All Questions
Tagged with optimization microeconomics
98 questions
0
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0
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71
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Comparing amount of investment under two situations
I am studying the following problem -
Investor has $1 and he has to split it between two assets. These are risk-free assets.
Under scenario 1 - These are normal times so he don't have to worry about ...
- 499
2
votes
1
answer
115
views
Find the Pareto Efficient set for 3 Leontiefs
I'm struggling with the following General Equilibrium exercise:
Find the Pareto Efficient set for this Pure Exchange Economy;
The consumers are $i = 1,2,3$ with these Leontief utilities:
$u_i(x_{1i},...
- 2,351
0
votes
1
answer
45
views
How to solve for demand?
Hi I am fairly new to these kinds of optimization problems and I am not clear how from equation (1) the authors derive the demand function for a drug j.
I mean I think they used the Lagrangian and as ...
- 143
1
vote
1
answer
63
views
Cost minimisation for the production function $f(L,K) = L^\lambda + K^\gamma$
For the function: $f(L,K) = L^\lambda + K^\gamma $. The value for $\lambda$ and $\gamma$ is not given.
What type of production function is this (quasi-linear, CES)?
Is it true for there to be ...
- 41
1
vote
0
answers
73
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Convexity of indirect utility in probabilities
I am interested in the concavity in $p$ of the indirect utility function
$$V(p,W)=max_{x,y,z} pf_1(x,y)+(1-p)f_2(x,z)$$
under the constraint
$$x+py+(1-p)z=W$$
where $0<p<1$ and where $f_1,f_2$ ...
- 11
0
votes
0
answers
17
views
Breakeven analysis for computer upgrade decision making
I need to perform a break-even analysis of moving from one system design to another.
Definitions
$M_0, M_1$ = one-time initial manufacturing cost of currently deployed system design and new system ...
- 1
0
votes
1
answer
100
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How to find marshallian demand for Leontief Utility with 3 goods. u(x 1 ,x 2 ,x 3 )=min{2x1 + x3, x2/2}
I have a utility function
$$u(x_1, x_2, x_3) = \min \{2x_1 + x_3, x_2/2\} $$
I would have assumed that the relationship established is $2x_1 + x_3 = x_2/2$ but my solution manual has it as $$x_1 + ...
- 1
1
vote
0
answers
52
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Help with Deriving Hicksian Demand in the Monocentric City Model?
I have a fairly standard Alonso-Muth-Mills model, but struggling to derive the Hicksian demand.
Starting with the basic utility function:
And this Budget Constraint:
Housing Floor-space is ...
- 11
2
votes
2
answers
120
views
Do standard consumer theory axioms rule out corner solutions?
By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.
If these axioms are not sufficient ...
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0
answers
33
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Derivation of Euler Equation in presence of the Dixit-Stiglitz Aggregator
I reading the working paper of Sebastian Banz (2012). I have an issue with the derivation of the Euler equation.
The author models the demand side of the economy as follows
The representative consumer ...
- 430
2
votes
2
answers
155
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Proving quasi-concavity for a utility function
I have a utility function, and I want to prove that it is a quasi-concave function:
$$ u(x_1,x_2)= 2x_1x_2+x_1+2x_2 $$
I do this by showing that the set of points where the utility is larger than or ...
- 65
2
votes
1
answer
99
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Using lagrange on a quasi-concave utility function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I have maximized his utility function, and found its demand functions, for $x_1$ and $x_2$, using Lagrange. However, is it ...
- 65
2
votes
1
answer
352
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Solving utility maximization, and finding demand function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I want to maximize his utility function.
$$max: 2x_1x_2+x_1+2x_2. uc:p_1x_1+p_2x_2=y_A$$
Using Lagrange, I get
$$L(x_1,...
- 65
1
vote
2
answers
76
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Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)
Could you please help me evaluate this logic?
I've been told that "if preferences are strongly monotonic, duality holds."
In the case of utility u(x,y) = x^2 + y^2, we will get a corner ...
