity) Qd = Qs (equilibrium condition) Y = a + bX0 (behavioral equation) Y: endogenous variable X0: exogenous variable a: constant b: parameter / the coefficient of

exogenous variable X0

Variabel endogen dan eksogen, konstanta, parameter

= TR - TC (identitas) Qd = Qs (kondisi ekuilibrium) Y = a + bX0 (persamaan perilaku) Y: variabel endogen X0 : variabel eksogen a: konstanta b: Parameter / koefisien dari

variabel eksogen X0

FUNCTION IN ECONOMICS

FUNGSI DALAM EKONOMI

Functions and RelationsFunctions and Relations Function: a set or ordered

pairs with the property that for (x, y) any x value uniquely determines a single y value

Relation: ordered pairs with the property that for (x, y) any x value determines more than one value of y

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Fungsi dan Hubungan

Fungsi: satu set atau pasangan dipesan dengan sifat bahwa untuk (x, y) setiap nilai x secara unik menentukan nilai y tunggal

Hubungan: memerintahkan pasangan dengan properti yang untuk (x, y) setiap nilai x menentukan lebih dari satu nilai dari y

General FunctionsGeneral Functions Y = f (X) Y is value or dependent variable

(vertical axis) f is the function or a rule for

mapping X into a unique Y X is argument or the independent

variable (horizontal axis)

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Fungsi umum

Y = f (X) Y adalah nilai atau variabel

dependen (sumbu vertikal) f adalah fungsi atau aturan untuk

pemetaan X ke Y yang unik X adalah argumen atau variabel

independen (sumbu horisontal)

Specific FunctionsSpecific Functions

Algebraic functionsY = a0 (constant: fixed costs)

Y = a0+ a1 X (linear: S&D)

Y = a0 + a1X + a2X2 (quadratic: prod.)

Y = a0 + a1X + a2X2 + a3X3 (cubic: t. cost) Y = a/X (hyperbolic: indiff.)Y = aXb (power: prod. fn)lnY = ln(a) + b ln(X) (logarithmic: easier)

Transcendental functions Y = aX (exponential: interest)

(Chiang & Wainwright, p. 22, Fig. 2.8)11

Fungsi spesifik

fungsi aljabarY = a0 (konstan: biaya tetap)Y = a0 + a1 X (linear: S & D)Y = a0 + a1X + a2X2 (kuadrat: prod.)Y = a0 + a1X + a2X2 + a3X3 (kubik: t. biaya)Y = a / X (hiperbolik:. Indiff)Y = axb (kekuatan:. Prod fn)lnY = ln (a) + b ln (X) (logaritma: mudah)

fungsi transendentalY = AX (eksponensial: bunga)

(Chiang & Wainwright, hal 22, Gambar. 2.8)

Digression on exponentsDigression on exponents Rules for exponents Xn = (X*X*X*...*X) n times

Rule I: Xm * Xn = Xm+n

Rule II:

Rule III: X-n = Rule IV: X0 = 1

Rule V: X1/n =nx Rule VI: (Xm)n = Xmn

Rule VII: Xm * Ym = (XY)m

nmn

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XX

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1

X n

Levels of generalityLevels of generality

Specific function 1: specific form and specific parameters

Y = 10 - .5X Specific function 2:

specific form and general parameters

Y = a – bX General function:

general form and no parameters Y = f (X) f maps X into a unique value of

Y

Tingkat umum

Fungsi Khusus 1: bentuk khusus dan parameter yang spesifik

Y = 10-0,5 X Spesifik fungsi 2: bentuk khusus dan parameter

umum Y = a – bX

Umum fungsi: bentuk umum dan tidak ada parameter

Y = f (X)f peta X ke nilai unik Y

EQUATION IN ECONOMICS

PERSAMAAN DALAM EKONOMI

INTRODUCTION An equation is a statement that two

expressions are equal to one another. In economic modelling we express

relationships are equations and then use them to obtain analytical result. Solving the equations gives us values for which the equations are true.

We can express the condition for market equilibrium as an equation in terms of price, P, solving the equation for P tells us the price at which the market is in equlibrium

PENDAHULUAN

Persamaan adalah suatu pernyataan bahwa dua ekspresi yang sama satu sama lain.

Dalam pemodelan ekonomi kita menyatakan hubungan persamaan dan kemudian menggunakannya untuk mendapatkan hasil analisis. Memecahkan persamaan memberi kita nilai-nilai yang persamaan benar.

Kita dapat mengungkapkan kondisi ekuilibrium pasar sebagai sebuah persamaan dalam hal harga, P, memecahkan persamaan untuk P memberitahu kita harga di mana pasar berada dalam equlibrium

REWRITING AND SOLVING EQUATIONS When rewriting equation:

1. Add to or subtract from both sides.2. Multiply or divide through the whole or each side (but don’t

divide by 0).3. Square or take the square root of each side.4. Use as many stages as you wish.5. Take care to get all the signs correct.

Example: Plot the equations y = -5 + 2x and y = 30 - 3x. At what values of x and y do they cross ? Find also the algebraic solution by setting the two expressions in x equal to one other.

We are asked to plot two linier functions, so plotting two points on each then connecting them will suffice.The table shows x value of 0 and 10 and the corresponding y values for each line. These points are used to plot the lines shown in figure. Notice that the line cross at x = 7, y = 9.

MENULIS DAN PEMECAHAN PERSAMAAN

Ketika menulis ulang persamaan:1. Menambah atau mengurangi dari kedua belah pihak.2. Mengalikan atau membagi melalui seluruh atau setiap sisi (tetapi tidak         membagi dengan 0).3. Persegi atau mengambil akar kuadrat dari setiap sisi.4. Digunakan sebagai banyak tahapan yang Anda inginkan.5. Berhati-hatilah untuk mendapatkan semua tanda-tanda yang benar.

