All Questions
Tagged with optimization microeconomics
98 questions
12
votes
2
answers
44k
views
Marshallian Demand for Cobb-Douglas
When trying maximize the utility having a cobb-douglas utility function $u=x_1^ax_2^b$, with $a+b = 1$, I found the following formulas (Wikipedia: Marshallian Demand):
$x_1 = \frac{am}{p_1}\\
x_2 = \...
- 245
8
votes
3
answers
2k
views
Does the Marshallian demand function always include prices and income?
I have the following utility function:
$$U(x_i)=x_1x_2+x_3$$
with budget constraint:
$$p_1x_1+p_2x_2+p_3x_3\leq I$$
I use the Kuhn-Tucker method to find the optimal choices of the Utility ...
- 1,044
8
votes
4
answers
898
views
Can $u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$ be solved by Kuhn–Tucker conditions?
Consider
$\max_{x_1, x_2, x_3, x_4} u(x) = \sqrt{x_1 x_2} + \sqrt{x_3 x_4}$
s.t. $\; p_1x_1 + p_2x_2 + p_3x_3 + p_4x_4 \le w$
I know we can solve the max problem through separately considering ...
- 101
8
votes
1
answer
98
views
Overlaping jurisdictions model, Alesina Spolaore: Proof of Lemma 1; The Size of Nations
I've been reading the book 'The Size of Nations' by Alberto Alesina and Enrico Spolaore (can be found on the net if you know where to look) and I'm having trouble following their "proof" of ...
- 81
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
5
votes
2
answers
4k
views
Transformation Function
In Mas-Colell microeconomics textbook I have found that profit maximization problem (as well as many further optimization tasks) could be represented with application of some transformation function (...
- 195
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
5
votes
1
answer
450
views
Estimating the second derivative of function from optimizers
Consider the following optimization
$$x^*(s) = \max_{x\in X} \big(\,f(x)-sx\,\big)$$
where $f$ is assumed to be a strictly concave function and $X$ is an interval constraint, e.g $X = [0,b]$. We do ...
- 265
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
1k
views
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
3
votes
2
answers
818
views
Solving a maximization problem by substitution when the constraint is in implicit form
I am trying to understand how the first order conditions for an interior solution of a maximization problem were derived using the substitution method.
The problem is:
$$\max\limits_{x\ge0,y\ge0}P(a-...
- 55
3
votes
3
answers
637
views
The formula for expansion path
Is there a way how to precisely compute the expansion path?
I know a consumer's utility function $U(\boldsymbol{x})$, I know the budget constraint $\sum P_i x_i \leq M$, I am able to compute the ...
- 834
3
votes
2
answers
680
views
setting of Lagrangian function
Consider a simple consumer's problem:
Max $u(X)$ s.t. $\sum_i^l p_i x_i\leq \sum_i^l p_i w_i$
$w$ is initial endowment.
We can set the Lagrangian function to solve this problem.
$L=u(X)+\lambda ( \...
- 245
3
votes
1
answer
173
views
Constrained Optimization with Multiple Constraints: Do multiple strictly positive multipliers imply a solution at a vertex?
This might be a bit of a silly question but I am interested in solving standard economic problems with many constraints and am wondering if there are any shortcuts.
To preface suppose we have the ...
- 8,847
3
votes
2
answers
201
views
Utility maximization for a household consisting of a woman and a man, with gender discrimination
Consider a household consisting of a woman and a man, with preferences over leisure and consumption given by:
$U(\overrightarrow{c},\overrightarrow{l}) = \ln{c} + \ln{l^F} + \ln{l^M}$
where $\...
- 2,351
3
votes
2
answers
439
views
Indirect changes in Marshallian Demand
Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$.
It can be shown that the Marshallian demand for $x$ ...
- 448
3
votes
1
answer
1k
views
Externalities - First order conditions
I am currently reading the book "Microeconomics: Principles and Analysis" by Cowell on my own. I'm reading the externalities chapter, and i found an interesting example:
There are just two firms: ...
- 103
3
votes
2
answers
2k
views
Why is instantaneous utility of current period discounted?
Consider a two period model of consumption.
I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the ...
- 31
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
3
votes
0
answers
52
views
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
3
votes
1
answer
631
views
Calculate optimal discount for product bundling
So recently I made some rules with my transaction data. Based on it I can determine which products are profitable to bundle it together.
But even though I know e.g. product A→ product B, are there ...
- 31
3
votes
0
answers
387
views
Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$ [duplicate]
Suppose preferences are represented by the following utility function
\begin{equation}
u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}
\end{equation}
Write the
...