- 53
2
votes
1
answer
77
views
Suppose $A$ is a $2x2$ matrix and ${\bf x}=(x_1, x_2)$. What does "$f(Ax)$ is supermodular" mean?
Suppose $A$ is a $2x2$ matrix, e.g., $A=\begin{vmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{vmatrix}$, and ${\bf x}=(x_1, x_2)$. Suppose $f()$ is continuous and twice differentiable.
...
- 23
1
vote
1
answer
112
views
Cost function from a weighted CES production function
I want to find the cost function given the CES production function:
$$
Y = F(x_1,x_2) = (\lambda x_1^ \rho+(1-\lambda)x_2^\rho)^\frac{1}{\rho}
$$
with $0<\rho<1$.
So far I have set up the ...
- 11
0
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1
answer
188
views
Assumption of interior solution in the Lagrangian method
Why do we need to assume an interior solution before using Lagrangian method for utility maximization problems?
1
vote
1
answer
249
views
Conditions for an interior solution to the UMP
I was wondering under what set of conditions one is allowed to assume an interior solution to the Utility Maximisation Problem. In most of my classes and lecture notes, interior solutions are assumed ...
- 13
1
vote
1
answer
103
views
FOCs for profit maximization using a transformation function
I'm (still) reading the microeconomics textbook of Mas-Colell et al. On p. 135, the profit maximization problem (PMP) for producers is introduced; characterizing the technology as $Y = \{ y \in \...
- 335
0
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0
answers
189
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why is the MRS same for everyone?
If the consumers are optimizing and at interior solutions and facing the same prices, then the MRS=p1/p2 will be the same for everyone no matter the preferences and income. but why? I don't understand ...
- 1
2
votes
1
answer
100
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Help with a proof for an quite intuitive Utility optimization problem
Assume $U(x,y,a,c )= - c x + B(x,y,a)$, with $\frac{\partial B(x,y,a)}{\partial c }=0$, and with $a$ and $c\geq 0$ being parameters, and with $x$ and $y$ being variables. Further, $B(x,y,a)$ is ...
- 105
1
vote
0
answers
391
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Question About Non-Degenerated Constraint Qualification (NDCQ)
I am studying constrained optimization using Mathematics for Economists by Simon and Blume, and I have some difficulties understanding the Non-Degenerated Constraint Qualification (NDCQ). I would like ...
- 505
2
votes
2
answers
293
views
Question About Implicit Function Theorem and Comparative Statics - Mathematics for Economists by Simon and Blume Chapter 15 Exercise 32
I am studying Implicit Function Theorem and its application on comparative statics using Mathematics for Economists by Simon and Blume. Here is the question:
Consider a pure exchange economy with two ...
- 505
3
votes
0
answers
52
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Is there a labor vs leisure model with work experience?
I find the labor-leisure model with utility functions interesting, but I find it lacks the factor of work experience, which is very important in the real life labor market.
This is a reason people why ...
- 2,351
1
vote
2
answers
131
views
Question About Non-Discriminating Monopolist - Mathematics for Economists by Simon and Blume Chapter 17 Exercise 7
I am working on Mathematics for Economists by Simon and Blume Exercise 17.7. I know there is an Answers Pamphlet. However, the solution to this question does not make any sense to me. It seems that ...
- 505
1
vote
2
answers
363
views
How to derive the short run cost function
Given the production function $f(K, L)=\min\{3K,2L\}$, the procedure to find the long-run cost function would be to use the condition: $3K=2L=Y$ where $K=\frac{\overline{Y}}{3}$ and $L=\frac{\overline{...
- 45
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
1
vote
1
answer
112
views
Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
Encountered the following problem during self study:
My take on the problem is that if we can show that the equation of the income expansion path is $x_1=x_2$ for all such $U(x_1,x_2)$ then we have ...
- 667
3
votes
3
answers
637
views
The formula for expansion path
Is there a way how to precisely compute the expansion path?
I know a consumer's utility function $U(\boldsymbol{x})$, I know the budget constraint $\sum P_i x_i \leq M$, I am able to compute the ...
- 834
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
2
votes
2
answers
80
views
Why does $\frac{MU_x}{P_x}=\frac{MU_y}{P_y}$?