Contoh: Plot persamaan y = -5 + 2x dan y = 30 - 3x. Pada apa nilai-nilai dari x dan y yang mereka salib? Temukan juga solusi aljabar dengan menetapkan dua ekspresi di x sama dengan satu lainnya.

Kami diminta untuk merencanakan dua fungsi linier, sehingga merencanakan dua titik pada masing-masing kemudian menghubungkan mereka akan cukup.Tabel ini menunjukkan nilai x dari 0 dan 10 dan nilai y untuk setiap baris. Titik-titik ini digunakan untuk merencanakan garis yang ditunjukkan dalam gambar.Perhatikan bahwa garis salib di x = 7, y = 9.

-10

-5

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0 2 4 6 8 10 12

X

Y

y=-5+2x

y=-30-3x

x 0 2 4 6 8 10 12

y=-5+2x -5 -1 3 7 11 15 19y=30-3x 30 24 18 12 6 0 -6

For algebraic solution, the two y value are equal so equate the right hand sides of the expressions and solve for x:

-5 + 2x = 30 – 3x

We want term in ix on the left-hand side but not on the right, so add 3x to both sides since -3x + 3x = 0. We then have:

-5 + 2x + 3x = 30 or -5 + 5x = 30

To remove the constant term from the left side we now add 5 to each side, giving:

Lines intersect at (7, 9)

5x = 35

And so, dividing by 5, we have x = 7

We then find the value for y by substi tuting x=7 in either of the equations. Using y=30 – 3x gives:

y = 30 – 21 = 9

Which confirms the graphical result. The solution is x = 7 and y = 9

Untuk solusi aljabar, nilai y kedua adalah sama sehingga menyamakan sisi tangan kanan dari ekspresi dan memecahkan untuk x:-5 + 2x = 30 - 3xKami ingin istilah dalam ix di sisi kiri tapi tidak di sebelah kanan, sehingga menambah 3x untuk kedua belah pihak sejak-3x + 3x = 0. Kita kemudian memiliki:-5 + 2x + 3x = 30 atau -5 + 5x = 30Untuk menghapus istilah konstan dari sisi kiri sekarang kita tambahkan 5 untuk setiap sisi, memberikan:

5x = 35Dan, membaginya dengan 5, kita memiliki x = 7Kami kemudian menemukan nilai y dengan x substi tuting = 7 di salah satu persamaan. Menggunakan y = 30 - 3x memberikan:y = 30-21 = 9Yang menegaskan hasil grafis. Solusinya adalah x = 7 dan y = 9

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x 0 2 4 6 8 10 12

y=-5+2x -5 -1 3 7 11 15 19y=30-3x 30 24 18 12 6 0 -6

SOLUTION IN TERMS OF OTHER VARIABLES Not all the equations you deal with have numerical solutions.

Sometimes when you solve and equation for x you obtain and expression containing other variables.

Use same rules to transpose the equation. Remember that in the solution x will not occur on the right-hand side

and will be on its own the left-hand side. If you are given a relationship in the form y=f(x), rewriting the

equation in the form x=g(y) is called finding the inverse function. To be able to find the inverse there must be just one x value

corresponding to each y value. For non linier function there can be difficulties in finding an inverse,

but we may be able to do so for restricted set values. The function y=x2 has two x values (one positive and one negative)

corresponding to every y value, but if we consider the restricted function y=x2, x>0 this function has the inverse x=y.

For the linier functions often use in economic models inverse functions can always be found. One reason for finding the inverse function can always be found. One by y is conventionally plotted in economic on the horizontal axis.

Demand and supply equations provide examples of this.

SOLUSI DALAM KETENTUAN LAIN VARIABEL

Tidak semua persamaan Anda berurusan dengan memiliki solusi numerik. Kadang-kadang ketika Anda memecahkan dan persamaan untuk x Anda mendapatkan dan ekspresi yang mengandung variabel lain.Menggunakan aturan yang sama untuk merefleksikan persamaan.Ingat bahwa dalam larutan x tidak akan terjadi pada sisi kanan dan akan sendiri sisinya kiri-tangan.Jika Anda diberikan suatu hubungan dalam bentuk y = f (x), menulis ulang persamaan dalam bentuk x = g (y) disebut fungsi invers menemukan.Untuk dapat mencari invers harus ada hanya satu nilai x yang sesuai untuk setiap nilai y.Untuk fungsi linier non bisa ada kesulitan dalam mencari invers, tetapi kita mungkin dapat melakukannya untuk nilai-nilai yang ditetapkan dibatasi.Fungsi y = x2 memiliki nilai-nilai x dua (satu positif dan satu negatif) yang sesuai untuk setiap nilai y, tapi jika kita mempertimbangkan fungsi dibatasi y = X2, x> 0 Fungsi ini memiliki invers x = y.Untuk fungsi linier sering digunakan dalam model ekonomi fungsi invers selalu dapat ditemukan. Salah satu alasan untuk menemukan fungsi invers selalu dapat ditemukan. Satu demi y konvensional diplot dalam ekonomi pada sumbu horisontal.Permintaan dan pasokan persamaan memberikan contoh ini.

Solve for x in term of zx = 60 + 0.8x + 7z

At first glance you seem to already have a solution for x, but notice that x occurs also on the right-hand side of the equation. We must collect terms in x on the left-hand, so we subtract 0.8x from both sides and obtain:

x - 0.8x = 60 + 7zsince both left-hand side terms contain x we may write:

(1 – 0.8)x = 60 + 7zwhich gives

0.2x = 60 + 7zTo get x with a coefficient of 1 we divide both sides by 0.2 = 1/5, which is the same thing as multiplying both sides by 5. This gives:

x = 300 + 35z

Selesaikan untuk x dalam jangka zx = 60 + 0.8x + 7z

Pada pandangan pertama Anda tampaknya sudah memiliki solusi untuk x, tapi perhatikan bahwa x juga terjadi pada sisi kanan dari persamaan. Kita harus mengumpulkan istilah dalam x di sisi kiri, jadi kita kurangi 0.8x dari kedua belah pihak dan memperoleh:x - 0.8x = 60 + 7zkarena kedua kiri sisi istilah mengandung x kita bisa menulis:(1 - 0,8) x = 60 + 7zyang memberikan0.2x = 60 + 7zUntuk mendapatkan x dengan koefisien 1 kita membagi kedua sisi dengan 0,2 = 1 / 5, yang merupakan hal yang sama seperti mengalikan kedua sisi dengan 5. Hal ini memberikan:x = 300 + 35z

Given y = x + 5, obtain an expression for x in term of y.

