Difficulty: Intermediate | Prerequisites: Utility functions, MRS, and indifference curves (see companion notes)

TL;DR

The budget constraint shows what a consumer can afford, and the optimal bundle is the best they can do within that constraint. For Cobb-Douglas preferences, the optimum is an interior tangency where MRS = Px/Py. For perfect complements, it is on the kink of the indifference curve. For perfect substitutes, it is usually a corner solution. Changing the price of X while holding everything else fixed traces out the consumer's demand curve for X.

Key Terms

Budget constraint

The set of all bundles (X, Y) a consumer can afford given their income M and the prices Px and Py. Written as Px * X + Py * Y <= M. The boundary (the budget line) is the equation Px * X + Py * Y = M.

Think of it as: everything you could buy if you spent all your money.

Budget line

The outer edge of the budget constraint, where the consumer spends all income. Rearranging: Y = M/Py - (Px/Py) * X. The slope is -Px/Py, the Y-intercept is M/Py, and the X-intercept is M/Px.

In simple terms, the line shows the trade-off the market forces on you: to get one more X, you must give up Px/Py units of Y.

Optimal bundle

The bundle on the budget constraint that gives the consumer the highest utility. How you find it depends on the utility function type.

Tangency condition

For smooth, convex preferences (Cobb-Douglas), the optimal interior bundle is where the indifference curve is tangent to the budget line: MRS = Px/Py. This is the condition that the consumer's personal trade-off rate equals the market's trade-off rate.

Corner solution

An optimum where the consumer spends all income on one good only. This happens with perfect substitutes when MRS differs from Px/Py.

Interior solution

An optimum where the consumer buys positive quantities of both goods. Cobb-Douglas always produces interior solutions.

Demand curve

The relationship between the price of a good and the quantity the consumer chooses at that price, holding income and other prices fixed. Found by solving for X* at different values of Px.

Budget Constraint Basics

The budget constraint equation is Px * X + Py * Y = M.

• Rearranging for Y: Y = (M/Py) - (Px/Py) * X.

• Slope of the budget line: -Px/Py. This tells you the market trade-off: to buy one more X, you must give up Px/Py units of Y.

• Y-intercept: M/Py (spend everything on Y).

• X-intercept: M/Px (spend everything on X).

The budget set (the affordable region) is the triangle below and to the left of the budget line, including the line itself. Any bundle inside or on the line is affordable. Any bundle outside is not.

Cobb-Douglas Optimisation

For U(X, Y) = X^a * Y^b, the optimal bundle is found using the tangency condition MRS = Px/Py, combined with the budget constraint.

The spending-share shortcut

With Cobb-Douglas utility U = X^a * Y^b, the consumer always spends a fraction a/(a+b) of income on X and b/(a+b) on Y. This gives:

• X* = [a/(a+b)] * M/Px

• Y* = [b/(a+b)] * M/Py

This shortcut works regardless of the specific prices or income, which is one reason Cobb-Douglas is so popular in textbooks.

Worked example from the homework

U(X, Y) = X^(1/6) * Y^(5/6), M = $24, Px = $2, Py = $1.

• a = 1/6, b = 5/6, so a/(a+b) = (1/6)/1 = 1/6 and b/(a+b) = 5/6.

• Spending on X: (1/6)(24) = $4, so X* = 4/2 = 2.

• Spending on Y: (5/6)(24) = $20, so Y* = 20/1 = 20.

• Optimal bundle: (X*, Y*) = (2, 20).

• Check: 2(2) + 1(20) = 4 + 20 = 24. Correct.

How to graph it

• Budget line: Y = 24 - 2X, from (0, 24) to (12, 0).

• Mark the optimal bundle (2, 20) on the budget line.

• Draw a smooth indifference curve tangent to the budget line at that point, curving away from the line on both sides.

Perfect Complements Optimisation

For U(X, Y) = MIN(aX, bY), the optimal bundle is always at the kink of the highest affordable indifference curve. The kink sits on the line aX = bY.

Method

1. Set the kink condition: aX = bY, so Y = (a/b)X.

1. Substitute into the budget constraint: Px * X + Py * (a/b) * X = M.

1. Solve for X, then Y.

Worked example from the homework

U(X, Y) = MIN(4X, Y), M = $56, Px = $4, Py = $1.

• Kink condition: 4X = Y.

• Budget constraint: 4X + 1(Y) = 56.

• Substitute: 4X + 4X = 56, so 8X = 56, giving X* = 7.

• Y* = 4(7) = 28.

• Optimal bundle: (7, 28).

• Check: 4(7) + 1(28) = 28 + 28 = 56. Correct. And MIN(28, 28) = 28.

How to graph it

• Budget line: Y = 56 - 4X, from (0, 56) to (14, 0).

• The kink line is Y = 4X.

• The optimal bundle (7, 28) is where Y = 4X crosses the budget line.

• Draw an L-shaped indifference curve with the corner at (7, 28).

Perfect Substitutes Optimisation

For U(X, Y) = aX + bY, the MRS = a/b is constant. The consumer compares this to the price ratio Px/Py.

• If MRS > Px/Py (i.e. a/b > Px/Py): the consumer values X more than the market charges. Buy only X. Corner solution: X* = M/Px, Y* = 0.

• If MRS < Px/Py (i.e. a/b < Px/Py): the consumer values X less than the market charges. Buy only Y. Corner solution: X* = 0, Y* = M/Py.

• If MRS = Px/Py: the consumer is indifferent. Any bundle on the budget line is optimal.

Worked example from the homework

U(X, Y) = X + 3Y, M = $24, Px = $2, Py = $4.

• MRS = 1/3.

• Price ratio Px/Py = 2/4 = 1/2.

• Compare: MRS = 1/3 < Px/Py = 1/2.

