All Questions
Tagged with consumer-theory optimization
32 questions
1
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1
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37
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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
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2
answers
120
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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 ...
1
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2
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74
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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
1
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1
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102
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Logarithmic Utility function Algebra
Question:
I'm told the following (by an exam mark scheme):
Using $a + b =1$
$a[ln(\frac{am}{p_1})] + b[ln(\frac{bm}{p_2})] = ln(m) - aln(p_1) - bln(p_2)$
I can't get this to hold without the ...
- 1,011
1
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1
answer
112
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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
2
votes
0
answers
82
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Convex Combination of pairs of points
Is it appropriate/meaningful to write vector/points $(a,b) \le (c,d)$, where i would mean component wise each component is $\le$
Specifically is my example below with reference to concavity ...
- 1,011
1
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0
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113
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Testing for Concavity - Local Maximum & Global Maximum
My question is under which contexts Negative Definitness (ND) vs Negative Semi-Definitness (ND) is required for classifying a global maximiser. And also Global vs Local.
I also want to understand what ...
- 1,011
3
votes
3
answers
635
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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
4
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2
answers
351
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Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Complementary slackness condition (CSC) state that
$\lambda_j[g_j(x) − c_j] = 0 \hspace{5pt} \text{for} \hspace{5pt} j = 1, ..., m.$ Therefore, every constraint either needs to be an equality ...
- 172
0
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1
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74
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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
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2
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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
2
votes
1
answer
114
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Arguments of the Marshallian demand system of a Cobb-Douglas utility function
For a utility function of the form $U(x_1,x_2) = x_1^\alpha x_2^\beta$ and the standard budget constraint, the utility maximisation problem gives us a demand system characterised by:
$x_1(\alpha, \...
- 57
1
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3
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4k
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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
1
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1
answer
125
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Optimal consumption for infinite number of periods and exogenous income
I have the following optimization problem:
$\max_{\{c_t, s_{t+1}\}} \Pi_{t=0}^\infty c_t^{\beta^t}$
$\text{subject to } \space c_t + s_{t+1} = y_t + (1 + r) s_t \text{ and } s_0 = 0$
How do I find ...
- 167
3
votes
2
answers
678
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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
2
votes
1
answer
71
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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
1
vote
1
answer
741
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Generalizing demand for perfect substitutes utility function
I have the utility function:
$U(x_1,...,x_n)=a_0+\sum_{i=1}^{n}a_ix_i\;\;\;\;\;\;\;\;\;a_j\in\mathbb{R}_+ \;\;\forall j=\{0,...,n\}$ (maybe $a_0$ could be zero)
$\sum_{i=1}^{n}a_i\in (0,K)\;\;\;$ ...
- 943
2
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2
answers
456
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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
1
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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
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1
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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
0
votes
2
answers
1k
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Corner solution of the maximization problem
Answer
Hello, I upload the actual question with my 8-pages answer. Please can you check it. Is there a corner dissolution for $c=\gamma$. Please share your ideas. Thanks.
- 192
0
votes
0
answers
284
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Finding the optimal consumption bundle given the strictly concave utility function $v(x,y) = U(x) +y$?
I am also finding it difficult to understand what are the fundamental differences between analysing optimal bundles between concave and convex functions ?
Does it also happen that the optimal bundle ...
- 11
1
vote
0
answers
375
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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)
$$\max_{\{c_t(s^t),a_{t+1}(s^t,s_{t+1})\}_{t=0}^{\infty}}\...
- 121
2
votes
2
answers
3k
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Show that First order conditions are necessary and sufficient for utility maximization
I have a budget set
$$B=\{x=(x_1,x_2)\in R^2_+ \mid 2\sqrt{x_1}+x_2\le y\}$$
where $y>0$ is income.
Assuming the preferences are strictly monotonic and convex, I want to show that first order ...
- 63
1
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1
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227
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Find Indifference curve/s and Marginal Rate/s of Substitution given only one point
Arka likes fries. She wants to consume as much as possible. She consumes either regular (1 oz) or large sizes (5 oz).
Draw her indifference curve through $(x_R, x_L) = (10,0)$ and her indifference ...
- 370
1
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0
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78
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Does this conditional increase in income affect a budget line in the same way as an unconditional increase in income would?
Note: It's been awhile since I've taken introductory microeconomics. I remember increases in income move budget line outward. What if the increases have some condition?
The problem:
Jill has $I$ to ...
- 370
3
votes
0
answers
387
views
Constrained optimization for $u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}$ [duplicate]
Suppose preferences are represented by the following utility function
\begin{equation}
u(x_1,x_2,x_3,x_4)=\alpha \min \{a x_1, b x_2\} + \beta \min \{c x_3, d x_4 \}
\end{equation}
Write the
...
- 189
8
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3
answers
2k
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Does the Marshallian demand function always include prices and income?
I have the following utility function:
$$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
2
votes
2
answers
2k
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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
8
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1
answer
6k
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Leontief preferences
I can solve most utility maximization problems using my mathematical knowledge .... but not when it comes to Leontief preferences. I do not have a book to lean on (am self-studying), so would really ...
- 81
2
votes
2
answers
1k
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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
12
votes
2
answers
44k
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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 = \...
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