Begin by interchanging the side so that the sides with x is on the left of the equation. We then have:

x + 5 = yNext subtract 5 from both sides, giving:

x = y – 5To find x we must square both sides. This means that the whole of the right-hand side is multiplied by itself, so use brackets. We obtain:

x = (y – 5)2

Squaring out the bracket we may also write:x = y2 – 10y + 25

Jika diketahui y = x + 5, memperoleh ekspresi untuk x dalam jangka y.

Mulailah dengan mempertukarkan sisi sehingga sisi dengan x adalah di sebelah kiri dari persamaan. Kita kemudian memiliki: x + 5 = ySelanjutnya kurangi 5 dari kedua belah pihak, memberikan: x = y - 5Untuk menemukan x kita harus persegi kedua belah pihak. Ini berarti bahwa seluruh sisi kanan dikalikan dengan sendirinya, jadi gunakan tanda kurung. Kita memperoleh:x = (y - 5) 2Mengkuadratkan keluar braket kita juga dapat menulis:x = y2 - 10Y + 25

SUBSTITUTION When two expression are equal to one another, either can

be substituted for the other. The technique is used the effect of the imposition of a per

unit tax on a good and to solve simultaneous equations. When substituting, always be sure to substitute the whole

of the new expression and combine it with the other term in exactly the same way the expression it replaces was combined with them.

For example, if y = x2 + 6 and x = 30 - , find an expression for y in term of . Substituting 30 - for x we obtain:y = (30 - )2 + 6which on multiplying out and collecting term becomesy = 900 – 54 + 2

SUBSTITUSI Ketika dua ekspresi yang sama satu dengan lainnya, dapat diganti

untuk yang lain.Teknik ini digunakan ï † efek dari pengenaan pajak per unit pada yang baik dan untuk memecahkan persamaan simultan.Ketika mengganti, selalu pastikan untuk menggantikan seluruh ekspresi baru dan menggabungkan dengan istilah lainnya dengan cara yang persis sama ungkapan itu menggantikan dikombinasikan dengan mereka.Sebagai contoh, jika y = x2 + 6 dan x = 30 - , menemukan ekspresi untuk y dalam jangka . Mengganti 30 - - untuk x kita mendapatkan: y = (30 - )2 + 6 yang pada keluar dan mengumpulkan mengalikan panjang menjadi y = 900 – 54 + 2

DEMAND AND SUPPLY Demand and supply function in economics express

the quantity demanded or supplied as a function of price, Q = f (P).

According to mathematical convention the dependent variable (Q) should be plotted on the vertical axis.

Economic analysis, however, use the horizontal axis as the Q and for consistency we follow that approach. So that we can determine the points on graph in the usual way, before plotting a demand or supply function we first find its inverse function giving P as a function of Q.

PERMINTAAN DAN PENAWARAN

Permintaan dan fungsi penawaran dalam ekonomi mengekspresikan kuantitas yang diminta atau diberikan sebagai fungsi dari harga, Q = f (P).Menurut konvensi matematika variabel dependen (Q) harus diplot pada sumbu vertikal.Analisis ekonomi, bagaimanapun, menggunakan sumbu horizontal sebagai Q dan untuk konsistensi kita mengikuti pendekatan itu. Sehingga kita dapat menentukan titik-titik pada grafik dengan cara yang biasa, sebelum merencanakan fungsi permintaan atau penawaran pertama kita menemukan fungsi invers P memberikan sebagai fungsi dari Q.

Find the inverse function for the demand equation Q = 80 – 2P and sketch the demand curve.

Adding 2P to both sides of the demand equation we get2P + Q = 80

Subtracting Q from both sides we obtain:2P = 80 – Q

Dividing each side by 2 gives the inverse functionP = (80 – Q)/2 = 40 – (Q/2)

The demand function is linier, so it suffices to plot two point. Selected values of Q are shown in the table together with corresponding value for P.

Q 0 20 40 60 80P = 40 - (Q/2) 40 30 20 10 0

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0 20 40 60 80

Q

P

Cari fungsi invers untuk persamaan permintaan Q = 80 - 2P dan sketsa kurva permintaan.

Menambahkan 2P kedua sisi persamaan permintaan kita2P + Q = 80Mengurangkan Q dari kedua sisi kita memperoleh:2P = 80 - QMembagi setiap sisi oleh 2 memberikan fungsi inversP = (80 - T) / 2 = 40 - (Q / 2)Fungsi permintaan linier, sehingga cukup untuk plot dua titik. Nilai yang dipilih Q ditunjukkan dalam tabel bersama-sama dengan nilai yang sesuai untuk P.

