Argand diagram VS cartesian coordinate

For those who lacks knowledge, the problems on complex numbers can be quite hard (Like those who're trying old Pure without some old ref book). That's why we need some simple transformation to make it equivalence to a 2D-co-geom problem.
Basic knowledge:
Complex number can be written in form of $z=x+iy$ where x and y are real numbers. Then $Z(x,y)$ represents the point z on the argand diagram.
The absolute value of the complex number: $|z|=|OZ|=\sqrt{x^2+y^2}$
The principal of complex number: $\operatorname{Arg}(z)=\tan ^{-1}(m_{OZ})$ where OZ is a vector such that angle beyond 180 degrees are considered. (upon sign and obviously fails for pure imaginary numbers)
Slope between two points $u=a+bi$ and $v=c+di$:
Complex version: $\operatorname{Arg}(u-v)$
Cartesian version: $\frac{d-b}{c-a}$
Transform:
Complex version: $z\operatorname{cis}\theta$
Cartesian version: $Z\left( \begin{array}{cc}\cos\theta &\sin\theta \\-\sin\theta &\cos\theta\end{array} \right)$

Example 1: HKALE 1983 Q3
a) For complex number w, show that $|w-i|=|w+i|$ if and only if w is real.
Standard ans:
If part: Let $w=x+iy$
$x^2+(y+1)^2=x^2+(y-1)^2$, $y=0$.
Only if part: Let w be real. both sides trivially equal.

Co-geom analyse:
For circle $r=|w+i|$, t's symmetrical along y-axis/Im axis. With $w-i$ being another point on the circle and the line joining w+i and w-i is perpendicular to the symmetrical axis, therefore the midpoint lies on the line joining center and perpendicular to the constructed line. i.e., w is on Re-axis, so w is real.

b) Sketch the locus of $|2u-i|=1$
I think complex number method would be faster here.

Answer:
$|u-\frac{i}{2}|=\frac{1}{2}$, therefore the locus is a circle with center i/2 with radius 1/2.

Co-geom analysis:
$|2x+(2y-1)i|=1$
$x^2+y^2-y=0$
$x^2+(y-\frac{1}{2})^2=\frac{1}{4}$, the same result is given.

c) Assume $u \neq i$. Show that $|2u-i|=1$ if and only if $v=\frac{iu}{i-u}$ is real, hence show that u,v and i are colinear.
The first part of question is easy when you "rationalize" v, you will have:
$v=\frac{-y+ix}{-x-(y-1)i}=a+b(x^2+y^2-y)i$, Im(v) =0 by our deduction before.

The second part is much more difficult, we should analyse the question in a geometrical manner:

Choose a point on $|2u-i|=1$, say U. The center of the circle C1 is X.
rotate the circle along the origin by 90 degress anticlockwisely to make the transform "iu" (Circle C2).

Now UX intersect the circle C1 again at U'. Assume iU intersect Re-axis at V'.
U' represents the point i-u. Note that OU is perpendicular to OU', i.e., the point U''(iu) lies on the line OU'. Therefore U', U'' and O are colinear.
Furthermore, by simple geometry, we can show that U'O is parallel to the line iU and iUOU' forms a rectangle. Then we can show that triangle U'OU and iOV' are in sprial rotation relationship. (similar triangle)

^ Actually we proved the first part of the question here.
Now for the second part of question:
By trigonometry means, and U'O^2 + UO^2 = 1, we can show that U''O:U'O = U'O : UO = 1: OV'. Therefore V lies on iU, Q.E.D.

Example 2: HKALE 1993 Q11
a) Show that $\Delta W_1W_2W_3\sim \Delta Z_1Z_2Z_3$ iff $\frac{z_3-z_1}{z_2-z_1}=\frac{w_3-w_1}{w_2-w_1}$.
b) Show that a triangle is equilateral iff $z_1+\operatorname{cis}(\frac{2\pi}{3})z_2+\operatorname{cis}(\frac{4\pi}{3})z_3=0$.
c) Disprove the existance of equilateral triangle on argand diagram with all integral point vertices.

