nhi 111 Lecture_Choice3

Consumer Decision Model

• Dr. Leon Vinokur

• ECN 111 – Microeconomics 1

• Lecture 4

Objectives for Today

• Model the individual’s choice.

  • Focus question: What does a person choose to buy given their monetary income, the prices of the goods, and their preferences?

Optimal Choice of the Consumer

Introduction

• Calculators away: Life’s big choices call for gut instinct.

• Reference: Tim Harford, Undercover Economist, Financial Times.

• Model of how individuals make choices:

  • It is simplified and abstracts from reality.

  • Includes considerations of uncertainty and the trade-off between future and present.

  • Still a useful tool for policy evaluation (e.g., food stamps).

Economic Rationality

• Principal behavioral postulate: A decision maker chooses its most preferred alternative from available options.

• The available choices construct the choice set.

• The decision maker seeks to locate the most preferred bundle in the choice set mathematically, expressed as:

  • $ext{Maximize } U(x_1, x_2) ext{ subject to } p_1x_1 + p_2x_2 = m$

Rational Constrained Choice

• The solution to the constrained optimization problem yields the consumer's demand function.

• The most preferred affordable bundle is termed the consumer’s ORDINARY DEMAND at given prices and budget.

• Denoted as $x_1^<em>(p_1, p_2, m)$ and $x_2^</em>(p_1, p_2, m)$.

Optimal Bundle Identification

• Definition of optimal bundle (point A):

  • At equilibrium, the individual has no incentive to alter the bundle since higher utility cannot be achieved due to budget constraints.

  • Increasing good x would enhance utility, but given the budget, it is unfeasible. The same applies for good y.

• Analysis of Point B:

  • Switching some of x for more of y leads to higher utility and is feasible, indicating Point B is not in equilibrium.

Conditions for Rational Constrained Choice

• Condition 1: For $(x_1^<em>, x_2^</em>)$ to be considered an interior solution:

  • The budget must be exhausted: $p_1x_1^* + p_2x_2^* = m$.

• Condition 2: At $(x_1^<em>, x_2^</em>)$, the slope of the indifference curve equals the slope of the budget constraint.

First-Order Conditions (FOC)

• The Lagrangian for this constrained maximization problem can be set up as follows:

  • $ext{Lagrangian } \boldsymbol{ ext{L} }(x_1, x_2, \boldsymbol{ heta})=U(x_1, x_2) + heta(m - p_1x_1 - p_2x_2)$.

• The FOCs are:

  1. $MU_1 - heta p_1 = 0$

  2. $MU_2 - heta p_2 = 0$

  3. $p_1x_1 + p_2x_2 = m$

Marginal Utility Equalities

• From the FOCs:

  • Rearranging gives us: $MU_1/p_1 = MU_2/p_2 = heta$

  • By equalizing these, we derive: $rac{MU_1}{MU_2} = rac{p_1}{p_2}$.

Equilibrium Condition

• The tangency of the indifference curve and budget constraint indicates equilibrium:

  • Equal slopes mean the individual and market appraise goods similarly.

  • The condition for choice: Marginal Rate of Substitution (MRS) equals the rate at which goods can be traded in the market.

  • Mathematically:

  • $MRS = rac{MU_x}{MU_y} = - rac{p_x}{p_y}$.

Marginal Utility Principle

• The bundle that maximizes total utility occurs when:

  • $MU_x/P_x = MU_y/P_y$.

• Justification: If you reallocate the last dollar spent from x to y, due to diminishing marginal utility, you'd lose more utility than you gain, invalidating optimization.

Computing Ordinary Demands - A Cobb-Douglas Example

• Given Cobb-Douglas preferences represented by:

  • $U(x_1, x_2) = x_1^a x_2^b$

  • Income $m$ and prices $p_1$ and $p_2$.

Solving the Demand Functions

• Start with two essential conditions:

  1. The budget constraint: $p_1x_1 + p_2x_2 = m$.

  2. Indifference curve tangency to the budget constraint.

• Results in:

  • $rac{MU_1}{p_1} = rac{MU_2}{p_2}$ which aligns with earlier derivations.

Solutions for Demand Functions

• Deriving demand functions for the Cobb-Douglas utility structure:

  • The analysis allows for understanding the effect of changing prices or income on the optimizing bundle.

  • Derived functions ensue:

  • $x_1^* = rac{a}{a+b} rac{m}{p_1}$

  • $x_2^* = rac{b}{a+b} rac{m}{p_2}$.

Summary of Rational Constrained Choice

• Conditions for achieving ordinary demands:

  • Interior solutions: x_1^* > 0 and x_2^* > 0 .

  • Budget constraint constraint: $p_1x_1^* + p_2x_2^* = m$.

  • Slopes of the budget constraint and the indifference curve are equal at optimal bundle point:

  • $- rac{p_1}{p_2} = MRS$.

Example of Corner Solutions

• Scenario evaluating ordinary demand functions where preferences can be outlined by utilities such as:

  • $U(x,y) = 2xy^2$.

Perfect Substitutes Case: Corner Solutions

• Example: $(U(x,y) = x_1 + x_2)$ leading to the following analyses and conclusions depending on pricing:

  • If p_1 > p_2 , purchase only good $x_2$.

  • If p_1 < p_2 , purchase only good $x_1$.

  • If $p_1 = p_2$, all bundles in the budget are equally acceptable.

Non-Convex Preferences: Corner Solutions

• Analysis does not yield clear preferences due to non-convexities leading to multiple potential bundles.

Perfect Complements Case: Kinky Solutions

• When considering perfect complements represented by utility functions such as:

  • U(x_1,x_2) = ext{min}igra x_1, ax_2igra , which leads to kink solutions.

• Kink solutions refer to scenarios where the utility is defined up to certain limits of the consumed goods.

Food Stamp Program Impact

• Analysis of food stamp issuance in terms of optimal bundle adjustment in consumer choices.

• Two distinct preferences lead to different evaluations between food stamps and cash transfers in varying levels of need:

  • Strong preferences for food lead to increased consumption of food due to optimal bundle adjustments benefiting from stamps.

  • Normal preferences might leverage a cash transfer for better utility.

Key Points to Remember from the Lecture

1. Optimal choice and conditions for optimization

2. Optimal bundle with Cobb-Douglas: well-behaved interior solutions

3. Optimal bundle with perfect substitutes: corner solutions

4. Optimal bundle with concave preferences: corner (boundary) solutions

5. Optimal bundle with perfect complements: kinky solutions