Q 0 20 40 60 80P = 40 - (Q/2) 40 30 20 10 0

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P

MARKET EQULIBRIUM Market equilibrium occurs when the quantity supplied

equals the quantity demanded of a good. The supply and demand curves cross at the equilibrium

price and quantity. If you plot both the demanded and supply curves you

can of approximate equilibrium values from the graph. Another approach is to solve algebraically for the point

where the demand and supply equation are equal. This gives exact value. Suppose we wish to find the equilibrium price and quantity when demand is given by

Demand: Q = 96 – 4PAnd the supply equation is

Supply : Q = 8P

PASAR EKUILIBRIUM

Ekuilibrium pasar terjadi ketika kuantitas yang ditawarkan sama dengan kuantitas yang diminta dari yang baik.Kurva penawaran dan permintaan silang pada harga keseimbangan dan kuantitas.Jika Anda plot kedua kurva penawaran menuntut dan Anda dapat nilai ekuilibrium perkiraan dari grafik.Pendekatan lain adalah untuk memecahkan aljabar untuk titik di mana permintaan dan persamaan penawaran adalah sama. Hal ini memberikan nilai yang tepat. Misalkan kita ingin mencari harga keseimbangan dan kuantitas ketika permintaan diberikan olehPermintaan: Q = 96 - 4PDan persamaan pasokanSupply: Q = 8P

For an algebraic solution we can use the equation in this form. Since in equilibrium then quantity supplied equals the quantity demanded, the right-hand side of the supply equation must equal the right-hand side of the demand equation. This gives an equation in P:

Supply Q = Demand Q (in equilibrium), so 8P = 96 – 4P

Adding 4P to both sides gives12P = 96

Dividing by 12 we find P = 8 which is the equilibrium price. We can then substitute this into either equation, say the equation. This gives:

Q = 8 x 8 = 64Which is the quantity supplied in equilibrium and therefore also the quantity demanded. Market equilibrium occurs at Q = 64, P = 8.

Untuk solusi aljabar kita dapat menggunakan persamaan dalam bentuk ini. Sejak dalam kesetimbangan maka kuantitas yang ditawarkan sama dengan kuantitas yang diminta, sisi kanan persamaan harus sama dengan pasokan sisi kanan dari persamaan permintaan. Ini memberikan persamaan dalam P:Pasokan Q = Q Permintaan (dalam kesetimbangan), sehingga8P = 96 - 4PMenambahkan 4P untuk kedua belah pihak memberikan12p = 96Membaginya dengan 12 kita menemukan P = 8 yang merupakan harga ekuilibrium. Kami kemudian dapat menggantikan ini ke dalam persamaan baik, katakan persamaan. Hal ini memberikan:Q = 8 x 8 = 64Yang merupakan kuantitas yang ditawarkan dalam kesetimbangan dan karena itu juga kuantitas yang diminta. Ekuilibrium pasar terjadi di Q = 64, P = 8.

Tugas 1A

Rewriting these equations expressing P as a function of Q then plot them on a graph

Supply : Q = 4PDemand : Q = 280 – 10P

Menulis ulang persamaan ini menyatakan P sebagai fungsi dari Q kemudian plot mereka pada grafik

Supply : Q = 4PDemand : Q = 280 – 10P

TOTAL AND AVERAGE REVENUE

Total revenue (TR) is price (P) multiplied by quantity (Q)TR = P . Q

Average revenue (AR) per unit of output is TR + Q = PAR = TR/Q

A market demand curve is assumed to be downward sloping.

JUMLAH PENDAPATAN DAN RATA-RATA

Total pendapatan (TR) adalah harga (P) dikalikan dengan kuantitas (Q)TR = P. QPendapatan rata-rata (AR) per unit output adalah TR = P + QAR = TR / QKurva permintaan pasar dianggap miring ke bawah.

If average revenue is given by:

P = 72 – 3Q

Sketch this function and also, on a separate graph, the total revenue function.

The average revenue function has P on the vertical axis and Q on the horizontal axis. The general form of linier function is y = a + bx. Comparing our average revenue function we see that it take this linier form with y = P, a = 72, b = -3 and x = Q. We therefore need find only two points on our function to sketch the line and can the extend it as required. For simplicity we choose Q = 0 and Q = 10. The corresponding P values are listed, the two points are plotted and the line is extended to the horizontal axis.

Chosen value Q = 0 and Q = 10

substituting in P = 72 – 3(0) = 72 and P = 72 – 3(10)= 42

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0 2 4 6 8 10 12 14 16 18 20 22 24

Q

P AR = 72-3Q

Jika pendapatan rata-rata diberikan oleh:P = 72 - 3QSketsa fungsi ini dan juga, pada grafik yang terpisah, fungsi total pendapatan. Fungsi pendapatan rata-rata memiliki P pada sumbu vertikal dan Q pada sumbu horizontal. Bentuk umum dari fungsi linier adalah y = a + bx. Membandingkan fungsi pendapatan rata-rata kita, kita melihat bahwa mengambil bentuk linier dengan y = P, a= 72, b = -3 dan x = Q. Karena itu kita perlu menemukan hanya dua titik pada fungsi kita untuk membuat sketsa garis dan dapat memperpanjang itu seperti yang diperlukan. Untuk kesederhanaan kita memilih Q = 0 dan Q = 10. P yang sesuai nilai-nilai yang terdaftar, dua poin diplot dan garis ini diperpanjang untuk sumbu horisontal.Dipilih nilai Q = 0 dan Q = 10mengganti dalam P = 72 - 3 (0) = 72 dan P = 72 - 3 (10) = 42

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TR

We next find and expression for TR:TR = P . Q = (72 – 3Q) . Q = 72Q – 3Q2

so,

Q 0 2 4 6 8 10 12 14 16 18 20 22 2472Q 0 144 288 432 576 720 864 1,008 1,152 1,296 1,440 1,584 1,728 3Q^2 0 12 48 108 192 300 432 588 768 972 1,200 1,452 1,728 TR 0 132 240 324 384 420 432 420 384 324 240 132 -

TR = 72Q – 3Q2

Kami selanjutnya mencari dan ekspresi untuk TR:TR = P. Q = (72 - 3Q). Q = 72Q - 3Q2demikian,

TOTAL AND AVERAGE COST A firm’s total cost of production (TC) depends on its

output (Q). The TC function may include a constant term, which

represent fixed cost (FC). The part of total cost that varies with Q is called variable

cost (VC). We have, then, that TC = FC + VC Remember: FC is the constant term in TC

VC = TC – FC AC = TC/QAVC = VC/Q

TOTAL RATA-RATA DAN BIAYA

Total biaya sebuah perusahaan produksi (TC) tergantung pada output (Q).Fungsi TC mungkin termasuk istilah konstan, yang merupakan biaya tetap (FC).Bagian dari total biaya yang bervariasi dengan Q disebut biaya variabel (VC).Kami memiliki, kemudian, bahwa TC = FC + VCIngat: FC adalah istilah konstan dalam TCVC = TC - FCAC = TC / QAVC = VC / Q

For a firm with total cost given by: TC = 120 + 45Q – Q2 + 0.4Q3

Identify it AC, FC, VC and AVC functions. List some values of TC and AC, correct to the nearest integer. Sketch the total cost function and on a separate graph the AC function.