We will do the question seperately.

a) In geometrical approach, what can you see from the given information?
Information I: The quotient are equal. It implies that $\angle Z_3Z_1Z_2 = \angle W_3W_1W_2$.
Information II: The quotient are equal. It implies that $\frac{|Z_3Z_1|}{|Z_2Z_1|}=\frac{|W_3W_1|}{|W_2W_1|}$.
Now apply the knowledge learnt in secondary 1. They are similar because angle equal, 2 sides proportional!!!!
We just get 4 mark easily from the AL Pure Maths paper XD

b) Consider another equilateral triangle with vertex $\operatorname{cis}(\frac{2k\pi}{3}),k=0,1,2$.
Now $\frac{z_3-z_1}{z_2-z_1}=\frac{\operatorname{cis}(\frac{2\pi}{3})-\operatorname{cis} 0}{\operatorname{cis}(\frac{4\pi}{3})-\operatorname{cis} 0}$, since cis 0 = 1,
$(z_3-z_1)(\operatorname{cis}(\frac{4\pi}{3})-\operatorname{cis} 0)=(z_2-z_1)(\operatorname{cis}(\frac{2\pi}{3})-\operatorname{cis} 0)$
By simple rearranging, we can obtain the above result.

c) Standard solution:
By equation in b, we have:
Equating real part: $\operatorname{Re}(\Delta)=\operatorname{Re}(z_1-\frac{z_2}{2}-\frac{z_3}{2})=0$
Equating imaginary part: $\operatorname{Im}(\Delta)=\operatorname{Im}(z_1+\frac{\sqrt{3}z_2}{2}-\frac{\sqrt{3}z_3}{2})=0$
Due to the closure of rational numbers under multiplication, we can show that the two equation does not have ALL rational solution. Hence there does not exist ALL integral solution.

Disclaimer: The above skills AREN'T answering skills, they are POSSIBLE solutions, but NOT PERFECT PRESENTATIONS. A RIGOROUS PROOF should be respected.

Exercise:
1) Consider example 2a:
a) Proof or disproof the existance of the general form proofing similar triangle on Argand diagram, showing the properties for i) AAA, ii) 3 sides proportional.
b) Write down one more requirement such that two triangles are congruent.
c) Suppose the two triangles are similar, write a general transformation between two triangles in terms of the complex numbers representing the points given.
d) Let triangle A,B,C,D,E be (1+i), (2+i), (3+3i), (4+5i), (-2+4i) respectively. If triangle ABC ~ triangle DEF where F is a point on the Argand diagram, find F and the transformation between two triangles.
2) Consider exmaple 3b:
a) Is the expression symmetrical for all three points of the triangle? Prove your assertion.
b) Equilateral triangle ABC has circumcircle |2u-i|=1. Write the general expression for the triangle.
c) Let A,B represent the point (1+i), (3+0i) respectively.
i) Find two points C and D such that ABC and ABD are equilateral.
ii) Show that CD is perpendicular to AB.
3) Consider example 1, in circle |2u-i|=1,
a) Show that (i-u), (0+i/2) and (u) are colinear.
b) Prove that (i,u), (u), (0) is an right-angled triangle.
c) Show that (iu) lies on line passing through (i-u) and 0.
4) Account the following FACT in terms of complex numbers:
a) Parallel lines do not intersect each other.
b) Angle bisectors of triangle are concurrent.
c) Ptolemy's theorem for both equality and inequality case. i.e. for quadrilateral ABCD, $(AB)(CD)+(AD)(BC)\geq (AC)(BD)$, equality hold iff ABCD is concyclic.
5) HKALE 1984 Q6b: f(z) is defined by $f(z)=\frac{z-a}{bz-1}$ are defined on $D=C/\left\{\frac{1}{b}\right\}$ where $a,b\in C$. Suppose |f(1)|=|f(-1)|=|f(i)|=1.
a) Show that b is the conjugate of a, |f(z)| = 1 for all $z\in D,|z|=1$.
b) Show that f(z) is a constant function if |a|=1.

Agressive bidding

(1) Non vul, IMP pair.
C J3
D AKJT983
H K82
S 6
RHS opened 3S, what should you do?
A 3H implies a solid H weak jumping call, you can imagine his/her hand as H AKJxxx with no/useless side power. Then your H K might be a trick and a control. Considering the chance to win on hand, I called a [gambling] 3NT.
LHS called 4S and your partner called a non-invitative 4NT and all pass.
Your partner's hand is
C AK765
D 7
H 7654
S QT9
LHS lead C2, now you'll have to decide the route:
It seems easy but an uneven distribution will make the game risky because you have one and only one H control.The correct way is to keep finessing the D Q. It's safe because LHO probably holds S AK so your S Q is safe. You found the D is in a bad 4-1 distribution, but luckily LHO lacks Heart as well so 4NT is in justmade.
Besides this meaningless play, the controversal point is that whether we should bid this 3NT with two dangerous suit on opponent's hand:
- Gambling 3NT: solid 7 card suit; no side power
- Power 3NT: 25-27points, 8+ tricks with control on every suit
- Unusual NT is for 2NT/4NT, so it's not our interest.
And for me, a defensive gambling 3NT here make sense:
- Against strong opening: very solid 7+ solid suit with 1+ side control, 7.5 tricks
- Against weak opening: solid 7+ solid suit promising control on the weak-opened suit or 2+ side control, 7.5 tricks.
We can make this conclusion becoz the other case fits with a 4X jumping call like:
C K6
D 2
H AKJT9873
S Q2
Such a solid hand deserves a 4H opening.