• The consumer values X relatively less than what the market charges. Buy only Y.

• X* = 0, Y* = 24/4 = 6.

• Optimal bundle: (0, 6).

How to graph it

• Budget line: Y = 6 - (1/2)X, from (0, 6) to (12, 0).

• Indifference curves are straight lines with slope -1/3 (flatter than the budget line slope of -1/2).

• The highest indifference curve touching the budget set touches at the Y-intercept: (0, 6).

• Mark the corner solution at (0, 6).

Deriving Demand Curves

To find the demand curve for X, solve for the optimal X* as a function of Px, holding M and Py fixed. Each row in the tables below is one point on the demand curve.

A) Cobb-Douglas: U(X, Y) = X^(1/3) * Y^(2/3), M = $24, Py = $1

Spending share on X: a/(a+b) = (1/3)/1 = 1/3. So X* = (1/3)(24)/Px = 8/Px.

The demand curve is X = 8/Px, a standard downward-sloping hyperbola.

B) Perfect complements: U(X, Y) = MIN(X, 2Y), M = $35, Py = $1

Kink condition: X = 2Y, so Y = X/2. Budget: Px * X + 1 * (X/2) = 35, giving X(Px + 1/2) = 35, so X* = 35/(Px + 0.5).

C) Perfect substitutes: U(X, Y) = 4X + Y, M = $30, Py = $1

MRS = 4/1 = 4. Compare to Px/Py = Px/1 = Px.

• If Px < 4 (MRS > Px/Py): buy only X. X* = 30/Px.

• If Px > 4 (MRS < Px/Py): buy only Y. X* = 0.

• If Px = 4: any bundle on the budget line works. X* is anything from 0 to 30/4.

Formulas Reference

Common Misconceptions

• Students often try to use the tangency condition (MRS = Px/Py) for perfect complements. This does not work because the indifference curve has a kink, not a smooth tangent. For Leontief, always use the kink condition aX = bY.

• With perfect substitutes, students sometimes assume the consumer always buys some of both goods. Unless MRS happens to equal Px/Py exactly, the optimum is a corner solution with all spending on one good.

• A common arithmetic slip in the Cobb-Douglas spending-share formula: students divide by (a + b) but forget that a and b may not sum to 1. With U = X^(1/6) * Y^(5/6), they do sum to 1, but with U = X^(1/3) * Y^(2/3) they also sum to 1. Be careful with forms like X^2 * Y^3 where a + b = 5.

• Students sometimes confuse the slope of the budget line (-Px/Py) with the MRS. They measure different things: the budget line slope is the market's trade-off rate, while the MRS is the consumer's personal trade-off rate. At the optimum, for Cobb-Douglas, these happen to be equal.

Why It Matters / Exam Flags

• The three-utility-type optimisation (Cobb-Douglas, Leontief, linear) is the bread and butter of intermediate micro exams. Expect at least one of each.

• Being able to fill a demand-curve table quickly (Q3-style) requires knowing the formula for each type cold. Practise until the Cobb-Douglas spending-share and the Leontief kink-plus-budget substitution are automatic.

• Graphing matters: the budget line must have correct intercepts, the optimal bundle must be on the budget line, and the indifference curve must be tangent (Cobb-Douglas), kinked at the point (Leontief), or flatter/steeper than the budget line at the corner (linear).

Quick Self-Test

1. True or false: For Cobb-Douglas utility U = X^a * Y^b, the consumer always spends fraction a/(a+b) of income on good X. ___

1. Fill in the blank: For U = MIN(3X, Y), the kink condition is Y = ___X.

1. True or false: With perfect substitutes U = 2X + 5Y, if Px = $3 and Py = $1, the consumer buys only Y. ___

1. Fill in the blank: The slope of the budget line Px * X + Py * Y = M is ___.

1. True or false: The tangency condition MRS = Px/Py works for all utility function types. ___

Answers: 1. True. 2. 3. 3. True (MRS = 2/5 = 0.4 < Px/Py = 3, so buy only Y). 4. -Px/Py. 5. False (it fails for perfect complements and perfect substitutes).

Practice Q&A

Q: U(X, Y) = X^(1/4) * Y^(3/4), M = $100, Px = $5, Py = $2. Find the optimal bundle.

A: Spending on X: (1/4)(100) = $25, so X* = 25/5 = 5. Spending on Y: (3/4)(100) = $75, so Y* = 75/2 = 37.5. Optimal bundle: (5, 37.5).

Q: U(X, Y) = MIN(2X, 3Y), M = $60, Px = $3, Py = $2. Find the optimal bundle.

A: Kink condition: 2X = 3Y, so Y = (2/3)X. Budget: 3X + 2(2/3)X = 3X + (4/3)X = (13/3)X = 60. X* = 180/13 ≈ 13.85. Y* = (2/3)(180/13) = 120/13 ≈ 9.23.

Q: U(X, Y) = 5X + 2Y, M = $40, Px = $4, Py = $1. What is the optimal bundle?

A: MRS = 5/2 = 2.5. Px/Py = 4/1 = 4. Since MRS = 2.5 < 4, buy only Y. X* = 0, Y* = 40.

Q: For U(X, Y) = X^(1/3) * Y^(2/3), M = $24, Py = $1, find X when Px = $3.*

A: X* = (1/3)(24)/3 = 8/3 ≈ 2.67.

Connections to Other Topics

This material flows directly from the MRS and indifference curves covered in the companion notes. The optimisation techniques here are the foundation for everything that follows in ECON 302: income and substitution effects, labour supply, and intertemporal choice all reuse the same tangency/kink/corner logic with different labels on the axes.

The demand curves derived here connect to market demand (horizontal sum of individual demands) and to elasticity, which measures how responsive quantity demanded is to price changes.

Related Terms / Search Tags

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