TC = 120 + 45Q - Q2 + 0.4Q3

AC = TC/Q = 120/Q + 45 – Q + 0.4Q2

FC = 120 (the constant term in TC) VC = TC – FC = (120 + 45Q - Q2 + 0.4Q3) – (120) = 45Q - Q2 +

0.4Q3

AVC= VC/Q = (45Q - Q2 + 0.4Q3) / Q = 45 – Q + 0.4Q2

Q 0.00 0.30 1.00 3.00 5.00 8.00 10.00 12.00 15.0045Q 0.00 13.50 45.00 135.00 225.00 360.00 450.00 540.00 675.00Q^2 0.00 0.09 1.00 9.00 25.00 64.00 100.00 144.00 225.000.4Q^3 0.00 0.01 0.40 10.80 50.00 204.80 400.00 691.20 1350.00

TC 120 133 164 257 370 621 870 1207 1920AC 445 164 86 74 78 87 101 128

Untuk perusahaan dengan total biaya yang diberikan oleh: TC = 120 + 45Q - Q2 + 0.4Q3Identifikasi itu AC, FC, VC dan fungsi AVC. Daftar beberapa nilai dari TC dan AC, benar ke integer terdekat. Sketsa fungsi biaya total dan pada grafik memisahkan fungsi AC.TC = 120 + 45Q - Q2 + 0.4Q3AC = TC / Q = 120 / Q + 45 - Q + 0.4Q2FC = 120 (istilah konstan dalam TC)VC = TC - FC = (120 + 45Q - Q2 + 0.4Q3) - (120) = 45Q - Q2 + 0.4Q3AVC = VC / Q = (45Q - Q2 + 0.4Q3) / Q = 45 - Q + 0.4Q2

TC = 120 + 45Q – Q2 + 0.4Q3

AC = 120/Q + 45 – Q + 0.4 Q2

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PROFIT Profit is difference between a firm’s total revenue and its

total costs. Using the symbol as the variable name for profit we have

= TR – TC

A firm has the total cost function: TC = 120 + 45Q – Q2 + 0.4Q3

And faces a demand curve given by: P = 240 – 20Q What is its profit function ?

TR = P . Q = (240 – 20Q) . Q = 240Q – 20Q2

= TR – TC = (240Q – 20Q2) – (120 + 45Q – Q2 + 0.4Q3)

= -120 + 195Q – 19Q2 – 0.4Q3

LABA

Profit is difference between a firm’s total revenue and its total costs.

Using the symbol as the variable name for profit we have = TR – TC

A firm has the total cost function: TC = 120 + 45Q – Q2 + 0.4Q3

And faces a demand curve given by: P = 240 – 20Q What is its profit function ?

TR = P . Q = (240 – 20Q) . Q = 240Q – 20Q2

= TR – TC = (240Q – 20Q2) – (120 + 45Q – Q2 + 0.4Q3)

= -120 + 195Q – 19Q2 – 0.4Q3

PRODUCTION FUNCTIONS, ISOQUANTS AND THE AVERAGE PRODUCTS OF LABOUR

The long run production function shows that a firm’s output (Q), depends on the amount of factors it employs (always assuming that whatever factor are employed are used efficiently)

If a production process involves the use of labour (L) and capital (K), we write Q = f (L, K)The dependent variable Q is function of two independent variables, L and K.

Average product of labour (APL) = Q + L

FUNGSI PRODUKSI, isokuan DAN PRODUK RATA-RATA DARI BURUH

Fungsi produksi jangka panjang menunjukkan bahwa output perusahaan (Q), tergantung pada jumlah faktor yang mempekerjakan (selalu dengan asumsi bahwa faktor apa pun yang dipekerjakan digunakan secara efisien)Jika proses produksi melibatkan penggunaan tenaga kerja (L) dan modal (K), kita menulis Q = f (L, K)Variabel dependen Q adalah fungsi dari dua variabel independen, L dan K.Produk rata-rata tenaga kerja (APL) = Q + L

A firm has the production function Q = 25 (L . K)2 – 0.4(L . K)3. If K = 1, find the value of Q for L = 2, 3, 4, 6, 12, 14 and 16. Sketch this short run production function putting L and Q on the axes of your graph. Next suppose the value of K is increased to 2. On the same graph sketch the new short run production function for the same values of L. Add one further production function to your sketch, corresponding to K = 3, using the same L values again.

For the short run production function with K = 3, find and plot the average product of labour function.

K\L 2 3 4 6 12 14 161 96.8 214.2 374.4 813.6 2,908.8 3,802.4 4,761.6 2 374.4 813.6 1,395.2 2,908.8 8,870.4 10,819.2 12,492.8 3 813.6 1,733.4 2,908.8 5,767.2 13,737.6 14,464.8 13,363.2

Sebuah perusahaan memiliki fungsi produksi Q = 25 (L K.) 2 - 0,4 (L K.) 3. Jika K = 1, menemukan nilai Q untuk L, = 2 3, 4, 6, 12, 14 dan 16. Sketsa ini fungsi produksi jangka pendek meletakkan L dan Q pada sumbu grafik Anda. Selanjutnya misalkan nilai K meningkat menjadi 2. Pada sketsa grafik yang sama menjalankan fungsi baru produksi pendek untuk nilai yang sama L. Tambahkan satu fungsi produksi lebih lanjut untuk sketsa Anda, sesuai dengan K = 3, menggunakan nilai L yang sama lagi.Untuk fungsi produksi jangka pendek dengan K = 3, menemukan dan produk rata-rata plot fungsi tenaga kerja.