(2) NS vul, IMP pair
Your partner (N)
C AKJ
D T
H AT842
S AK43
You (S)
C Q9763
D AKJ83
H 975
S ----

A rare 5-5-3-0 distribution; 6C is a good contract, Considering the misfit and poor C AKJ on your partner's hand, it would not be easy to be the final bid.
After N's 1H opening (no doubt for this), how can you describe the 5-5 minor powerful hand? If your partner have some good major power, that will be a 3NT; and with some good minor like:
C AKxxx
D xxxx
H Ax/Kx
S xx
Then 5 or 6C/D can be played.

My approach:
1H - 2D [1]
2S [2] - 3C [3]
4NT [4] - 6C
[1]: 10+P, 5+D, positive
[2]: reversal bid, game force, 4+S
[3]: At least 4-5 minor; do not prefer 3NT; opening power
[4] quantitative invitation, 19~21P
A splinter would not be suitable as your hand still lacks power from a 5X to slam force. Due to the lack of power, other slow approach might not be able to make such calls and stop the bidding at 4H.

P.S. I'm not an expert in bridge but a relaxing player, so I'll need more opinions : D

Chemistry note: Qualitative method to identify chemical species

To access the document version of this chapter of notes click the "Notes Corner" above.

Qualitative analysis on different compounds

Ionic compound: cation (Al³⁺, Ca²⁺, Cu²⁺, Fe²⁺, Fe³⁺, Pb²⁺, Mg²⁺, K⁺, Na⁺, Zn²⁺, Ag⁺, NH₄⁺)

1) Appearance, e.g. Cu²⁺ is blue-green, Fe³⁺ is brown.

2) Solubility: Nitrates and hydrogencarbonate are all soluble

Halides are all soluble except silver halides AgX

Sulphates are all soluble except BaSO₄, PbSO₄, CaSO₄ (slightly soluble)

Carbonates and sulphites are all insoluble except Na⁺, K⁺, (NH₄)₂CO₃

3) Flame test: use a clean Pt wire to dip conc. HCl followed by the sample solid, burn under non-luminous flame (luminous flame has its own colour) to see the characteristic colour. Principle: the compound forms metal chloride under conc. HCl (H₂SO₄ can’t be used as metal chlorides are more volatile than metal sulphates), and the excited metal cation emit photon at visible spectrum to give the characteristic colour.

Na⁺: [persistent] golden yellow, K⁺: lilac, Ca²⁺: brick red, Ba²⁺: green, Cu²⁺: bluish green

4) Actions on dilute HCl/H₂SO₄ followed by solubility test, etc.

5) Actions on NaOH: Metal ion [except Na⁺, K⁺] forms hydroxide ppt under NaOH, Al³⁺, Pb²⁺, Zn²⁺ re-dissolve in excess NaOH to give hydroxyl complex.

6) Actions on NH₃: Metal ion [except Na⁺, K⁺] forms hydroxide ppt under NH₃, Cu²⁺, Ag⁺, Zn²⁺ re-dissolve in excess NH₃ to form amine complex. Note:

I) [Cu(NH₃)₄]²⁺ is in deep blue colour, not the blue colour given by Cu²⁺.

II) Ag form Ag₂O rather than AgOH by dehydration 2AgOH → Ag₂O + H₂O, but it forms diamminesilver(I) ion under excess NH₃, i.e., [Ag(NH₃)₂]⁺

III) Al(OH)₃ and Pb(OH)₂ can be distinguished either by actions in H₂SO₄ or heat until dryness (Al₂O₃ VS PbO), PbO change colour when heated.

7) Test for ammonium: Warm with NaOH to give pungent small/turn wet red litmus paper blue.

Test for gases (NH₃, CO₂, SO₂, HX)

1) CO₂: turning lime water milky and become clear when CO₂ is in excess; note that SO₂ has similar effect by Ca(OH)₂ + SO₂ → CaSO₃ + H₂O but CO₂ don’t have reducing power.

2) SO₂: It act as an reducing agent so it reduces Cr₂O₇^2- (orange) to Cr³⁺

3) SO₃: equivalent to SO₄^2-

4) Cl₂: turn wet blue litmus paper red then bleached (white) since Cl₂ + H₂O and HCl + HOCl are in equilibrium. HCl, HOCl are both acidic while OCl^- has bleaching effect.

5) NH₃: pungent smell, form dense white fumes (NH₄Cl) with HCl(g), there are test each other.