- 189
3
votes
0
answers
141
views
Appropriate economic/econometric tools to analyze segmented promotion optimization problem
I'm trying to determine which micro-economic/econometrics concepts, models, and/or tools are appropriate for an analysis of promotions.
Below I
Describe the problem in general terms
Give ...
- 380
3
votes
1
answer
148
views
Can be the duality theorem applied to not locally non-satiated utility functions?
I have the following not locally non-satiated utility function:
$$U(x,y)=-(x-1)^2-(y-2)^2$$
where $U(x,y): \, \!R^n_+ \rightarrow \!R $
The 3D plot of this function is an infinite paraboloid; ...
- 1,044
2
votes
1
answer
270
views
Two-Stage Utility Maximization Problem
Actually I don't know how to solve such utility maximization problem, only know using FOC and budget constraint to solve for demand. I will appreciate it if someone tell me the procedure facing such ...
- 23
2
votes
2
answers
120
views
Do standard consumer theory axioms rule out corner solutions?
By standard consumer theory axioms I mean (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves.
If these axioms are not sufficient ...
2
votes
2
answers
3k
views
Utility maximization question setting up.
Consider a consumer whose preferences can be represented by the following utility function: $$u(x_1,x_2)=\dfrac{x_2}{(1+x_1)^2}.$$
Assume the agent's income is $y=5$. The price of one unit ...
- 192
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
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
2
votes
2
answers
456
views
In an intertemporal (2-period) consumption model, why is the investment rate independent of discount factor?
In lecture, my professor defined the following 2-period consumption model:
$c_i = $ consumption in period $i$.
$y =$ endowed income in period 1.
$r = $ interest rate in perfect credit markets.
$h = $ ...
- 21
2
votes
1
answer
402
views
CV, EV for additive utility; confirm or deny
I'm currently a TA for a class and recently graded a midterm. I gave the answer key back to the teacher, after going over part of the exam in a study hall. I was going to go over the rest of it ...
- 6,678
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
2
votes
1
answer
100
views
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
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
2
votes
1
answer
1k
views
Is this Cost function concave or convex?
Given the following cost function, where t is the quantity of some product.
$$C(t) = 1/3t^3 - 7t^2 +11t + 50$$
here is a graph between $t= 0$ and $t = 25$
We are asked if this function is convex or ...
- 135
2
votes
2
answers
3k
views
Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
- 63
2
votes
1
answer
612
views
Adding a non-binding constraint to the objective function
I am dealing with a constrained optimization problem found in Tirole's Theory of corporate finance. My question is not related to the details of this model, but just to provide some context, we are ...
- 55
2
votes
1
answer
99
views
Using lagrange on a quasi-concave utility function
A consumer has the following utility function
$$u(x_1,x_2)=2x_1x_2+x_1+2x_2$$
I have maximized his utility function, and found its demand functions, for $x_1$ and $x_2$, using Lagrange. However, is it ...
- 65
2
votes
1
answer
352
views
Solving utility maximization, and finding demand function
A consumer has the following utility function
$$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
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
2
votes
2
answers
2k
views
Intermediate macroeconomics: optimal bundle for quasilinear utility?
How would I go about solving this question:
Assuming consumer's utility function is $U(C,L)=c+2l^{0.5}$, consumer earns a wage of 0.5/hour, $h=24$ and there is no real dividend and tax is $T=11$. ...
- 133
2
votes
2
answers
155
views
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
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
2
votes
1
answer
56
views
Have I found the correct Emission Price
Let's say that there is a hotel owner $(H)$ and a woodworker $(W)$ working in close proximity to one another.
The woodworker produces $x$ units to sell at market at $p_{x}=6,5$. From the woodworking ...
- 147
2
votes
2
answers
1k
views
Editing formula for finding Marshallian Demand with Cobb-Douglas utility function
Suppose a utility function $u=x_1^ax_2^b$ with $a+b=1$. The following formula finds the values for $x$:
$x_1 = \frac{am}{p_1}\\
x_2 = \frac{bm}{p_2}$
But what if the utility function looks like $u=...
- 245
2
votes
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
2
votes
0
answers
32
views
Perfect complement outputs with each output being composed of substitutable inputs
How does one solve the following maximization problem?
$\underset{K_1, K_2, L_1, L_2}{\text{maximize }} min\{K_1 + L_1,K_2 + L_2\}$
subject to $c(K_1 + \mu K_2) + \beta c(L_1 + \mu L_2)$
where $c(...
- 21
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
1
vote
2
answers
1k
views
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
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