K\L 2 3 4 6 12 14 161 96.8 214.2 374.4 813.6 2,908.8 3,802.4 4,761.6 2 374.4 813.6 1,395.2 2,908.8 8,870.4 10,819.2 12,492.8 3 813.6 1,733.4 2,908.8 5,767.2 13,737.6 14,464.8 13,363.2

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For K = 3, we have:

Q = 25(3L)2 – 0.4(3L)3

= 225L2 – 10.8L3

APL = Q/L = 225L – 10.8L2

L 2 3 4 6 8 10 12 16APL 406.8 577.8 727.2 961.2 1108.8 1170 1144.8 835.2

APL=225L – 10.8L2

Tugas 1B1. Sketch the total cost function: TC = 300 + 40Q – 10Q2 +

Q3, write expressions for AC, FC, VC and AVC !2. If the firm in question 1 faces the demand curve P = 100 – 0.5Q Find an expression for the firm’s profit function and

sketch the curve !3. A firm in perfect competition sells it output at a price Rp

12.000. Plot it total revenue function (TR) = 12Q !1. Sketsa fungsi biaya total: TC = 300 + 40Q - 10Q2 + Q3, menuliskan ekspresi untuk AC, FC, VC dan AVC!2. Jika perusahaan dalam pertanyaan 1 menghadapi kurva permintaan P = 100 - 0.5Q Menemukan ekspresi untuk fungsi keuntungan perusahaan dan sketsa kurva!3. Sebuah perusahaan dalam persaingan sempurna menjualnya output pada harga Rp 12.000. Plot itu fungsi total pendapatan (TR) = 12Q!

EFFECT OF A PER UNIT TAX The information contained in the supply equation about

how much producers will supply is based on the prices that they receive.

If a per unit tax (t) is imposed, although buyers still pay P for each unit of the good, the suppliers receive only P – t.

The difference between the price paid and the price received is the per unit tax (t), which is paid to the government.

A per unit tax therefore change the supply equation and causes the supply curve to shift.

Whatever the form in which the supply equation is written, we can alter it to incorporate a per unit tax by writing P – t in place of P wherever it occurs.

PENGARUH PAJAK UNIT PER Sebuah

Informasi yang terkandung dalam persamaan penawaran tentang berapa banyak produsen akan memasok didasarkan pada harga yang mereka terima.Jika pajak per unit (t) dikenakan, meskipun pembeli tetap membayar P untuk setiap unit yang baik, para pemasok hanya menerima P - t.Perbedaan antara harga yang dibayarkan dan harga yang diterima adalah pajak per unit (t), yang dibayar kepada pemerintah.Pajak per unit sehingga mengubah persamaan penawaran dan menyebabkan kurva penawaran bergeser.Apapun bentuk di mana persamaan penawaran ditulis, kita bisa mengubahnya untuk memasukkan pajak per unit dengan menulis P - t di tempat P mana pun itu terjadi.

For example, if when there is no tax the supply equation is given by

Q = -3 + 4Pthen when a per unit tax of t is imposed the supply equation becomes

Q = -3 + 4(P – t) We rewrite them expressing P as a function of Q. The original supply

equation becomes: P = Q/4 + ¾

Writing P – t for P in the equation, the post tax equation isP – t = Q/4 + ¾

Adding t to both sides this becomes:P = Q/4 + 3/4 + t

The post tax values for P are t more than the original one, so when we plot the two supply equation with P on the vertical axis the post tax curve is higher by the amount of the tax.

Sebagai contoh, jika ketika ada pajak ada persamaan penawaran diberikan olehQ = -3 + 4Pmaka ketika pajak per unit t dikenakan persamaan penawaran menjadiQ = -3 + 4 (P - t) Kami menulis ulang mereka mengekspresikan P sebagai fungsi dari Q. pasokan asli           persamaan menjadi:            P = Q / 4 + ¾Menulis P - t untuk P dalam persamaan, persamaan pajak posP - t = Q / 4 + ¾Menambahkan t untuk kedua sisi ini menjadi:P = Q / 4 + 3 / 4 + tNilai pajak posting untuk P adalah t lebih dari yang asli, jadi ketika kita merencanakan persamaan penawaran dua dengan P pada sumbu vertikal kurva pajak pos lebih tinggi dengan jumlah pajak.

If demand and supply in a market are described by the equation below, solve algebraically to find equilibrium P and Q.

Demand : Q = 120 – 8PSupply : Q = -6 + 4P

If now a per unit tax 4.5 impose, show the equilibrium solution changes. How is the tax shared between producers and consumers ? Sketch a graph showing what changes ensue when the tax is imposed ?

In equilibrium Supply Q = Demand QSo equating the right-hand sides of the equation gives:

-6 + 4P = 120 – 8PAdding 8P + 6 to each side we have: 12P = 126Dividing by 12 gives: P = 10.5Substituting in the supply equation gives: Q = -6 + 4(10.5) = 36The equilibrium values are P = 10.5 and Q = 36 When a tax of 4.5 is imposed the supply curve becomes:

Supply: Q = -6 + 4(P – 4.5) = -24 + 4P In equilibrium this new quantity supplied equal the quantity demanded, giving:

-24 + 4P = 120 – 8P

Adding 8P + 24 to each sides gives: 12P = 144

and so dividing by 12 we find : P = 144/12 = 12 From the new supply equation we obtain: Q = -24 + (4 x 12) = 24

