South 고등학교 3학년 Mathematics for Economics
Chapters: 4
1. Numbers and economic life
Economic statistics · Percent change · Exchange rates · Gains and losses from rates · Taxes
- Macroeconomic Indicators: GDP, Inflation and Unemployment – Governments judge how well an economy is doing with a few key numbers called macroeconomic indicators. The main ones are real GDP growth (is output rising?), inflation (are prices rising, and how fast?), unemployment (are people who want work able to find it?) and the current account balance (is the country paying its way with the rest of the world?). Each has a clear formula and each has limits.
- Percentages – Per cent means "out of 100", so x% = x/100. To find x% of an amount, multiply by x/100. Percentage change = (change ÷ original) × 100. To increase by r%, multiply by (1 + r/100); to decrease by r%, multiply by (1 − r/100). Successive changes multiply their multipliers. For reverse percentage, divide the new value by the multiplier.
- Exchange Rates: Converting Money Between Currencies – A currency is the money used in a country or group of countries. An exchange rate is the price of one currency in terms of another, for example 1 USD = 83 INR. To convert into the second currency, multiply by the rate; to convert back, divide. Banks and money changers sell foreign currency at a higher rate and buy it at a lower rate, and may charge commission, so each swap costs you a little. Rates change with demand and supply. When a currency appreciates it buys more foreign money: imports get cheaper and exports dearer. When it depreciates, the opposite happens. People and firms gain or lose when rates move between buying and selling.
- Taxation: Types of Taxes and How to Calculate Them – A tax is money people and businesses must pay to the government. The government uses it for public goods (roads, schools, hospitals, defence), to help poorer people and to manage the economy. Direct taxes are paid straight from income or wealth (income tax, corporate tax, property tax). Indirect taxes are added to the price of goods and services (GST, VAT, excise, customs). A tax is progressive if richer people pay a bigger share of income, proportional (flat) if all pay the same share, and regressive if poorer people pay a bigger share. Income tax often uses slabs: each slab of income has its own rate. Effective rate = total tax ÷ income × 100.
2. Sequences and finance
Simple and compound interest · Interest and discount rates · Present value · Continuous compounding · Using continuous compounding · Annuities · Present value of annuities
- Compound Interest – Interest is money paid for using money. Simple interest is paid only on the starting amount (the principal), so it grows by the same step every year. Compound interest is paid on the principal plus all the interest earned so far, so the steps get bigger: interest earns interest. The amount after n years is A = P(1 + r/100)ⁿ. If interest is added k times a year, use rate r/k and k × n periods: A = P(1 + r/(100k))^(kn). Adding it every instant gives continuous compounding, A = Pe^(rt). The same idea run backwards gives depreciation: A = P(1 − r/100)ⁿ. Compound growth is exponential growth.
- Annuities: Equal Payments, Future Value and Present Value – An annuity is a series of equal payments made at equal time gaps, like a monthly saving or a loan instalment. Each payment earns compound interest, so the payments form a geometric series. Future value of an ordinary annuity (payments at the end of each period): FV = P[(1+i)ⁿ − 1]/i. Present value (what the payments are worth today): PV = P[1 − (1+i)⁻ⁿ]/i. Loans use PV: the loan amount equals the present value of all instalments.
3. Functions and economy
Production and cost functions · Demand and supply · Utility · Equilibrium · Tax and income effects · Decisions with utility functions · Regions of inequalities
- Functions: Composite, Inverse and Standard Graphs – A function is a rule that gives exactly one output for each allowed input. The allowed inputs are the domain; the outputs are the range. Two functions can be joined: g(f(x)) means do f first, then g. An inverse function f⁻¹ undoes f, and its graph is the mirror image of the graph of f in the line y = x. Only one-to-one functions have an inverse.
- Demand, Supply and Market Equilibrium – The law of demand says buyers want less when the price rises; the law of supply says sellers offer more. The market price settles at equilibrium, where quantity demanded equals quantity supplied, and shifts in demand or supply move it. Some goods break the usual laws (Giffen, Veblen, panic buying). A price ceiling set below equilibrium causes shortages. Markets can also fail, for example with pollution or public goods like street lights, so the government steps in.
- Consumer's Equilibrium: Utility, Budget Line and Indifference Curves – A consumer is in equilibrium when she gets the most satisfaction from her fixed income at given prices, and has no reason to change her purchases. By utility analysis, with one good she buys until MU (in rupees) = price; with two goods MUx/Px = MUy/Py. By indifference curve analysis, the best bundle is where the budget line just touches the highest indifference curve: MRS = P1/P2, with MRS falling.
- Linear Programming (Class 12): find the best answer with a graph – Linear programming finds the biggest profit or the smallest cost when you must obey some rules. The rules are straight-line inequalities (constraints). Together they cut out a region of allowed points (the feasible region). The goal, Z = ax + by (the objective function), is always best at a corner of that region. So: draw the lines, shade, find the corners, put each corner in Z, pick the largest or smallest. If the region is open (unbounded), check once more that the answer really holds.
4. Differentiation and economy
Meaning of derivatives · Graphs with derivatives · Marginal product · Elasticity
- Continuity and Differentiability – A function is continuous at a point when its graph has no break there: the left limit, the right limit and the value are all equal. The derivative is the slope of the tangent line. With the chain rule, implicit differentiation, the rules for eˣ, ln x and inverse trig functions, logarithmic differentiation and parametric forms, you can differentiate almost any Class 12 function. The second derivative tells how the slope itself changes, that is, how the curve bends.
- Curve Sketching Using Derivatives – To sketch y = f(x): find the domain and intercepts, solve f'(x) = 0 for stationary points, use the sign of f'(x) to see where the curve rises or falls, use f''(x) for concavity and points of inflection, check asymptotes and end behaviour, then join everything smoothly. Example: y = x³ − 3x has a maximum at (−1, 2), a minimum at (1, −2) and an inflection at (0, 0).
- Price Elasticity of Demand and Supply – Elasticity measures how strongly quantity reacts when price changes. Price elasticity of demand (PED) = % change in quantity demanded ÷ % change in price; it is negative but usually written without the sign. PED > 1 is elastic (buyers react a lot), PED < 1 is inelastic (buyers hardly react), PED = 1 is unit elastic. Demand is more elastic when there are close substitutes, the good is a luxury, it takes a big share of income, or buyers have more time. If demand is elastic, a price cut raises total revenue; if inelastic, a price rise raises total revenue. Price elasticity of supply (PES) = % change in quantity supplied ÷ % change in price; supply is more elastic with spare capacity, stocks, easy inputs and more time.