utility maximization problem

Question 1

how we make sure that the utility maximization problem is well-behaved?

Answer 1

To ensure the problem is well behaved (= has a unique solution), we will assume that preferences are:

1. Rational, meaning the consumer makes consistent choices
2. Continuous, meaning small changes in options lead to small changes in preferences
3. Strictly monotonic (or locally nonsatiated), meaning more is always better
4. Strictly convex, meaning the consumer prefers balanced combinations of goods rather than extremes

Question 2

what does the fact that the problem is well-behaved imply?

Answer 2

• The consumer having a utility function that is continuous, always increasing, and strictly quasi-concave
• The consumer spends all their wealth

Question 3

what do we assume about prices in this market economy?

Answer 3

We assume that prices are:
- Strictly positive, meaning all prices are greater than zero
- Taken as fixed by the consumer, meaning the consumer takes the prices as they are and can’t change them.

Question 4

what is the budget set?

Answer 4

• The consumer has a fixed amount of wealth , which is also strictly positive, and they can’t spend more than this amount.
• The set of possible bundles the consumer can afford is described by the budget set, this means the consumer can only choose bundles of goods that cost less than or equal to their wealth)

Question 5

what is the UMP?

Answer 5

The consumer’s problem of choosing his most preferred bundle, given prices p≫0 and wealth level w>0.
max u(x) (x≥0)
s.t. p*x≤w
In this utility maximization problem, the consumer aims to choose a consumption bundle, in their budget set with the goal to achieve the highest possible utility level.

Question 6

if p ≫0, how can you describe the budget set?

Answer 6

The budget set is closed and bounded.

• Bounded: For any good l=1,…,L we have x≤w/p this means that the quantity of each good cannot exceed the maximum amount that the consumer can afford given their wealth and the prices
• Closed: The budget set includes all the bundles that meet the budget constraint, which makes it closed in the commodity space

Compactness: since is bounded and closed, it is a compact subset of R^L

Question 7

Do the perfect choice for the consumer actually exists given the budget constraint?

Answer 7

According to the Weierstrass theorem, a continuous function defined on a compact set reaches its maximum value. Therefore, we can conclude that there exists a maximum of u(*) over the budget set B_{p,w}

(because the utility function is continuous and the budget set is compact, the Weierstrass theorem guarantees that there is a consumption bundle X within the budget set that maximizes the consumer’s utility. In other words, there is a best possible consumption choice for the consumer)

We also know that since the budget set is compact and the objective function is strictly quasi-concave, the solution is unique (strict quasi-concavity ensures that there are no flat areas at the top of the utility function. The function only has one peak)

Question 8

what is the MRS?

Answer 8

Rate at which the consumer is willing to give up one good in exchange for another good, keeping utility constant. (= Marginal Rate of Substitution)

At an interior optimum, the absolute value of the MRS between any two goods must equal the ratio of the goods’ prices.

MRS=-MU1/MU2

Question 9

what is the interior optimum?

Answer 9

This is a point where you’re getting the best possible mix of goods to maximize satisfaction, given your budget.

Question 10

what does the utility function u(x1,x2) tell us?

Answer 10

The utility function describes satisfaction (utility) from consuming quantities of good 1 and good 2 .

Question 11

what do the partial derivatives tell us?

Answer 11

for example the partial derivative of L with respect to x1 represents how utility changes when you change the amount of good 1 while holding good 2 constant. MU1 (marginal utility)

At the optimum, the marginal utility is proportional to the price for all goods.

Question 12

what is the total differential?

Answer 12

shows how a small change in x1 and x2 affects the total utility du.
The formula is:
MU_{1}dx_{1}+MU_{2}dx_{2}

Question 13

when do we have du=0?

Answer 13

An indifference curve represents all combinations of x1 and x2 that provide the same level of utility, meaning du=0 along the curve

du= total utility change represents a small change in total utility when you change the amounts of the two goods. So:
- if du is positive, utility increases
- if du is negative, utility decreases
- if du=0, there is no change in utility → MU_{1}dx_{1}+MU_{2}dx_{2}=0 →. MU_{1}dx_{1}=-MU_{2}dx_{2}

Question 14

What are the optimally conditions and how do we recognize the optimal point?

Answer 14

In the two-good case, the optimality conditions require:
1. The slope of the indifference curve at the optimal point x* must match the slope of the budget line.
2. The optimal point x* should lie on the budget line itself, not inside it.

The indifference curve must be tangent to the budget line at an interior optimum.
- x* is at the point of tangency
- preferences are strictly convex → we got only one optimal point

Question 15

what x* actually represents?