Jika permintaan dan penawaran di pasar dijelaskan oleh persamaan di bawah ini, memecahkan aljabar untuk menemukan keseimbangan P dan Q.Permintaan: Q = 120 - 8PSupply: Q = -6 + 4PJika sekarang pajak per unit 4,5 memaksakan, menunjukkan perubahan ekuilibrium solusi. Bagaimana pajak dibagi antara produsen dan konsumen? Sketsa grafik yang menunjukkan perubahan apa yang terjadi ketika pajak dikenakan? Dalam ekuilibrium Pasokan Q = Q PermintaanJadi menyamakan ruas kanan dari persamaan memberikan:-6 + 4P = 120 - 8PMenambahkan 8P + 6 untuk setiap sisi kita memiliki: 12p = 126Membaginya dengan 12 memberikan: P = 10,5Dengan mensubstitusikan persamaan penawaran dalam memberikan: Q = -6 + 4 (10,5) = 36Nilai-nilai ekuilibrium adalah P = 10,5 dan Q = 36 Ketika pajak sebesar 4,5 dikenakan kurva penawaran menjadi:Supply: Q = -6 + 4 (P - 4.5) = -24 + 4PDalam keseimbangan ini kuantitas baru diberikan sama kuantitas yang diminta, memberikan:-24 + 4P = 120 - 8PMenambahkan 8P + 24 untuk setiap sisi memberikan: 12p = 144dan membaginya dengan 12 kita menemukan: P = 144 / 12 = 12Dari persamaan penawaran baru kita mendapatkan: Q = -24 + (4 x 12) = 24

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The equilibrium values are P= 12 and Q= 24. Although the tax is 4.5, price has risen by only 12 – 10.5 = 1.5.

On third of the tax has been passed on to consumers as a price increase, but the remainder has been absorbed by the producers.

The quantity traded has fallen from 36 to 24. To plot the curves we write the inverse function expressing P in term of Q. We find:

Demand, D: P = 15 – Q/8 Original Supply, S : P = 3/2 + Q/4

Supply after tax, S : P = 3/2 + Q/4 + 4.5 = 6 + Q/4

S2 P= 6 + Q/4

S1 P= 3/2 + Q/4

D P= 15 – Q/8

Price changes = 1.5

COST – VOLUME – PROFIT ANALYSIS

Cost – volume – profit analysis is a method use by accountants to estimate the desired sales level in order to achieve a target level of profit.

Two simplifying assumptions are made: namely that price and average variable cost are both fixed.

= TR – TC , TR = P . Q TC = FC + VC AVC = VC/Q Substituting in the profit function for TR and TC

= P . Q - (FC + VC) = P . Q – FC – VCMultiplying both sides of the expression for AVC by Q we obtain

AVC . Q = VCSo we may substitute for VC in the profit equation and get

= P. Q – FC – AVC . Q

Adding FC to both sides gives: + FC = P . Q - AVC . Q

Interchanging the sides we obtain: P . Q – AVC . Q = + FC Q is a factor of both term on the left so we may write:

Q(P – AVC) = + FC Q = ( + FC) / (P – AVC) If the firm accountant can estimate FC, P and AVC, substituting

these together with the target level of profit (), gives the desired sales level.

For a firm with fixed cost of 555, average variable cost of 12 and selling at a price of 17, find an expression for profit in terms of its level of sales, Q. What value should Q be to achieve the profit target of 195 ? At what sales level does this firm break even ? Illustrate your algebraic analysis with diagram.

= TR – TC = P . Q – FC – VCwriting VC = AVC . Q gives: = P . Q – FC – AVC . QSubstituting cost and price we find: = 17Q – 555 – 12QSo = 5Q – 555Which is the required expression for profit. Rewriting this to give Q in term of add 555 to both sides so:

+ 555 = 5Q, interchanging the sides gives: 5Q = + 555 and dividing by 5 we have : Q = ( + 555) / 5 Substituting the profit target of 195 gives:

Q = (195 + 555)/5 = 150 For the break even value of Q we substitute instead = 0, so Q = (555/5) = 111

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LINIER EQUATION

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A horizontal line has zero slope

As x increases, y does not change

Slope = 0

Slope = y/x = (distance up)/(distance to right)

Y = 18

As x increases,

y increases

y = 9x

Slope = 9

Line passes through the origin

Positive slope, intercept at zero

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Target Profit

Cost – Volume – Profit Analysis

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Negative slope, positive intercept Positive slope, negative intercept

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QUADRATIC EQUATION In economic modelling we work with the simples function that

can adequately represent a relationship, but sometimes a curve rather than a straight line is required.

We used quadratic functions to represent total revenue and average variable cost

General form : y = ax2 + bx + cwhere a, b, and c are constants. When you sketch a quadratic function you find it has either a hill or U shape and that generally two values of x give the same value of y.

A quadratic function has a term in x2 but no higher powers of x. A quadratic equation you can solve it graphically or

sometimes by factorizing it or by using the formula. We first write it in the form ax2 + bx +c = 0. The value(s) of

x for which this equation is true can be found graphically by plotting

y= ax2 + bx + c and looking y = 0.

Algebraic methods are more accurate than graphical ones, but the

squared term means we need a special technique. Sometimes factori- zing the expression helps. Consider for example:

5x2- 20x = 0Since each term is divisible by 5x we can factorize the left hand side and write: 5x(x-4) = 0There are now two term multiplied together, 5x and (x-4). For their product to be 0, one of term must be zero. This mean either 5x = 0 or (x-4) =0.If 5x = 0, dividing by 5 we find that x = 0 and this one possible solution to the equation.If x – 4 = 0, adding 4 to both sides we find that x = 4, which is the other possible solution.The graph of the function is plotted It also shows that x = 0 and x = 4 are two solutions to the equation, since they are the values of x at which at which the curve cuts the x axis

-50

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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Y = 5x2 – 20x

A more general method for solving the quadratic equation ax2+bx +c =0 uses a formula:

-b b2 – 4acx = ------------------- 2a

Where a is the coefficient of x2, b is the coefficient of x and c is the constant term.

