6
votes
Accepted
Lagrange Multiplier Dual Meaning?
The Lagrange multiplier is both of the above, both statements are equivalent.
In the classic consumer problem
$\max U(x,y)$
s.t. $p_x x + p_y y = I$
whose Lagrangian is
$\mathcal{L}(x,y,\lambda) = U(x,...
- 2,006
3
votes
Accepted
Arguments for Concavity or Quasi-concavity
I will try to address your queries in the order you've asked them by providing the necessary definitions and procedures to find the answers you're looking for.
The first approach is valid and it is ...
- 439
3
votes
Accepted
Utility maximization for a household consisting of a woman and a man, with gender discrimination
In principle, using Lagrange multipliers instead of substitution should be the same. Lagrange multipliers are important when the constraint functions can't be explicitated, otherwise they are not ...
- 2,717
3
votes
Lagrange Multiplier Dual Meaning?
I'm purposefully handwavy here; getting things formally right can be very subtle.
Suppose you could ignore the constraint. Then you do your usual unconstrained maximization problem, and the marginal ...
- 11.2k
3
votes
Lagrange Multiplier Dual Meaning?
Consider the following constraint optimization with an equality constraint:
$$\begin{align}
\max_{(x,y)\in\mathbb{R}^2} \quad & f(x,y) \\
\textrm{s.t.} \quad & g(x,y)=c
\end{align}$$
The ...
- 439
3
votes
Lagrange Multiplier Dual Meaning?
In general the Lagrange multipliers from a constrained optimization problem takes on the interpretation of the value of adjusting the "size" of the constraint in question.
There can be many ...
- 8,034
3
votes
Accepted
CRS, Homothetic Functions, and constant MRTS
I think I’m generally missing the added value of a homothetic
function vs homogenous function.
The added value of homothetic functions is that they are a wider class with respect to homogeneous ...
- 2,717
2
votes
Accepted
Inappropriate use of Calculus in estimating ΔCost?
If this is correct, is it unsurprising that it is wildly different
from the estimate given that the change in capital was a 50% increase,
and calculus is designed to approximate small changes? ...
- 2,717
2
votes
Accepted
Mixed Partial Derivatives in Profit Function
In my classes, they’re called Marshallian or uncompensated demands, as in consumer theory.
The partial derivative on the right hand side means to directly differentiate the demand for $z$ (the ...
- 2,006
2
votes
Perfect Complement Utility Function Maximisation
Proof:
Suppose there is an optimal bundle $x^\star$ such that $p \cdot x^\star < m$. By continuity there is a neighborhood $N$ of $x^\star$ such that $p\cdot x<m$ and $u(x)\le u(x^\star)$ for ...
- 6,157
2
votes
Accepted
Quadratic Form Single Summation notation
The sum $\sum_{i \le j}=a_{ij}x_ix_j$ goes over all indices $(i,j)$ with $i\leq j$. We can do that by either summing over all $i$ and then for each $i$ all $j\geq i$, which would give us
$$\sum_{i=1}^...
- 11.2k
1
vote
How to derive the short run cost function
Since $L$ is fixed at some $\overline{L} > \frac{\overline{Y}}{2}$, then $\overline{Y} < 2L$.
This implies the $\min$ term equals $3K$.
Therefore, $f(K,\overline{L}) = 3K$.
With this, the ...
- 2,006
1
vote
Accepted
Perfect Complement Utility Function Maximisation
Suppose there is an optimal bundle $x^\star = ({x_1}^\star,{x_2}^\star)$ such that $p \cdot x^\star < m$.
Since $p \cdot x^\star < m$, there exists a $t > 1$ such that $t (p \cdot x^\star) = ...
- 2,006
1
vote
Accepted
Logarithmic Utility function Algebra
$U(x^\star,y^\star) = a \ln( \frac{am}{p_1}) + b \ln(\frac{bm}{p_2}) $
$= a \ln(a) + a \ln(m) - a \ln(p_1) + b \ln(b) + b \ln(m) - b \ln(p_2) $
Since $a + b = 1$,
$= \ln(m) - a \ln(p_1) - b \ln(p_2) + ...
- 2,006
1
vote
Accepted
Non-Negativity Constraints KKT
The way I’d do it is to simply optimize the usual Lagrangian
$\mathcal{L} = f(x,y) + \lambda (k - g(x,y))$.
If all variables end up being non-negative, that is your solution.
If any variable, let’s ...
- 2,006
1
vote
Utility maximization for a household consisting of a woman and a man, with gender discrimination
I found BakerStreet's answer useful to intuitively make conclusions about the model, however I was then able to explicitly solve for the household's agents' optimal time allocations myself with a ...
- 2,006
1
vote
Accepted
Missing Non-Negativity Constraint?
The only thing that changes is the $k$ first order condition.
The one you got with the non-negativity constraint Lagrange multiplier is
$-r + \lambda \alpha L^{1-\alpha} + \mu_k \leq 0$
Substracting $\...
- 2,006
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