expenditure minimization problem Flashcards
what is the goal to the EMP?
The EMP ,Expenditure Maximization Problem, provides another way of studying the consumer’s optimal choices, it answer the question: “What’s the smallest amount of wealth needed to reach a specific utility level u?”
min p*x x≥0
s.t. u(x)=u
It computes the minimum level of wealth required to reach a given utility level.
is about minimizing spending to reach a certain level of utility (happiness).
what are iso-expenditure lines?
- Iso-expenditure lines represent all the bundles (x1,x2) of goods that cost the same amount at given prices (p1,p2)
- Each line is described by the equation:
e=p1x1+p2x2 - Here, e is the total expenditure (money spent)
- All the lines have the same slope (-p1/p2)
- Lines farther from the origin represent bundles that cost more. e4>e3>e2>e
Iso-expenditure lines are like “budget lines” that show all the ways you can spend a fixed amount of money. Lines farther out mean higher spending.
what does the indifference curve tell us?
Indifference curve u: this curve shows all the combinations of goods (x1,x2) that give the consumer the same level of happiness u
what is the correct iso-expenditure line to pick?
- To minimize spending, we look for the lowest iso-expenditure line that just touches the indifference curve u
- This means we are looking for the point where the consumer spends the least amount of money while still reaching their desired utility level.
what is the solution to this problem?
The solution to the problem, the consumption bundle x^h, is known as the Hicksian (or compensated) demand function.
The Hicksian demand function gives the cheapest bundle of goods x^h that allows the consumer to reach a specific level of happiness (utility u) , given prices p.
x^h changes when prices p or the required utility level u change
How does the Hicksian demand function work?
We decide on a specific level of happiness that the consumer must maintain
No matter how prices change, we ensure the consumer’s utility stays the same
We basically adjust wealth as needed:
- If prices go up, we compensate the consumer by giving them more money
- If prices go down, the consumer needs less money to maintain the same happiness.
This is why it is also called “compensated demand”
There isn’t just one Hicksian demand curve. There’s a different one for every utility level, each shaped by the consumer’s preferences.
How would it be the Hicksian demand for good 1?
To minimize spending, the consumer chooses the optimal quantity of good 1, which is written as:
x₁^h(p₁⁰ , p₂⁰ , u)
It can shows how much of good 1 the consumer would buy at different prices of good 1, while keeping:
- the price of good 2 fixed
- the utility level constant
what does the expenditure function tell us?
The minimum expenditure necessary to reach utility u at prices p, denoted by e(p,u), is the expenditure function.
For all p≫0 the
expenditure function is defined by
e(p,u) = p · x^h (p,u).
the expenditure function shows us the minimum income required to reach a certain utility level.
What is the hicksian demand function suppose to solve analytically?
min p・x st u(x)≥u
x≤0
lagrangian method
if u(*) is well-behaved, the constraint will bind, meaning the consumer will be on the indifference curve corresponding to utility level u.
what is an interior optimum?
where the consumer buys positive amounts of all goods
What is the condition satisfied at an optimal point?
At an optimal choice (interior solution), the consumer chooses goods in such a way that their willingness to trade goods(MRS) matches the market trade-off (price ratio). This condition is the same as the tangency condition we found earlier when solving the utility maximization problem (UMP).
main differences between Indirect Utility Function and Expenditure Function
Indirect Utility Function:
This gives the highest level of utility the consumer can achieve, given their prices and wealth.
v(p,w)=u(x(p,w))
This is the utility the consumer achieves when they choose the optimal bundle x(p,w) from their budget (prices and wealth).
Expenditure Function:
This shows the minimum expenditure needed to achieve a specific utility level u, given prices p
e(p,u)=p・x^h(p,u)
It tells you how much money is required to reach a specific utility at prices .
mail differences between UMP and EMP
Utility Maximization Problem (UMP):
- The consumer’s goal is to maximize utility given their budget (wealth) and the prices of goods.
- The optimal consumption bundle is found at the tangency point between the budget line and the highest indifference curve.
- The utility achieved at this optimal bundle is the indirect utility function v(p,w)
Expenditure Minimization Problem (EMP):
- The consumer is given a target utility level and needs to find the minimum expenditure required to achieve that utility at given prices.
- The optimal bundle in the EMP is the one that achieves the target utility at the lowest cost, found at the tangency point between the target utility’s indifference curve and the lowest iso-expenditure line.
- The expenditure function e(p,u) shows the minimum expenditure required to reach utility level u
When do the UMP and the EMP lead to the same consumption bundle?
When the target utility in the EMP is set to the utility level achieved in the UMP (u=v(p,w), both problems yield the same optimal consumption bundle x*(p,w).
In short, the two problems are like two sides of the same coin: solving one gives you the answer to the other. Whether you’re working within a budget or trying to hit a happiness goal as cheaply as possible, the solution (the combination of goods) remains the same
What properties follows the Hicksian demand?
Suppose that u() is a continuous utility function representing locally nonsatiated preferences. Then, for any p≫0, the Hicksian demand function has the following properties:
1. Homogeneity of degree zero in p: If you scale all prices by the same amount (e.g., double them), the best choice of goods doesn’t change.
2. No excess utility :
The chosen bundle of goods always gives exactly the level of happiness (utility) you’re aiming for, no more, no less.
If the bundle gave more happiness, you could spend less and still hit the target, which means the original bundle wasn’t truly cost-minimizing.
This ensures the “cost-minimizing” condition is met: every penny spent serves its purpose to reach the desired happiness level.
3. Convexity/uniqueness: if preferences are convex, then x^h(p,u) is a convex set; if preferences are strictly convex (u() is strictly quasi-concave), then x^h(p,u) is single-valued.
What properties woul follows the espenditure function if u(*) is continuous and strictly increasing?
- Homogeneous of degree one in p
- Strictly increasing in u , non-decreasing in p_l for any l
- Concave in p
- Continuous in p and u
what is Shephard’s Lemma?
There’s a connection between the expenditure function e(p,u) and the Hicksian demand x^h(p,u).
This connection is called Shephard’s Lemma.
What Shephard’s Lemma Says:
If u( *) is continuous and strictly quasi-concave, then:
The Hicksian demand for a good l is the same as the derivative of the expenditure function with respect to that good’s price pl
