1  Step by Step: The Consumer’s Optimal Choice

1.1 Setup

A consumer chooses a bundle \((x, y)\) of two goods. Preferences are represented by a utility function:

\[u(x, y)\]

The price of good \(x\) is \(p_x\) and the price of good \(y\) is \(p_y\):

\[p_x = 4 \qquad\qquad p_y = 1\]

The consumer’s income is \(I\):

\[I = 36\]

The budget constraint is:

\[p_x\,x + p_y\,y = I \;\;\Longrightarrow\;\; 4x + y = 36\]

The consumer chooses the bundle that maximizes utility subject to the budget constraint.

Three preferences:

\[u = x^{1/3}y^{2/3} \qquad\quad u = \tfrac13 x + \tfrac23 y \qquad\quad u = \min\left\{\tfrac13 x,\ \tfrac23 y\right\}\]

1.2 Cobb-Douglas: \(u = x^{1/3}y^{2/3}\)

1.2.1 Tangency condition

Marginal utilities:

\[MU_x = \tfrac13 x^{-2/3}y^{2/3} \qquad\qquad MU_y = \tfrac23 x^{1/3}y^{-1/3}\]

Dividing:

\[MRS = \frac{MU_x}{MU_y} = \frac{y}{2x}\]

Setting equal to \(p_x/p_y\):

\[\frac{y}{2x} = \frac{4}{1} \;\;\Longrightarrow\;\; y = 8x\]

1.2.2 Optimal choice

Substituting \(y = 8x\) into the budget constraint:

\[4x + 8x = 36\]

Solving:

\[x = 3 \qquad\qquad y = 8\cdot 3 = 24\]

1.2.3 Utility attained

Evaluating at \((3, 24)\):

\[u(3, 24) = 3^{1/3}\cdot 24^{2/3} = 3^{1/3}\cdot 4\cdot 3^{2/3} = 12\]

1.2.4 Indifference curve and budget constraint

1.3 Perfect substitutes: \(u = \tfrac13 x + \tfrac23 y\)

1.3.1 Comparing \(MRS\) and \(p_x/p_y\)

Marginal utilities:

\[MU_x = \tfrac13 \qquad\qquad MU_y = \tfrac23\]

Dividing:

\[MRS = \frac{1/3}{2/3} = \frac12 \qquad\qquad \frac{p_x}{p_y} = 4\]

Comparing:

\[MRS < \frac{p_x}{p_y} \;\;\Longrightarrow\;\; x = 0\]

1.3.2 Optimal choice

Substituting \(x = 0\) into the budget constraint:

\[4\cdot 0 + y = 36\]

Solving:

\[x = 0 \qquad\qquad y = 36\]

1.3.3 Utility attained

Evaluating at \((0, 36)\):

\[u(0, 36) = \tfrac13\cdot 0 + \tfrac23\cdot 36 = 24\]

1.3.4 Indifference curve and budget constraint

1.4 Perfect complements: \(u = \min\left\{\tfrac13 x,\ \tfrac23 y\right\}\)

1.4.1 Fixed-proportions condition

Without wasting either good:

\[\tfrac13 x = \tfrac23 y \;\;\Longrightarrow\;\; x = 2y\]

1.4.2 Optimal choice

Substituting \(x = 2y\) into the budget constraint:

\[4\cdot 2y + y = 36 \;\;\Longrightarrow\;\; 9y = 36\]

Solving:

\[y = 4 \qquad\qquad x = 2\cdot 4 = 8\]

1.4.3 Utility attained

Evaluating at \((8, 4)\):

\[u(8, 4) = \min\left\{\tfrac13\cdot 8,\ \tfrac23\cdot 4\right\} = \min\left\{\tfrac83,\ \tfrac83\right\} = \tfrac83\]

1.4.4 Indifference curve and budget constraint