Budget Constraints qStart to Build a Theory of Consumer Choice ...

Microeconomics 

Dr. Dmitri M. Medvedovski 

Budget Constraints 

●Start to Build a Theory of Consumer Choice 

●Describe the Axioms of Human Behavior 

●Show how axioms map into a concept called The Utility Function 

Axioms of Behavior 

●Insatiability 

–More is better 

●Continuity 

●even just a little bit more is better 

●Rationality 

–Consumer can judge and choose between options 

●Transitivity 

–If A is better than B and B is better than C, the consumer must prefer A to C 

Footnote On Arrow 

● Arrow has shown that it is impossible for a collection of individuals to 

possess a representation that follows all of these axioms. 

●This is called the Impossibility Theorem 

●The most conspicuous culprit is transitivity 

The Representation Theorem 

●Given the axioms of behavior, there exists a utility function that represents 

the consumer’s tastes 

●This utility function is ordinal not cardinal 

●Like a thermometer, scale is irrelevant 

●Utility function is consistent to a monotone increasing transformation 

Review: 

. Consumer preferences 

. Indifference curves 

. Utility 

●Derive the Budget Constraint 

●Derive the Principles of Utility Maximization 

Page 1 out of 7 Lec. 5



Principles of Optimization – Technical foundation for most problems 

in economics 

1. Definition of the objective function 

- utility function for consumers 

- production function for producers 

- profits 

- cost function 

2. Want either maximize or minimize the objective function. Your 

objective function must be well specified and conceptually 

measurable. 

3. Choice variables – they are imbedded in the objective function. 

Example: you have a consumer who is self interested with utility 

function that has numerous arguments as U=U(X1, X2,….Xn), the Xi 

are different things that make you happy. Start with things 

consumables from which you derive pleasure. 

4. Objective function has a goal. 

5. Be as large as possible (maximize your utility function) 

6. Introduce constraint 

7. Constrained optimization – optimization problems do not have to be 

constrained. However, the most interesting optimization problems in 

economics are constrained. 

- objective function 

- set of constraints (n-1 is the maximum number of constraints) 

- the optimizer chooses the levels of the choice variables. 

Assume a consumer picks Xi to maximize Utility. 

1) People have unlimited desires (wants) – which is not economics if 

there is no constraint. 

2) Budget constraint (Income is limited) 

M=I=P1X1+P2X2+P3X3+….PnXn (price times quantity that you 

choose) or M=∑ 

n 

i= 

1 

P i 

X i 

3) Each consumer has idiosyncratic parameters such as: 

U=U(X1,X2…..Xn) is subject to constraint M=∑ 

dU\dx is the marginal utility of the i-th good. 

n 

i= 

1 

P i 

X i 




U(X1, X2) 

X2 

U1 

U2 

X1 

It is impossible for indifference curves to cross each other because that 

would violate the axiom of transitivity. Prove follows. 

A 

C 

B 

One way to solve it is to use Mr. Lagrange 

n 

1 

, X 

2, 

X 

3... 

X 

n 

) + ( M −∑ 

Pi 

X 

i 

) 

i= 

1 

L = U ( X 

λ 




The Consumer’s Problem 

●Attain the highest possible level of utility 

–Scale utility mountain 

●The available budget constitutes the constraint 

●Think of the budget as a fence on the mountain 

●Find the highest place on the fence 

The Budget Constraint 

●The consumer has a sum of money to spend each period, M 

This money is spread across many goods, X i 

●Thus the expenditure on the first good, X 1 

, is 

●And similarly for all other goods 

Graphical Construction of the Budget Constraint 

The Budget Line in Mathematical Terms 

●Imagine the consumer spends all income on good Y 

●The maximum amount of Y that can be purchased is M/(P Y 

) 

●If all income is spent on good X then the maximum quantity is M/(P X 

) 

●Join these two points to make a straight line 

●Any combination on the line exhausts the entire budget M 

The Slope of the Budget Constraint 

●The slope is the ratio of the two prices: derive: (M/Py)/(M/Px) 

●-P Y 

/P X 

The Consumer Choice 

●Attain highest possible utility 

●Don’t violate the budget constraint 

● 

Budget: I = P 1 X 1 + P 2 X 2 , where X 1 and X 2 are goods. 

X 2 

50 

I 

P 2 

I 

P 1 

= X 1 

+ P 2 

P 1 

X 2 

40 

30 

20 

Slope = -1 = − P 2 

P 1 

I 

X 1 

= I P 1 

− P 2 

P 1 

X 2 

10 

10 20 30 40 50 

X 1 

P 1 

constant 

slope 




Comparative Static 

X 2 

I 2 

X 1 

I 1 

X 2 

X 1 

What happens if I (income) and prices triple at the same time (inflation)? 

X 1 

= 3I 

3P 1 

− 3P 2 

3P 1 

• X 2 

nothing changes 

X 1 

= I P 1 

− P 2 

P 1 

• X 2 

Given U (X 1 , X 2 , X 3 ,… X U ) X i represents the 

goods you want with your goal being to 

maximize U within your budget constraint. 

I 1 

I 2 

I 3 

. I = money available, referred to as income. 

. Each good has a price; P i for good X i. 

N 

I = P 1 X 1 + P 2 X 2 +… ; E i = P i X i or ∑ xi p i 

=budget constraint 

i =1 

Lagrange (L) 

(L)=U(X 1 

, X 2 

, X 3 

… X u 

) − λ(I − P 1 

X 1 

− P 2 

X 2 

−…P i 

X i 

) 

. take derivatives with respect to each X i 




. they are partial derivatives because we do not let other variables ¨ 

Why can’t indifference curves slope upward? 

By axiom “more preferred to less” we disprove upward sloping in difference 

curves. 

Y 

U 0 = U(Y 0 ,X 0 ) 

U 1 = U(Y 0 ,2X 0 ) 

U 1 > U 0 & U 1 = U 0 

Y 0 

X 0 

2X 0 

X 

Y 

Y 0 

Why can we not, as consumers, choose point 

A for consumption with the given budget 

constraint? 

X 0 

X 

What combination exhausts the opportunity to improve? 

Y 

I 

P y 

*what axiom does this indifference 

curve violate? Under what 

circumstances, if any, 

is this indifference curve possible? 

I 

X 

P x 




Y 

(hat) 

What does this say about Y? Does more X 

make us happier? Does this graph have any 

conflict with one of the axiom? 

X 1 

0 

X 1 

I 

X 1 

II 

X 1 

III 

X 

(right shoes) 

Consumer choice: 

. must be located on the budget constraint. 

. Must give consumer the most preferred combinations of goods and 

services. 

Y 

MRS= P 1 

P 2 

means satisfaction ∆X 2 

∆X 1 

is maximized, given budget constraint P 1 

P 2 

What does this say about X? 

X

Budget Constraints qStart to Build a Theory of Consumer Choice ...

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