Unit 2: Understanding Functions in Economics
Wait, wait โ don’t run away! ๐ This is “maths without tears”. Every idea here is just a thali, an auto ride, or a vending machine in disguise. If you survived school till Class 10, you’ll cruise through this. Guaranteed.
Why is there maths in my economics?! ๐ค
Fair question! Here’s the honest answer: economics is full of relationships. Price goes up โ demand comes down. Income rises โ spending rises. Fuel gets costlier โ auto fares climb. A function is simply the mathematician’s way of writing “when THIS changes, THAT changes in a predictable way.”
So functions are not maths for maths’ sake โ they are the grammar of economics. Learn this small grammar once, and every demand curve, cost curve, and revenue curve in the next three years becomes readable. ๐
Sets: Fancy word for “a well-defined collection” ๐ฑ
Look at a South Indian thali: rice, sambar, rasam, curd, papad. That’s a set! A clearly defined collection of items. Your WhatsApp family group? A set of people. The IPL teams? A set of ten. You’ve been using sets since childhood โ we’re just giving them a uniform and a name tag today.
The definition + the dress code ๐
A set is a well-defined collection of distinct objects. “Well-defined” means there’s no confusion about who’s in and who’s out.
- โ “Vowels of English” = {a, e, i, o, u} โ crystal clear, so it’s a set.
- โ “Tasty dishes of India” โ tasty for whom? Your tasty โ my tasty. NOT a set (not well-defined).
Notation: Capital letters for sets, curly brackets for members: A = {2, 4, 6, 8}. We write 4 โ A (“4 belongs to A”) and 5 โ A (“5 doesn’t belong”).
Types of sets (with instant examples)
| Type | Meaning | Example |
|---|---|---|
| Empty / Null set (โ ) | No members at all | Set of Indian states starting with ‘Z’ = { } โ nothing qualifies! |
| Singleton set | Exactly one member | Set of India’s capital cities = {New Delhi} |
| Finite set | Countable members | Days of the week = 7 members, done |
| Infinite set | Never-ending members | All natural numbers {1, 2, 3, โฆ} |
| Equal sets | Exactly same members | {1,2,3} and {3,1,2} โ order doesn’t matter! |
| Subset (โ) | Every member of A is also in B | {CSK, MI} โ {all IPL teams} |
| Universal set (U) | The full “background” set for the discussion | Discussing IPL? U = all ten IPL teams |
Economics link: “Set of all consumers of Amul butter”, “set of goods in the CPI basket”, “set of taxpayers in AP” โ economists build sets first, then study them.
๐ฎ Interactive: Set Operations, the Chai-Coffee Way
In a hostel of 20 students: set A = chai lovers, set B = coffee lovers. Some drink both (the smart ones ๐). Click each operation and watch the shading change:
Make two sets from your own life โ A = apps you use daily, B = apps your best friend uses daily. Write A โฉ B (common apps) and A โ B (only yours). Congratulations, you just did set operations without a textbook!
Exam phrasing: Union = “either or both” ยท Intersection = “both” ยท Difference = “only in A”.Cartesian Product: The combo-meal maker ๐
Raju anna (from Unit 1!) upgrades his stall. Drinks: D = {chai, coffee}. Snacks: S = {samosa, vada, biscuit}. A customer asks: “Anna, what combos are possible?” Raju lists EVERY drink paired with EVERY snack. That complete list of pairs is the Cartesian product D ร S.
Definition, gently
A ร B = the set of ALL ordered pairs (a, b), where the first item comes from A and the second from B.
Raju’s full combo menu, D ร S:
Counting shortcut: if A has m members and B has n members, then A ร B has m ร n pairs. Here: 2 drinks ร 3 snacks = 6 combos. That’s literally why it’s called a “product”!
โ ๏ธ Order matters!
(chai, samosa) โ (samosa, chai). The pair (drink, snack) is not the pair (snack, drink) โ just like “Row 3, Seat 12” is a different cinema seat from “Row 12, Seat 3”. So in general, A ร B โ B ร A.
