All Questions
Tagged with optimization utility
63 questions
2
votes
1
answer
58
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Return to scale of a production function, $q = L^\lambda + K^\gamma$, is determining it possible in that general form?
Given the production function $q = L^\lambda + K^\gamma$, how do we determine the return to scale for different value of $\lambda$ and $\gamma$?
I know we have to determine the homogeneous degree of ...
- 41
1
vote
1
answer
37
views
Under what conditions would a quasilinear utility function in a function form exhibit diminishing marginal rate of substitution?
Let the utility function be: $U(x_1,x_2) = x_1 + x_2^\alpha$.
Diminishing MRS requires $\frac{dMRS}{dx_1} <0$, however, taking this derivative results in 0, as $MRS = \frac{1}{\alpha x_2^{\alpha -1}...
- 11
2
votes
2
answers
59
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Maximise Arbitrary Utility Subject to Budget Constraint
Suppose we have a general maximization problem of the form:
$$\max_{q_1,q_2} U(q_1,q_2) \text{ subject to } p_1 q_1 + p_2 q_2 = y$$
Suppose I allow $U$ to be concave, increasing and invertible. What ...
- 259
0
votes
1
answer
60
views
Why is incidence not included in social welfare maximization?
I am very confused on why incidence is not included in social welfare maximization of one good. Typically, I see the optimization over price done something like this:
$C$ ~ production cost function
$...
2
votes
2
answers
120
views
Do standard consumer theory axioms rule out corner solutions?
By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.
If these axioms are not sufficient ...
4
votes
2
answers
184
views
Intuition of sign used for Lagrange multiplier and corresponding constraint function in constrained optimization
It seems that in many applications there may be some economic interpretation for the Lagrange multiplier and thus it might be beneficial to ensure it's value takes on a specific sign.
If the above is ...
- 53
3
votes
1
answer
229
views
Intertemporal Utility Optimization For Multiple Goods
I'm building an economic simulation game and I'm trying to solve for the values that a person will spend on each good and the amount they will save in the current period, taking into account all ...
- 33
5
votes
1
answer
228
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Minimisation problem turned into Maximisation
My course always converts minimisation problems into maximisation. They give the following reason as outlined in the problem below.
$Min\; P_xx + P_yy \; s.t. \; u(x,y) \le x^{\frac{1}{2}} + y$
&...
- 1,011
2
votes
1
answer
369
views
Arguments for Concavity or Quasi-concavity
I'm faced with questions that want me to show that a utility or production function is either concave, or if not then quasi-concave so that we can apply the KKT conditions.
For example the production ...
- 1,011
1
vote
2
answers
162
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Perfect Complement Utility Function Maximisation
When we have the function $U(x_1,x_2) = min\{x_1,3x_2\}$
S.t. $p_1x_1 + p_2x_2 = m$
What's the economic, and mathematical intuition for assuming this constraint is binding, i.e. not having to ...
- 1,011
1
vote
0
answers
146
views
Is it possible to get back the consumer’s utility function from their demand functions?
I am curious about if it’s possible to reverse the utility maximization process, i.e. given the consumer’s Marshallian demand functions, find their utility function.
I was thinking of trying to find ...
- 2,351
3
votes
2
answers
201
views
Utility maximization for a household consisting of a woman and a man, with gender discrimination
Consider a household consisting of a woman and a man, with preferences over leisure and consumption given by:
$U(\overrightarrow{c},\overrightarrow{l}) = \ln{c} + \ln{l^F} + \ln{l^M}$
where $\...
- 2,351
3
votes
1
answer
276
views
Kuhn-Tucker(KT) conditons EMP
How should I formally solve the expenditure min.problem (EMP) by using KT conditions?
Since I should follow the notation of the Mas-Colell, I should write:
$\min~$ $p \cdot x$ , s.t. $u(x) \ge u$
...
- 319
0
votes
1
answer
218
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Quasiconvex Constraints in Maximisation
Why do we have to have Quasi-convex Constraints for constrained maximisation? I think i'm missing something pretty simple as this feels like a basic question:
My current Logic: If both the objective ...
- 1,011
2
votes
1
answer
256
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Quasi-convex constraints using monotonic functions
I believe i have a major misunderstanding surrounding quasi-convex constraints in maximisation, when using monotone functions. Can you help me spot my errors please?
The definition of a quasi-convex ...
- 1,011
1
vote
1
answer
53
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Are other 'variables' in demand functions always fixed?
My question is whether our demand functions e.g. Hicksian (compensated) demand, are ever functions of 3 or more variables, or if the other price variables and utility are always fixed, and hence just ...