6) HX: HCl can be tested by NH₃, HBr and HI are in equilibrium with H₂ + X₂ so they gives white fumes and brown liquid (Br₂) or black solid (I₂) as well. They can also be tested by actions on conc. H₂SO₄.

Ionic compound: anion (NO₃^-, CO₃^2-, SO₃^2-, SO₄^2-, OCl^-, S₂O₃^2-, X^-)

1) Actions of heat: Carbonates decompose upon Bunsen flame: Carbonate → oxide + CO₂

2) Sulphates are resistant under lab. flames, but iron(II) sulphates decomposes under heating: 2FeSO₄ → Fe₂O₃ + SO₂ + SO₃.

3) Actions on HCl: bleaching effect of OCl^-: OCl^- + H⁺ → Cl₂ + H₂O

4) Sulphites and thiosulphates gives SO₂ under acid:

SO₃^2- + 2H⁺ → SO₂ + H₂O, S₂O₃^2- + 2H⁺ → SO₂ + H₂O + S, so thiosulphates gives white ppt besides SO₂.

5) Barium test: sulphates, sulphites and carbonates forms ppt with Ba²⁺, common reagents includes BaCl₂, Ba(NO₃)₂. BaSO₃, BaCO₃ dissolve in HCl but BaSO₄ is resistant in acid. This serves as an unique test for sulphate ions.

6) Silver nitrate test: Dissolve the sample in HNO₃ followed by AgNO₃, X^- form ppt AgCl (white) / AgBr (pale yellow) / AgI (yellow). They can be further distinguished by:

(I) AgCl dissolve in NH₃, AgBr only dissolve in conc. NH₃, AgI is insoluble in NH₃

(II) AgCl turns grey in sunlight, AgBr turns yellowish grey, AgI remains yellow

Tests for organic compounds

1) Saturated hydrocarbons: blue/clear yellow flame under non-luminous flame

2) Alkene/alkyne: smoky flame under non-luminous flame since higher carbon-hydrogen ratio leads to more CO produced.

Test for addition: add a few drops of Br₂ (in organic solvents)/K₂Cr₂O₇/H⁺, it decolorize rapidly.

3) Haloalkanes: Boil with ethanolic KOH (KOH with EtOH), acidify with HNO₃ followed by AgNO₃, ppt of AgX is expected.

4) Alcohol:

(I) Esterification: Add acyl chloride (RCOOX) to give colourless/white fumes (HCl) (The smell of ester is probably covered by acid)

(II) Esterification: Add carboxylic acid and heat with a few drops of conc. H₂SO₄, a sweet fruity smell is given (ester)

(III) Lucas test: Add conc. HCl with ZnCl₂, cloudiness appears (NOT ppt; the mixture of liquid in different phase) according to the degree of carbon: primary alcohol (1hr), secondary alcohol (5 min) and tertiary alcohol (1 min).

(IV) Iodoform test for alcohol that the carbon connected to hydroxyl group also connected to a methyl group. Add iodine with NaOH, yellow ppt (CHI₃) with characteristic smell is given. The mechanism is as follows:

  

(V) Under K₂Cr₂O₇/H⁺, primary and secondary alcohol oxidizes but tertiary alcohol is resistant to oxidation since CH₃^- is a poor leaving group. Note that KMnO₄ can be used as the oxidizing power is too strong and it may oxidizing the tertiary alcohol.

5) Carbonyl group (aldehydes/ketones):

(I) Test by 2,4-dinitrophenylhydrazine: Aldehydes and ketones react with this reagent to give red~yellow ppt (2,4-dinitrophenylhydrazone). Accurate species can be determined by crystallization of ppt and determine the melting point.

(II) Iodoform test (for ketone CH₃COR): yellow ppt under I₂/NaOH

(III) Tollens’ reagent (silver mirror test): warm with [Ag(NH₃)₂]⁺, the reagent act as oxidizing agent that gives out silver, silver mirror is deposited in the inner surface of the tube

6) Carboxylic acid

I) Esterification with ethanol under conc. H₂SO₄ as catalyst to give sweet, fruity smell

II) RCOOH is the only organic compound to react with NaHCO₃. (Most acidic)

Economics note: International trade VI: Trading situation

To access the document version of this chapter of notes click the "Notes Corner" above.

Trading situation

1)  Globalization

Under economic aspect, globalization is the phenomenon of increasing international trade, goods and money flow among countries, due to the expansion of production line over countries (multinational corporations). Economical globalization is driven by:

-  Free trade policy, which attracts lots of investment.

-  Economic reform, especially changing from planned economy (closed/communist) to market economy (open/capitalism), which creates a large amount of job opportunities. The rapid economic growth attracts lots of foreign investment.