Answer 15

Because the utility-maximizing point x* will be unique for a given set of prices and wealth, we can think of the solution to the UMP as a function that maps the set of prices and wealth to the set of quantities.

We call x*=(p,w) walrasian (marshallian) demand function

Question 16

what is the ordinary demand function?

Answer 16

We call x*=(p,w) walrasian (marshallian) demand function

The ordinary demand function shows how the quantity demanded of a good changes when only its own price changes, while keeping wealth and the prices of other goods fixed.

Question 17

what are the properties of the demand function?

Answer 17

1. walras’ law
2. homogeneity of degree zero in (p,w)

Question 18

what does walras’ law states?

Answer 18

p*x=w
this comes from local nonsatiation
- If a consumer is choosing the most preferred bundle of goods (the one they like the most), the only way for them to be completely satisfied is if their total spending (based on prices) exactly matches their wealth. Otherwise, they would want to spend more or less to reach a better bundle.
- This is the condition that keeps them from wanting more or less of anything because they are already spending exactly what they can afford.

Question 19

what does homogeneity of degree zero in (p,w)?

Answer 19

• Homogeneity of Degree Zero means that if we change both the prices and the wealth by the same factor , the demand for goods stays the same.
• The budget set represents all the combinations of goods that the consumer can afford given their wealth and prices. If you multiply both the prices and wealth by the same factor the budget set doesn’t change: for example, if the price of everything doubles but the consumer’s wealth also doubles, they can still buy the same quantity of goods as before. The budget set remains the same.

So, since neither the budget constraint nor the optimality condition are changed, the optimal solution must not change either.

Question 20

In which cases the tangency conditions not hold?

Answer 20

1. When the MRS is constant: If the indifference curves are straight lines, the MRS (marginal rate of substitution) between the two goods is constant. This happens, for example, with perfect substitutes (where the consumer is willing to swap one good for the other at a constant rate). In such a case, you can’t really talk about “tangency” because the curves don’t bend—they’re straight, and there’s no clear point of tangency to the budget line.
2. When either the indifference curves or the budget constraint are non-differentiable: If the indifference curves or the budget constraint have kinks (sharp points where the slope suddenly changes), they are non-differentiable. For example, with perfect complements (like right shoes and left shoes), the consumer will only be happy with a 1:1 ratio of the two goods. The indifference curves for perfect complements are L-shaped, not smooth, and do not have a well-defined slope. This makes the tangency condition (which assumes smooth curves) break down.
3. When the individual prefers to consume zero of one of the goods (corner solutions): If the consumer ends up choosing to consume zero of one of the goods (because the price is too high or the utility is too low), this is called a corner solution. In this case, the optimal consumption point does not occur at the tangency between the indifference curve and the budget line, but rather at one of the corners of the budget set.

Question 21

what is the indirect utility function?

Answer 21

v(p,w)=u(x*(p,w))
hows the highest level of happiness (or utility) the consumer can achieve, based on their wealth and the prices of goods.

It’s called “indirect” because it depends on prices and wealth , rather than directly depending on the goods you consume.
Normally, utility comes from the specific goods you choose to consume.

Question 22

what happens to the indirect utility function if the direct utility function is continuous and strictly monotonic?

Answer 22

If u(*) is a continuous utility function that represents preferences where “more is always better” (strictly monotonic), then the indirect utility function v(p,w) has these properties:

1. Homogeneous of degree zero: If you scale both prices and wealth by the same amount, v(p,w) stays the same
2. Strictly increasing in w: if you have more wealth , your utility v(p,w) always increases, This happens because of local non-satiation, which means the consumer always prefers “a little more” of something if they can afford it.
3. Non-increasing in any price p: If the price of a good goes up, your utility will either stay the same or go down, if you’re consuming that good , (x>0) your utility will strictly decrease as p rises
4. Quasi-convex in (p,w): If you draw a “level curve” for v(p,w), the curve will form a convex shape. This means there’s a clear “best combination” of prices and wealth for a given utility level.
5. Continuous in p and w: Small changes in prices or wealth will only lead to small changes in v(p,w), no sudden jumps or breaks

Question 23

what is roy’s identity?

Answer 23

Roy’s Identity lets us find demand functions using the indirect utility function.
It shows how to calculate the demand for a good by taking partial derivatives of the indirect utility function:
x=-(1 deriv with resp to p/1 deriv with resp to w)

The negative sign, added afterwards, accounts for the fact that higher prices typically reduce demand.