I just started learning economics and the textbook says $\frac{MU_X}{P_X}=\frac{MU_Y}{P_Y}$ for a buyer with a fixed budget to spend on two goods, $X$ and $Y$.
Let's say goods $X$ and $Y$ both cost $\\...
- 121
0
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1
answer
74
views
Looking for a term I'm pretty sure exists
Let me describe the situation:
Company is selling a product; they buy it at x, sell it at some % over for profit. Taken on a monthly scale, you can see the profit of that particular object by ...
1
vote
1
answer
93
views
How to find the e(p,u) of u(x) = x1 + x2 + x3
If I do the LaGrangian for the Expenditure minimization problem, it comes as p1 = p2 = p3, how do I substitute it back in the constraint and find the Hicksian demand to find e(p,u)?
- 63
1
vote
1
answer
325
views
how to derive marshallian demand functions from leontief preferences?
For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a ...
- 141
0
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1
answer
43
views
How is production managed with respect to the long run vs the short run?
Assuming perfect competition, I think that firms are price takers in the labor/capital markets as well (in the short and long run), correct?
And I know that the Long-run total cost curve is derived by ...
- 61
2
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0
answers
218
views
Find cost function for given production function
I have the following production function
$$f(x_1,x_2,x_3,x_4)=max\{\min\{x_1, x_2), x_3+2x_4\}\}\ge q$$
And I want to find the cost function.
What I think
(1) $P_1+P_2 <P_3$ and $P_3/P_4<1/2$
...
- 192
4
votes
1
answer
692
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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
1
vote
0
answers
32
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How to define the market and the clearance conditions of a general sectoral computable equilibrium model?
I am trying to implement a general multiproduct market (partial/sectoral) computable equilibrium model, where "general" refers to the fact that the relation of complementarity/...
- 307
1
vote
0
answers
74
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Solving this budget constraint tangent to indifference curve without a graphical approach
Question:
Let $V(x, y) = (1-\overline{p}) U(x) + \overline{p} U(y) - \overline{p} U(F)$ where $U$ is a strictly concave function ($U'>0$ and $U''<0$) with $U(0)=0$ and $0<\overline{p}<1$ ...
- 349
1
vote
0
answers
36
views
Approximating Optimal Choices and Income Inequality
It seems that at least the basic microeconomic theory assumes we are optimizing over quantities that can take continuous magnitudes, but in practice one can often only purchase goods in discrete ...
- 111
1
vote
2
answers
1k
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Conditions to use the Lagrangian method
I have seen that the prices and $\text{MU}_{i}$ are assumed to be positive (or, the preferences monotonic). This is always mentioned when a utility maximization problem is being solved with the ...
- 244
3
votes
1
answer
173
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Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?
This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts.
To preface suppose we have the ...
- 8,847
2
votes
1
answer
270
views
Two-Stage Utility Maximization Problem
Actually I don't know how to solve such utility maximization problem, only know using FOC and budget constraint to solve for demand. I will appreciate it if someone tell me the procedure facing such ...
- 23
0
votes
1
answer
128
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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
2
votes
1
answer
142
views
Comparative Statics: Income Effect
Much of this is setting up the problem. So if you're familiar it's likely best to start from the very bottom and work up if needed. The question is asking about the income and substitution effect.
...
- 264
1
vote
3
answers
4k
views
Graphing indifference curves to visualize solutions?
I am having trouble with being able to graph indifference curves. This is a particularly important skill to have especially when trying to visualize corner solutions, and when the Lagrangian method ...
- 155
6
votes
1
answer
206
views
GE with an intermediate good
intro
I'm looking at a simple model with 1 consumer, 2 goods and 2 firms.
I'm trying to get a price vector [p0, p1] that makes it work.
By makes it work, I mean, ...
user34331
1
vote
1
answer
138
views
Calculating optimal level of negative externality
I am trying to solve the following question(s):
Let $h \geq 0$ represent a negative externality of a firm's production on one (representative) consumer. The consumer has a quasi-linear utility ...
- 71
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