-  Advancement in transportation and communication technology reduces transport and transaction cost, diversifies the market, and promotes global sourcing.

As the international trades are mutually beneficial, international trade increases and different countries become more interdependent. At the same time, increased adoption of free trade policy promotes higher degree of globalization.

However, globalization also lead to several negative effects, including closing of some firms which can't compete with foreign goods (and creating unemployed labour force), and the exploitation of labour (income inequality), and exhausting natural resources much faster.

2)  Trading pattern of Hong Kong

Hong Kong relies on importing since Hong Kong lacks natural resources. External trades are important to Hong Kong, and the total traded value is even larger than the GDP. By external trade, foreign currencies can be earned and for further investment or reserve assets.

Trading patterns: the largest trading partner is mainland China; raw materials and capital goods are imported and re-exported. Hong Kong has been an important port for China. Hong Kong exports several goods especially in the light industry like textile, machinery, etc. Comparing with goods exported, services exported are more important, including merchandising, transportation service, travel services and financial services. Besides China, US, Japan and Taiwan are also major trading partners of Hong Kong.

Challenges faced by Hong Kong including the low negotiation power against trade barriers, trade performance linked to mainland's domestic export, competition from other countries including quality of services and rising production cost, etc. Trade and Industry Department, Oversea Hong Kong Economic and Trade Offices and Hong Kong Trade Development Council (HKTDC) supports the trading in Hong Kong.

3)  Economic growth VS Economic development

Economic growth can be easily measured by per capita real GDP, but economic development includes more consideration like income equality and literacy rate, etc.

per capita GDP measures the productivity per person in the economy, but the cost of living or price level is not taken into account. Then Purchasing power parity (PPP) can be used to compare the living quality between countries.

PPP exchange rate = domestic price of a given basket of goods and services in domestic currency / price of the same basket of goods and services in foreign currency. If the PPP exchange rate < official exchange rate, then it implies that every unit of domestic currency should trade for more foreign currency than the official one, then the domestic currency is undervalued. Oppositely it's called as overvalued. It can measure the living quality/economic growth in a better way as it takes the cost of living into accounts.

The economic development (the living standard/economic welfare) can be measured by Human Development Index (HDI), including the following criteria:

-  per capita GNP(PPP) as it's more suitable for international comparison of national income.

-  Life expectancy at birth reflects a population's health conditions/health care services.

-  Educational attainment (mean/expected years of schooling) since higher education level generally implies higher productivity and hence higher level of economic development.

Economic growth can be promoted by:

-  Increasing investment, including sacrificing current consumption for saving (deposited and banks can used them to invest), foreign investment/aids/loans.

-  Promoting free trade/free competitions that under competition, the quality rises.

-  Improving human capital, protecting property rights and upholding rule of law

-  Develop R&D; specialize in several industries (industrial policy).

4)  Cost of economic development

- Lost in current consumption due to increase in investment

- Pollution and environmental damage; exhaustion of natural resources

- Unemployment and uneven income distribution (capital accumulation)

- Economic instability leads to income instability, worsen the problem of unemployment, and hence the redistributive effect is larger, than the income inequality is worsen.

- Loss in political autonomy due to dependence on trading with foreign countries

Sustainable development is the development that satisfies the current needs without compromising the needs in the next generations, is a way to develop the economy in a better way, including:

- Producing environmentally products

- 4Rs (renew, reuse, refuse, recycle)

- Donating to environmental groups for researches

- Government: law enforcement on related field

5)  Economic growth in different region and economic corporations

-  Developing countries develops slower due to the law of diminishing marginal return, while developing countries has higher growth rate; economy in transition (from command to market economy) has instable growth.

-  Economical integration occurs across countries like ASEAN Free Trade Area in Asia, and EU economic union.

-  There are several international organizations promoting free trade and economic growth, like UN (and WTO), IMF, World Bank and APEC, PECC in Asia.

Yeahhhhhh Economics is finished! >w<

Economics note: International trade V: trading and PPF

To access the document version of this chapter of notes click the "Notes Corner" above.

Trading and PPF

Production possibilities frontier (PPF) shows all combinations of two goods that a country can produce with all given resource and best technology.

Assumption:

1) Only two goods (X, Y) are produced.

2) Resource available and is given.

3) Technology is given.

4) The country uses all resource and best technology available.

Implication:

1) Scarcity: the possible combination is limited but the want is unlimited. The good produced can't satisfy all wants.

2) Choice: Since scarcity occurs, choices have to be made, the more the Y produced, the less the X is produced.

3) Cost: The cost of producing 1Y = number of X sacrificed, and in general OC = dY/dX. Negative slope implies one must be sacrificed to produce another good. When the PPF is a straight line/linear, then the marginal cost is constant. When the PPF is concave, the marginal cost is increasing. That implies that more cost have to paid for producing further more one kind of the good.