Example: 8x2 – 20x + 3 = 0We identify a= 8, b= -20 and c= 3. Notice that the sign of the coefficient must be included. Begin the calculating the expression the square root sign. This gives:b2 – 4ac = (-20)2 – (4 . 8 . 3) = 400 – 96 = 304

We take the square root of this value and obtain 17.436. substituting this result , -b and a into the formula gives: 20 17.436

x = -------------- 16x = 37.436/16= 2.34 or x = 2.564/16 = 0.16 The solution of the equation is x = 2.34 or 0.16.

INTERSECTION OF MC WITH MR OR AVC

Quadratic equation arise in economics when we want to discover where a quadratic function, say marginal cost, cuts another quadratic function, say average variable cost, or cuts a linier function, say marginal revenue.

We equate the two functional expressions, then subtract the right hand side from both sides so that the value on the right becomes zero.

After collecting term we solve the quadratic equation using one of the methods explained above.

A firm has the marginal cost function MC = 3Q2 – 32Q + 96 and marginal revenue function MR = 236 – 16Q. Find the firm’s profit maximizing output.

To maximize profits the firm chooses to produce where marginal cost equals marginal revenue. Equating the MC and MR functions we have that:

3Q2 – 32Q + 96 = 236 – 16QSubtracting the right hand side from both sides gives:

3Q2 – 32Q + 96 – (236 – 16Q) = 0Removing the bracket gives

3Q2 – 32Q + 96 – 236 + 16Q =0And by collecting term we obtain:

3Q2 – 16Q – 140 = 0 -b b2 – 4acWe now use the formula for solving a quadratic equation: x = -----------------

2awhere a= 3, b= -16, and c = -140. Calculating the expression within the square root sign gives: b2 – 4ac = (-16)2 – (4 . 3. -140) = 256 + 1680 = 1936.1936 = 44. We have them:x = (1644) / (2 . 3) = 60/6 or -28/6So x = 10 or x = -4.67. Only the positive value is economically meaningful, so profit maximization occurs when x = 10.

SIMULTANEOUS EQUATIONS When economists model how market operate, they often

use different equations to represent different aspects of the market.

For market equilibrium, they values of the variables are such that are true simultaneously.

Quantity demanded and quantity supplied are function pf price (P). In equilibrium these quantities are equal.

To solve for equilibrium values, we equate the two expression in P, thus eliminating Q. We obtain an equation which we can solve for P.

Another method of eliminating a variable is to subtract (or add) the left hand sides and the right hand sides of a pair of equations.

Solve the simultaneous equation 2x + 4y = 20 and 3x + 5y = 28 For ease of reference we number the equation

2x + 4y = 20 ….. (1)3x + 5y = 28 ……(2) we choose the variable to be eliminated, say x. We need to get x with the same coefficient in both equation. Using x’s coefficient in the other equation, we multiply through equation (1) by 3 and equation (2) by 2. This give:6x + 12y = 60 …… (3)6x + 10y = 56 ….. (4)Now that x has a coefficient of 6 in both equations we subtract the corresponding sides of equations (3) and (4). We obtain:0 + 2y = 4Since 2y = 4 y = 2 is solution for y. Now substitute it in to either equation, say (1). We get:2x + 4(2) = 20 2x + 8 = 20Subtracting 9 from both sides gives: 2x = 12 x = 6As a check, substitute x = 6, y = 2 in equation (2). The left hand side is3(6) + 5(2) = 18 + 10This equals the right hand side of 28, so the solution x = 6, y = 2 is correct !

SIMULTANEOUS EQUILIBRIUM IN RELATED MARKETS

Demand and supply in two related markets forms an example of an economics model using simultaneous equations.

Demand each market depend both on the price of the good it self and on the price of the related good.

To solve the model we use the equilibrium condition for each market and equate the quantity supplied to the quantity demanded in that market.

This give two equations in two unknows which we then solve.

The market for activity holidays is represented by the functions Demand : Qa = 600 – (Pa/3) + (Pb/4)Supply : Qa = -100 + Pa

and the market for beach holidays is represented by the functionsDemand : Qb = 1800 – 3Pb + (Pa/3)Supply : Qb = -400 + 3Pb

Where Qa and Qb are quantity of activity and beach holidays respec tively and Pa and Pb are the prices of each type of holiday. Find the equilibrium prices and quantities of each type of holiday.

Equating the quantity supplied and demanded in the activity holiday market and substituting, we get:

-100 + Pa = 600 – (Pa/3) + (Pb/4) or (4Pa/3) – (Pb/4) = 700 …… (1)

And equating the quantity supplied and demanded in the beach holiday market give:-400 + 3Pb = 1800 – 3Pb + (Pa/3) or (-Pa/3) + 6Pb = 2200 …… (2)

We now have two simultaneous equation equations to solve for Pa and Pb. Multiply equation (2) by 4, which gives:

(-4Pa/3) + 24Pb = 8800 …… (3)

Adding equation (1) and (3) we find:0Pa + 23.75Pb = 9500

so, 23.75Pb = 9500 Pb = 400We can now find Pa from equation (2)

Pa/3 + 6(400) = 2200Subtracting 2400 from each sides we have: (-Pa/3) = -200Multiplying by -3 gives : Pa = 600We can find the quantities of holidays most easily from the

supply equations. For activity holidays: Qa = -100 + PaWhich gives: Qa = -100 + 600 = 500

Using the beach holidays supply equation: Qb = -400 + 3PbWhich gives: Qb = -400 + 3(400) = 800The solution is:

Pa = 600, Pb = 400, Qa = 500, Qb = 800

Check by substituting in the demand equationsActivity holiday: Qa = 600 – (Pa/3) + (Pb/4)

The right hand side gives:600 –(600/3)+(400/4) = 500 = QaBeach Holiday: Qb = 1800 – 3Pb + (Pa/3)

The right hand sides gives:1800 – 3(400) + (600/3) = 800 = Qb

Therefore, the solution is correct