- 1,011
1
vote
1
answer
195
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Compensating Variation - Interpreting the formulae
Assume $U(x,y) = x^{1/2}y^{1/2}$ s.t. $P_xx + P_yy = m$
And a price increase from $P_x$ to $P'_x$:
$U_0 = \frac{M}{2(P_xP_y)^{1/2}}$
Compensation variation formulae is: $\frac{M + ∆M}{2(P_x'P_y)^{1/2}...
- 1,011
0
votes
2
answers
857
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Decoding Endogenous vs Exogenous - Parameter vs Decision Variable - and Independent vs Dependent
this is a topic that i feel is very implicit in a lot of economics, but is some times brushed over in interest of getting strait to the model or the maths. But often i realise i don't actually know ...
- 1,011
0
votes
1
answer
142
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Ruling out corner solution in portfolio maximization problem
I am new to the econometric world.
I have a portfolio maximization problem
$$
\max \sum_{i}^ n a_{i} x_{i} \quad \text{s.t.} \quad \sum_{i}^n a_{i}=1, a_{i} \geq 0.
$$
I solved the problem but I had a ...
4
votes
1
answer
164
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Calculating the Compensating Variation with $M^2$
We can calculate the compensating variation (CV), which (to my understanding) is the amount of money we would need to give back to a consumer to keep them at the same level of Utility after a price ...
- 1,011
1
vote
1
answer
33
views
Utility maximization across yield curves?
I'm attempting to solve a utility maximization problem for return-on-investment (ROI) across two different products, where each product experiences a different linear ROI curve.
For product one, the ...
- 11
1
vote
0
answers
61
views
Find the utility of each agent whenever the social welfare is maximized
Question:
Suppose that the utility possibilities curve of 2 people economy is given by the equation $u_1^2 + Au_2^2=20$ where $A\in R_+$ and the social welfare function of the economy is $W(u_1,u_2)=...
- 192
4
votes
1
answer
692
views
Concave utility functions solution example
In the following post an example is given of the corner solution for a concave utility function. I tried solving it but got stuck. I have no idea how these types of problems are solved so if you could ...
- 43
4
votes
1
answer
62
views
What assumptions can be made to ensure convexity in this optimization problem?
This question is a continuation of the question I asked at:
How can I show convexity of this value function?
Where I came to the conclusion that more assumptions are required to show that the ...
- 183
4
votes
1
answer
389
views
How can I show convexity of this value function?
I have set up an optmization problem as follows:
$$V(A)=\max_{l, C} \quad u(C,l)$$
Where the only constraint is as follows:
$$C=f(l,A)$$
Here $u$ is the utility function which captures social welfare. ...
- 183
3
votes
2
answers
210
views
How can I formulate the following optimization problem?
I want to set up an optmiization problem for global warming in which a planner determines how much carbon dioxide gas is emitted. Let's say we reduce this problem down to two periods, then I ...
user40867
2
votes
0
answers
528
views
How to derive utility function from indirect utility and Marshallian demand?
c is composite good with normalised price, q is good with price p. y is income.
I have this indirect utility function:
$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)}$$
And ...
- 244
0
votes
1
answer
128
views
How do I get to this demand function in the monocentric city model?
I need to get this resulting price and quantity (housing):
It's pretty clear that the denominator of the quantity function is just the price function.
From this utility function:
And this constraint:...
- 244
1
vote
1
answer
159
views
Visualizing the expenditure minimization problem
I can easily visualize the utility maximization problem
ie. $$v(\mathbf{p},m^{*})= \max_{\mathbf{x}} \ u(\mathbf{x}) \ \ s.t \ \ \mathbf{px}\leq m$$
Since it is pretty easy to graph the indifference ...
- 103
5
votes
2
answers
338
views
Arrow-Debreu Theorem of Existence: Non satiation
Let $n$ be the number of consumers and $m$ be the number of commodities.
The Arrow-Debreu theorem requires closed and convex consumption sets $X_i \subset \mathbb{R}^m$ for all buyers $i \in [n]$. ...
- 105
2
votes
1
answer
594
views
Lagrangian multiplier and optimal bundle
I would like to know where I am wrong (if I am) and why I am wrong here please:
If a consumer has an income of 600 euros to spend for good x (Px = 10 euros) and good y (Py = 5 euros).
What is the ...
- 123
1
vote
1
answer
353
views
Find Pareto optimal allocations and the core for the following economies
Find Pareto optimal allocations and the core for the following economies.
There are two consumers and two goods. Utility functions are $u_1(x_1,y_1)= 10x_1-(y_1-2)^2$ and $u_2(x_2,y_2) = 10y_2 − (x_2 −...
- 192
1
vote
1
answer
237
views
Essential goods: How does one restrict the utility function?