The increasing marginal cost is due to the heterogeneous nature of resources. Some resources are more suitable to produce X while some are better for Y. When more X need to be produced, the one better for Y is less efficient to produce X, hence more Y is sacrificed to produce later unit of X.

Inversely, constant marginal cost implies homogeneous nature of resources.

1) Efficiency: When the production point is along the frontier, all resources are best used so the economy is efficient. Otherwise the attainable combination is inefficient since the resources are not best used.

2) Improvement in technology: the curve extends on one side while the maximum output for another good remains unchanged.

3) Increase in resources: there's a parallel shift in production curve (because the OC remains unchanged)

Advantages and trading

Absolute advantages: when comparing unit of goods per unit of input (resources), the one with higher productivity has the absolute advantages.

Comparative advantages: when the slope is lower than that of another economy; i.e. dY_a/dX_a > dY_b/dx_b, the less of X sacrificed per Y produced states the comparative advantage. Note that it's independent of resources available.

For example, A has both the absolute advantage on good X and Y while A has comparative advantage on Y while B has comparative advantage on X.

Under constant marginal cost, total world output is maximized when they specialize in producing in the goods in which the economy has comparative. Then complete specialization is applied. Under a mutual beneficial TOT, the consumption possibilities frontier (CPF) increases when they trade.

Trading for concave PPF

For concave PPF, complete specialization is not preferred.

1) At Complete specialization of good X

The slope at the production point is greater than the TOT line. i.e., the cost of producing X is higher than the international price (or the cost to produce equivalent amount of Y to trade a specified amount of X) Therefore the economy will decrease production in X and increase production in Y.

2) At complete specialization of good Y

Similarly, at complete specialization on Y, the slope at the production point is smaller than the TOT line, the cost of producing Y is higher than the international price, so the economy will produce less Y, more X.

3) The production point that slope = TOT line

At that point, CPF is maximized and can satisfy most wants (indifference curve), so the production point is fixed at that point.

Gain and consumption determination

At the production point B, the economy can consume any point along the TOT=CPF (satisfaction maximization is not our interest). The amount of exported and imported goods is as shown. At point A, if the original production point is along the blue part of PPF, then A boosts both consumption of good X and Y.



A very tiring topic since many diagrams were drawn. The technical parts ends here so the last one would be the trading situation.

Economics note: International trade IV: BOP and national income identity

To access the document version of this chapter of notes click the "Notes Corner" above.

BoP is a statistical statement on economic transaction of an economy with the rest of the world in a specified period, usually a quarter or a year. Inflow of money is called credit (+) which improve BoP, outflow of money is called debit (-) which worsens the situation of BoP.

BoP is divided into two parts: the current account(CA) records the inflows and outflows of goods, services, factor incomes and current transfers which won’t generate further external returns; capital and financial account(KA) records the external inflows and outflows of assts or liabilities which are likely to generate future external returns like profits/dividends.

Current account

1)  Visible trade/ goods

It includes the transaction on visible physical goods, including [domestic] export, imports and re-export. We have total export of goods = export value + re-export value

Retained imports (imported goods that for own consumption) = import – ex-export

Balance of [visible] trade = total export of goods – total imports of goods

2)  Invisible trade/services

It includes the transactions on exports and imports of invisible services, including services, royalties and licences. We have invisible trade balance = export of services – import of services

3)  External factor income flows

It includes the income earned by resident aboard or the income earned by non-residents domestically. For example, when a local resident receives profits from foreign firms, it counted as the factor income from abroad. We have net income from abroad (NIFA) = factor income from abroad – factor income paid abroad. Note that GNP = GDP + NIFA.

4)  Current transfers

It includes the economy provides real, financial gifts/donations/remittance/aid, etc, in which are immediately/shortly consumed, which do not gives output in return. These transactions are unidirectional in nature. Net current transfers = inflow – outflow of current transfers.

Reserve assets are foreign assts and gold held by the central bank as foreign exchange reserve/official reserves. It can be used to settle BoP imbalance, supporting the currency or for emergency use.

Since the market BoP excludes the transactions made by the central banks, and the transactions are made by different economic agents, imbalance BoP may result.

[Marketing] BoP balance = CA + KA – net change in reserve assets = CA + net change in capital and financial non-reserve assets; When BoP > 0, it’s a BoP surplus, in opposite it’s a BoP deficit.

Accounting BoP must be balanced (Since the assets conserves and it includes transaction of the central bank) = CA + KA = 0 i.e., CA = -KA

Principle of reserve assets: the amount of imbalance is complemented by transactions between central bank and the market.