Why economists care: every graph you’ll ever draw โ price vs quantity, income vs consumption โ lives on a Cartesian plane, which is just โ ร โ: all possible (x, y) pairs. The Cartesian product is the floor on which all of economics’ pictures are drawn.
Relations vs Functions: The vending machine test โ๏ธ
Step 1 ยท What is a relation?
From the full combo list (Cartesian product), suppose we pick only the pairs that were actually ordered today: {(chai, samosa), (chai, vada), (coffee, biscuit)}. Any such chosen subset of A ร B is called a relation โ it “relates” some members of A to some members of B.
Relation = any subset of the Cartesian product. Easy!
Step 2 ยท What is a function?
A function is a special, well-behaved relation with one strict rule:
x = โน20
MACHINE
f(x) = rule
f(x) = 1 chai
The vending machine test ๐ฅค: Press button C3, get exactly one cold drink โ same one every time. That’s a function. Now imagine a haunted machine where pressing C3 sometimes gives Sprite and sometimes gives Thums Up. Unpredictable = NOT a function (just a relation). And a machine where button C3 gives nothing at all? Also not a function โ every input must produce an output.
| Situation | Function? | Why |
|---|---|---|
| Each student โ their roll number | โ Yes | Every student has exactly one roll number |
| Each price of petrol โ quantity demanded | โ Yes | One price, one quantity (the demand function!) |
| Each person โ their sibling | โ No | Some have none, some have three โ inputs with zero or many outputs |
| Each mobile number โ its owner | โ Yes | One number, one registered owner |
Golden line: All functions are relations, but not all relations are functions. (Like: all samosas are snacks, but not all snacks are samosas.)
Two words you’ll use for three years:
Independent variable (x) the input โ the one that changes freely (e.g., price).
Dependent variable (y) the output โ its value depends on x (e.g., quantity demanded). We write y = f(x), read “y is a function of x”.
Electricity bill and units consumed: which is the independent variable and which is dependent? Now think of one more (x, y) pair from your life where one clearly drives the other.
Quick check: ask “which one is the cause, which one responds?” Cause = independent.Linear Functions: Straight-line stories ๐
The auto-rickshaw fare. In your city, the meter starts at โน30 (minimum fare), then adds โน12 per kilometre. So:
Fare = 30 + 12 ร (kilometres) i.e. y = 30 + 12x
Congratulations โ that’s a linear function! You’ve been mentally computing linear functions every time you argued with an auto driver. ๐
The universal costume: y = a + bx
Every linear function wears the same outfit. Let’s name the parts using the auto example:
| Part | Name | Auto meaning | What it does on the graph |
|---|---|---|---|
| y | Dependent variable | Total fare | Measured on the vertical axis |
| x | Independent variable | Distance travelled | Measured on the horizontal axis |
| a | Intercept | โน30 minimum fare โ you pay it even for 0 km | Where the line CUTS the y-axis (the starting height) |
| b | Slope | โน12 extra per extra km | The STEEPNESS โ rise in y for each 1-step in x |
Slope in one line: slope = change in y รท change in x = ฮy/ฮx. If fare jumps โน24 when distance rises 2 km, slope = 24/2 = 12. That’s it. That’s the whole formula.
- Positive slope (b > 0): line climbs uphill โ โ like income and spending (earn more, spend more). Supply curves behave like this.
- Negative slope (b < 0): line goes downhill โ โ like price and demand (price up, purchases down). The famous demand curve!
- Zero slope (b = 0): flat line โ y doesn’t respond to x at all (your landlord’s rent this year, no matter how sweetly you ask).
Why “linear”? Because the slope is constant โ every extra km costs the same โน12, whether it’s the 2nd km or the 20th. Constant rate โ straight line. Remember this; non-linear functions will break exactly this rule.
