Budget line: intercepts, slope, equation and equilibrium

Data: price of X, Pₓ = ₹4; price of Y, Pᵧ = ₹8; money income M = ₹96. X is measured on the horizontal axis and Y on the vertical axis.

(i) Intercepts and slope

• Intercept on the X-axis = M ÷ Pₓ = 96 ÷ 4 = 24 units of X (all income spent on X).
• Intercept on the Y-axis = M ÷ Pᵧ = 96 ÷ 8 = 12 units of Y (all income spent on Y).
• Slope of the budget line = −Pₓ ÷ Pᵧ = −4 ÷ 8 = −0.5. Equivalently, −(Y-intercept ÷ X-intercept) = −12 ÷ 24 = −0.5. The market lets the consumer exchange 1 unit of X for 0.5 unit of Y: buying one more unit of X means giving up half a unit of Y.

(ii) Budget equation

Pₓ·X + Pᵧ·Y = M, that is 4X + 8Y = 96, or, dividing by 4, X + 2Y = 24. Solved for Y: Y = 12 − 0.5X, a straight line with vertical intercept 12 and slope −0.5.

Check of points on the line: (24, 0) costs 4 × 24 = ₹96; (0, 12) costs 8 × 12 = ₹96; (8, 8) costs 4 × 8 + 8 × 8 = 32 + 64 = ₹96.

(iii) Equilibrium

Diagram. Draw the budget line AB from A (0, 12) on the Y-axis to B (24, 0) on the X-axis, and an indifference map of convex curves. The consumer's equilibrium is at E, where AB is tangent to the highest attainable indifference curve, say IC₂. Lower curves such as IC₁ cut AB at two points, so the consumer can do better by moving along AB towards E; higher curves such as IC₃ lie wholly outside AB and cannot be reached.

Comment.

1. At E the consumer spends the whole ₹96 and reaches the highest satisfaction possible with this income and these prices.
2. At E the slope of the indifference curve equals the slope of the budget line: MRSₓᵧ = Pₓ/Pᵧ = 0.5.
3. The exact bundle at E depends on the consumer's preferences, which the question does not give. For illustration, if the consumer spends equal amounts on the two goods (₹48 each), E is at X = 48 ÷ 4 = 12 and Y = 48 ÷ 8 = 6; the bundle (12, 6) lies on the budget line, since 4 × 12 + 8 × 6 = 48 + 48 = ₹96.
4. Any point inside AB leaves income unspent and cannot be an equilibrium for a consumer who prefers more to less.

(iv) Slope of the indifference curve at equilibrium

At the point of equilibrium the indifference curve is tangent to the budget line, so its slope equals the budget line's slope: −0.5. In other words, MRSₓᵧ = Pₓ/Pᵧ = 4/8 = 0.5: at E the consumer is willing to give up half a unit of Y for one more unit of X, exactly the rate the market offers.

Conclusion

The budget line 4X + 8Y = 96 has intercepts of 24 units of X and 12 units of Y and a slope of −0.5. Whatever the consumer's tastes, at equilibrium the indifference curve touches this line, so its slope there is also −0.5.