1)  BoP surplus

The central bank buys an equivalent amount of foreign currency [debit to market] to offset the surplus make by market. If the BoP = +X, then the net change in reserve assets = +X. Then accounting BoP = CA + KA in non-reserve assets – X = (+X) – X = 0

2)  BoP deficit

The central bank sell an equivalent amount of foreign currency [credit to market] to offset the deficit made by the market. Then there’s a net decrease in reserve assets.

3)  BoP balanced

Central bank need not to do anything to balance the BoP.

National income identity

1)  The current account: CA = NX (net exports) = X – M

2)  The accounting BoP: CA + KA ≡ 0

3)  National income Y ≡ C + I + G + NX

4)  National income by consumption approach Y = C + S_p + T where S_p is private saving and T is the tax revenue for government.

5)  Government expenditure G = T - S_g where S_g is the public saving. The unused tax is saved by the government.

6)  National saving identity: national saving S = S_p + S_g

Implication:

Equating (3) and (4), NX = S_p + T – G – I = S_P + S_G – I = S – I.

It implies that in autarky (NX=0), saving = investment; when net export being constant, increase in saving also increase investment. This can be verified by the banking system: banks keep the minimum reserve and then lend out/invest the rest of the money.

We will have two more part for international trade:

V: Trading and PPF

VI: Trading, economic development and cost of economic development

Economics note: International trade III: Exchange rate system

To access the document version of this chapter of notes click the "Notes Corner" above.

Exchange rate system

Foreign currencies are not the medium of exchange in local market, so they are bought and sold in terms of local currency at the ratio of exchange rate. It refers to the price of foreign currency in terms of domestic currency.

For example, the exchange rate of 1 RMB is 1.2 HKD.

Exchange rate systems

1) Fixed (pegged) exchange rate system

The exchanged rate is fixed by central bank. i.e. central bank sell/buy foreign currency to the local market at a fixed rate. Revaluation happens when government rise the official exchange rate, while devaluation happens when government drops the official exchange rate.

2) Flexible exchange rate system

The exchange rate if controlled by market force. i.e., the exchange rate is determined by the equilibrium price from demand and supply of the market. When the price of the currencies rise under market force, it's called appreciation, in opposite it's called depreciation.

3) Linked exchange rate system (in HK)

Since the currency does not have intrinsic value, the value of currency under free market depends on the economic power like productivity, and the acceptability of the currency of small economy is usually small. As a result, the linked exchange rate system is adopted in 1983 to stabilize the exchange rate of HKD.

The official rate of USD $1 = HKD $7.8 while the actual rate floats between 7.75 and 7.85 HKD. When the exchange rate is out of the range, HKMA sell/buy HKD and buy/sell USD when the exchange rate is lower than 7.75(strong-side convertibility)/higher than 7.85(weak-side convertibility). Since it's a currency board system, the HKD is fully backed by USD.

It implies that the exchange rate of HKD against other currencies (except USD) is determined the exchange rate of USD against other currencies. For example, if Euro appreciates against USD, HKD depreciate against Euro as well.

Effects on change in exchange rate on trading

1) In terms of local currency

When local currency depreciates, in terms of local currency the price of good doesn't change (non-price factor) but the demand of goods (since price of goods in terms of foreign currency decreases), there's a net increase in total gain.

2) In terms of foreign currency

When local currency depreciates, in terms of foreign currency the price of goods rises (price factor), and the equilibrium quantity move along the demand, so the change in total expenditure is unknown, depending on the elasticity.

Economics note: International trade II: Quota and Tariff

To access the document version of this chapter of notes click the "Notes Corner" above.

Graph interpretation: tariff and quota

1) Effect of free trade

P_w: world price, D_f/S_f : demand/supply curve under free trade; bolded line = demand/supply under free trade

Domestic producer: The domestic equilibrium is higher than world price so domestic producer produce until Q₁ at price P_w. PS decreases from B+D to D so they lose in the trading.

Domestic consumer: They now buy goods at a lower price P_w < P₀ while consumption increased to Q₂. CS increased from A to A+B+C. The TSS increase from A+B+D to A+B+C+D, so the net gain is C as they can be consumed at a lower price than the price in which goods are domestically produced. The total expenditure is uncertain, depending on the elasticity.

Exporting producer: Selling at higher price (P_w) and more is produced (Q₂). (Q₂-Q₁) is exported while Q₁ is sold domestically. PS increases from D to B+C+D. Total gain increases.

Exporting consumer: They buy at higher price and consume less of it. CS decreases from A+B to A so they lose from the trading. Total expenditure is uncertain depends on elasticity.

