The budget line
With income m and prices pₓ and p_y, every affordable bundle satisfies pₓx + p_y y ≤ m. The budget line is where that holds with equality. It meets the axes at m/pₓ and m/p_y, and its slope is −pₓ/p_y: the rate at which the market lets you swap y for x.
In the explorer, income is £100, x costs £5 and y costs £4, so the line runs from 20 units of x to 25 units of y. A rise in income shifts the line out in parallel. A change in one price pivots it around the other axis, because the most you can buy of the other good hasn’t changed.
Indifference curves
An indifference curve joins bundles the consumer likes equally. Under the usual assumptions (more is better, preferences are complete and transitive, and averages are preferred to extremes) the curves slope down, never cross, and bow in towards the origin. Higher curves are better.
The slope of an indifference curve is the marginal rate of substitution: how much y the consumer would give up for one more unit of x and feel no worse off. It equals the ratio of marginal utilities, MUₓ/MU_y. Because curves are convex, the MRS falls as you move along them: the more x you have, the less y you will give for another unit.
The optimal bundle
The best affordable bundle is on the highest indifference curve that touches the budget line. Where both are smooth, that is a tangency: the MRS equals the price ratio pₓ/p_y. If MRS were above the price ratio, the consumer values x more than the market charges for it, and should buy more x.
Second-year micro modules turn this into a constrained optimisation problem and solve it with a Lagrangian. Maximise U(x, y) subject to pₓx + p_y y = m, set the derivatives with respect to x, y and the multiplier to zero, and the first two conditions divide to give MRS = pₓ/p_y.
Cobb-Douglas utility
The Cobb-Douglas utility function U = x^α y^(1−α) is the workhorse of exam questions because its demands are so clean: x* = αm/pₓ and y* = (1 − α)m/p_y. The consumer spends a fixed share α of income on x whatever the prices. With α = 0.5 in the explorer, that is £50 on each good: 10 units of x and 12.5 of y.
Two consequences are worth remembering. Demand for x doesn’t depend on the price of y, so the cross-price effect is zero. And both goods are normal, with an income elasticity of exactly one: raise income by 40% and both quantities rise by 40%.
Substitution and income effects
When pₓ doubles from £5 to £10, x falls from 10 to 5. The Slutsky decomposition splits that fall in two. First give the consumer just enough income to afford the old bundle at the new prices: 10 × £10 + 12.5 × £4 = £150. At those prices and £150, they would buy 7.5 units of x. The move from 10 to 7.5 is the substitution effect, caused only by x becoming relatively dearer.
Then take the extra income away again. The move from 7.5 to 5 is the income effect, caused by the price rise making the consumer poorer. For a normal good both effects point the same way. For an inferior good the income effect pushes back, and if it is strong enough to outweigh the substitution effect the good is a Giffen good, which exam questions like far more than real markets do.
Where this leads
The same diagram builds the demand curve: pivot the budget line through a range of prices, record the quantity of x each time, and plot price against quantity. It also sets up labour supply, where the goods are consumption and leisure, and intertemporal choice, where they are consumption today and tomorrow.
Hicksian decomposition holds utility, not purchasing power, constant, and leads to compensated demand and the expenditure function. Most courses teach both, and the Slutsky equation ties them together.