I understand that solutions on boundary of the set under consideration when doing constrained optimization are often problematical. Usually it is said that we assume that goods are essential to insure ...
- 165
3
votes
2
answers
679
views
setting of Lagrangian function
Consider a simple consumer's problem:
Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda ( \...
- 245
5
votes
1
answer
2k
views
Perfect substitutes and Lagrange
How does one solve utility maximization of perfect substitutes using Lagrangian function?
Consider the problem
$$\max_{x,y} ax +by $$
subject to the constraint that
$$px + qy \leq I$$
where $a,b,p,q,...
- 3,840
2
votes
1
answer
71
views
When the global optimal is outside of the constraint set, what will be the demand?
$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex.
$a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that ...
- 2,084
0
votes
1
answer
895
views
How to find the Utility Possibility Frontier when there are Perfect Substitutes?
I am trying to derive the Utility Possibility Frontier (UPF) when both utility functions display perfect substitutes (in an Edgeworth economy with to consumers and two goods).
The specific problem:
$...
- 3
0
votes
1
answer
320
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Utility Theory/Marginal Rate of Substitution: Can the marginal rate of substitution be calculated for a point of the budget line?
This a person's budget line with various points, and their consumption, C*, and their endowment e, which is worth $5000 (unimportant). Also shows is their initial indifference curve. The difference ...
4
votes
1
answer
1k
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Concavity of Cobb-Douglass Utility Function on Non-Open set
My textbook argues that the Cobb-Douglass utility function $u=(x1)^a(x2)^b$ with $a,b>0$ and $a+b<1$ is concave on $R2+$ by computing the Hessian and showing it to be negative semidefinite for ...
- 61
1
vote
0
answers
103
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On demand functions and Engel curves
A consumer has utility function $U(x,y)=(x−2)y$, where $x≥2$ and $y≥0$. The price of $x$ is $P_x$, the price of $y$ is $P_y$ and the consumer's income is $I>2P_x$. ($x$ and $y$ do not have to be ...
- 161
0
votes
1
answer
2k
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Budget Constraint in Utility Maximisation Problem with Lagrange Multipliers
Lets say we have a utility function $U: \mathbb{R}^{2} \to \mathbb{R}$ given by $U(x,y)$ and a binding budget constraint $p_{x} x + p_{y} y = m$, where $p_{x}, p_{y}$ are prices of goods $x,y$ and $m$ ...
- 131
1
vote
0
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189
views
derive value function from utility function
We have the utility function.
$$U_{t} = \ln{c_{t}} + E_{t}\sum_{s=1}^{\infty}(\beta^{s}\ln{c_{t+s}})$$
And I am trying to find the value function.
$U$ is utility function. $c_t$ is consumption at ...
- 11
1
vote
1
answer
65
views
Is anyone familiar with the following basic resource sharing model?
Here is a resource sharing model, I do not remember where I came across it, I am wondering if this is well known in econometrics.
Let $T > 0$ be the total quantity of resources. For example, ad ...
- 111
0
votes
1
answer
62
views
Why do we have to normalize the income of consumers when working with an Edgeworth Box in a simple trade model with Pareto optima?
I was studying microeconomics and I confess I am not the brightest person for maths and sorry if this is very dumb but I get that we CAN normalize the income and I get where it comes from and how it ...
- 1
1
vote
2
answers
551
views
Any interior solution for $u(x,y) = min\left \{ x,y \right \}^{2} + max\left \{ x,y \right \}$?
Will all the solutions be in the corner or will the cusp in the middle give us any interior solution? This is by the intersection of the budget line.
I am getting this type of a shape:
But I am not ...
- 285
3
votes
3
answers
1k
views
A question about Lagrange multiplier(when $\lambda=0$)
I need help in a maximization problem(finding the optimal investment portfolio).
where $R_s$ and $\Phi$ are $n$ by $1$, with other variables being scalars.
$C^s$ is consumption (or wealth) of an ...
- 131
3
votes
0
answers
55
views
Finding savings in an Overlapping Generations model
I have not seen this question asked anywhere, so I'm posing it here in case anybody else (hopefully) can help me get to the answer. In a nutshell, my question is: how do we arrive at the saving ...
- 31
1
vote
0
answers
375
views
Natural borrowing/debt limit and other borrowing constraints
When confronted with the simple household consumption maximization problem under uncertainty (and with Arrow security sequential trading)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
- 121
-4
votes
1
answer
756
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Question about budget constraint and utility maximization [closed]
I have also following budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where y is income.
Assume that there are two stories. The agent can shop in both of them. The first store ...
- 192
2
votes
2
answers
3k
views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
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