2) Trade barriers: Quota

When a quota is applied, the supply is upward slopping before Q₁ (because domestic cost<P_w) and after quota (more of the imported goods are banned). The new equilibrium price is P', amount of domestically produced goods is Q₂ while the imported quantity is (Q₃-Q₂).

Consumer: The CS is A+B, larger than in autarky and smaller than in free trade (A+B+C+D+E+F).The price increased and quantity consumed decreased.

Producer: The PS increase from G to C+G while more is sold in higher price so they gain.

The area E is the gain for importer since they buy goods at P_w and sell at P'. If government has the right of import, then E is the government revenue. D and F is the deadweight loss since people have to buy goods at higher price beyond the quota but originally they can buy it at P_w.

Quota is inefficient due to the existence of deadweight loss.

3) Trade barriers: tariff

When a tariff is applied, the domestic price rises as much as the tariff. Quantity consumed decrease but quantity produced increase from Q₁ to Q₂, quantity imported decreases from (Q₄-Q₁) to (Q₃-Q₂).

The change of gain for producer and consumer is same as the quota. Area E is the government tax revenue. D and F is the deadweight loss since people can originally buy goods at P_w but now at higher price (Q₁~Q₂), and they can originally buy goods at (Q₃~Q₄) in which they gain consumer surplus.

Tariff is also inefficient due to the existence of deadweight loss.

4) Effectiveness of trade barriers

Quota is ineffective when quota > imported volume under free trade. Tariff is ineffective when amount of tariff > price different in equilibrium price in autarky and world price.

Tariff is less effective when D and S is inelastic since a large change in price cause a small change in quantity. As a result (Q₂-Q₁) and (Q₄-Q₃) is small under a large tariff, then it's insignificant to reduce import volume. In this case quota is more certain to reduce the import amount.

5) Reasons behind trade barriers

Local government protect its local import-competing industries to protect its own economy (e.g when local industries close down, more labour is unemployed) as they don't have absolute/comparative advantages. By imposing trade barriers, price of foreign good rises and local goods are more competitive. This is called protectionism.

Also, trade barriers means that the importing quantity is decreased, that means exporter competes to export their own goods. As a result, the quality of goods is increased.

Other measures include subsidizing the local goods to export, embargo on a certain goods.

However, trade barriers hinder international trade and countries may take revenge by putting more barriers on other goods. As a result, a small economy (e.g. HK) won't impose trade barriers as it's hard for them to build up a autarky economy and they rely on importing necessary goods. World Trade Organization (WTO) is an organization to promote free trade.

Economics note: International trade I numerical analysis on international trading

To access the document version of this chapter of notes click the "Notes Corner" above.

International trade

1)  International trade is the exchange of goods and services across territories/boundaries.

2)  Free trade is the international trade that is not hindered by any restriction imposed by the government.

Absolute vs comparative advantages

Assumption:

1)  Only two countries [economies], A and B exist.

2)  Only two goods exist, say X and Y.

3)  The resource is homogeneous, i.e., the MC of production is constant.

4)  Trade involve zero transport and transaction cost.

Case: A produce 8X or 6Y while B produces 7X or 4Y with 1 unit of resources.

Absolute advantage: One country has higher productivity per unit of resources than another country. For the above case, A has an absolute advantage over B on both good X and Y.

Comparative advantage: One country forgone less goods to produce another good than another country. For the above case, opportunity cost (OC) of producing X for A is 6/8Y while that for B is 4/7Y. Since B has a lower OC, it has comparative advantage over A on X.

Note that a country can obtain absolute advantage for both goods but not comparative advantage. When A has the comparative advantage for X, then B has the comparative advantage for Y.

The principle of comparative advantage states that when two countries specialize in producing the good in which the country has a comparative advantage on it, the total world output increases.

Terms of trade (world price P_w) is the exchange ratio of goods between the two countries.

Gain of trade per unit of X for importer = OC of producing 1X - P_w (in terms of Y)

Gain of trade per unit of X for exporter = P_w – OC of producing X.

When deficit occur to one of the country no trade would occur. Therefore the term of trade (TOT) is fixed to OC of importer > TOT > OC of exporter. In the above case, the TOT is between X = 6/8Y and X=4/7Y.

Total gain of trade = OC of producing X for importer – OC of producing X for exporter.

When transaction cost occur, trade still exist when total gain of trade ≥ transaction cost, otherwise at least one of the country lost from the trade. When two countries share the transaction cost, the range of TOT is given by:

OC of producing X for importer – part of the transaction cost > TOT > OC of producing X for exporter + the rest of the transaction cost.

For the above case, assume the transaction cost is 0.2Y while A and B share the transaction cost equally, TOT is given by (6/8-0.1)Y > TOT > (4/7+0.1)Y, which lies between 0.671Y and 0.675Y.