๐ฎ The Line Lab: Drag a, drag b, watch the line obey
This is y = a + bx live. Move a (intercept) โ the whole line lifts or sinks. Move b (slope) โ it tilts. Try making the demand curve (negative slope)!
Your mobile plan: โน199 fixed + โน2 per GB extra data. Write it as y = a + bx. What is your ‘a’, what is your ‘b’, and what would the bill be for 5 extra GB? (Do it mentally โ no calculator!)
If you got โน209, you’ve officially done applied mathematics for economics. ๐Non-Linear Functions: When life isn’t a straight line ๐ข
Straight lines assume “every extra step costs/gives the same”. But real life? The first samosa is heaven, the fifth is okay, the tenth is punishment. Effects that speed up, slow down, or U-turn need curves, not lines. Enter the non-linear family โ three members you must meet.
Meet the family ๐จโ๐ฉโ๐ง
| Function | General form | Shape | Star role in economics |
|---|---|---|---|
| Quadratic | y = axยฒ + bx + c | Parabola โ one U-turn (โช or โฉ) | Total revenue curve: rises, peaks, then falls as price cuts go too far |
| Cubic | y = axยณ + bxยฒ + cx + d | S-shape / wave โ up to two bends | Total cost curve: rises fast, then gently, then fast again (the classic S) |
| Rectangular Hyperbola | xy = k (constant) | Smooth curve hugging both axes, never touching | Demand curve where total spending is fixed; average fixed cost curve |
Spot the difference from linear: the highest power of x. Power 1 = line. Power 2 = quadratic. Power 3 = cubic. x multiplied by y equal to a constant = rectangular hyperbola.
๐ข Quadratic in real life
A cinema hall cuts ticket price to sell more seats. At first, revenue climbs (more people!). But cut too much and revenue falls (tickets almost free!). Revenue rises โ peaks โ falls: an inverted-U parabola (โฉ). Quadratics are how economists find the “sweet spot” โ the top of the hill.
ใฐ๏ธ Cubic in real life
Total cost of running a small papad factory: the first few workers add cost quickly (setup pains), the middle workers are efficient (cost rises slowly), then overcrowding makes cost shoot up again. Fastโslowโfast = the S-shaped cubic. You’ll meet this exact curve in production theory later โ and you’ll smile because you already know it.
๐ช Rectangular hyperbola in real life
Suppose a family fixes its monthly milk budget at โน600, no matter what. If milk is โน60/litre they buy 10 litres; at โน50 they buy 12; at โน40 they buy 15. Notice: price ร quantity = โน600 always. That’s xy = k! The curve slides along but the rectangle’s area under it (price ร quantity) stays constant โ hence “rectangular”. It gets close to both axes but never touches: price never becomes zero, quantity never becomes zero.
๐ฎ The Curve Gallery: See all four shapes morph
Pick a curve, then drag the slider to bend it. Watch how each family behaves differently โ this visual memory will rescue you in exams!
Think of your exam marks vs hours studied. Is it linear (every hour adds equal marks) or non-linear (first hours help a lot, late-night 3 AM hours add little)? Which curve fits your life best, and why?
If you said “rises fast then flattens” โ you’ve just described diminishing returns. Unit 3 of microeconomics will feel like dรฉjร vu!Recap: Flip & remember ๐
Same drill as Unit 1: recall first, flip second. These 12 cards are your entire Unit 2 in pocket form.
Test Yourself โ๏ธ
10 MCQs, instant feedback, zero judgement. Keep a rough sheet handy for the two tiny calculations โ they’re auto-fare level, promise!
๐ Unit 2 conquered! What’s next?
Unit 3: Indian Economy โ A Basic Orientation. Back to full-on desi territory: primary/secondary/tertiary sectors, our demographic dividend, where India ranks globally, and climate change challenges. Zero maths, hundred percent India. ๐ฎ๐ณ
Suggested homework: this week, spot ONE straight-line relationship and ONE curved relationship in your daily life. Write them as y = f(x) in words. You’ll never see an auto meter the same way again.