8
votes
Accepted
Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Solving a non-linear programming (with inequality constraints) is about trial and error. You don't know a priori if a constraint is active. You consider all the possible cases satisfying your ...
- 665
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,...
- 2,008
5
votes
Accepted
What are the assumptions made about fixed points in the dynamics equations of Recursive macroeconomics?
On pages 53-55 of the Stokey, Lucas, with Prescott (1989) book they discuss the Contraction Mapping Theorem. This theorem guarantees existence and uniqueness of the solution (one fixed point).
The ...
- 487
5
votes
Accepted
Existence and uniqueness of demand, and symmetry implies equal demands given equal prices
Proof by contradiction. Suppose $(x_1^*, x_2^*)$ solves the problem and $x_1^* \neq x_2^*$ is true. By symmetry of the utility and equal prices, $(x_2^*, x_1^*)$ which is not the same bundle as $(x_1^*...
- 7,185
4
votes
Use of zero profit condition in determining unique solution
If the technology is CRS, and the parameters are such that 0 profit is both maximal and attainable with nonzero production, then the optimal solution is not unique, any multiple of it will also be ...
- 28.1k
4
votes
Accepted
The formula for expansion path
The answer depends on what you mean by "computing the expansion path".
Variant (i): For fixed $p_2$ and $M$, you want to compute the optimal points $\boldsymbol{x}=(x_1,x_2)$ as a function ...
- 6,202
3
votes
Accepted
Bellman Equation & Envelope Theorem
The key thing to note here is that in the optimum, $c_t$ will depend on $k_t$. Thus, the value function is
\begin{align}
V(k_t, t) &= \max\{u(c_t) + \beta V(f(k_t) - c_t, t + 1)\}\\
&= u(c_t(...
3
votes
Accepted
Quasiconvex Constraints in Maximisation
Consider the maximisation problem :
$$\max_x f(x) \text{ s.t. } g(x) \leq c$$
Note that
If $f$ is quasi-concave and $g$ is quasi-convex, then the set of solutions to the above problem is either an ...
- 7,185
3
votes
Accepted
Quasi-convex constraints using monotonic functions
Real-valued Monotonic functions defined on real line or subset of real line are both quasi-concave and quasi-convex, but that is not necessarily the case if the function is defined on $\mathbb{R}^n$ ...
- 7,185
3
votes
Accepted
Calculating the Compensating Variation with $M^2$
The left hand side expression corresponds to the new utility with the increased price and new budget $M - \Delta M$.
The income is now $M - \Delta M$ because we replaced the old income $M$ with $M - \...
- 2,008
3
votes
Accepted
Simplfying Euler equation under expectations
A formal assumption and twice ignoring Jensen's inequality can lead to the approximation.
FORMAL ASSUMPTION
$$
{\rm Cov} \left[C_{j,t+1}^{-\sigma}, \,\frac{R_{t+1}}{P_{t+1}}\right] =0.
$$
Technically, ...
- 33.1k
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?
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
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 ...
- 463
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
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
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 ...
- 463
2
votes
Utility maximization across yield curves?
Your maximization problem is
$$
\max_{x_1,x_2} x_1 \cdot (300 - 2x_1) + x_2 \cdot (200-1.25x_2) - x_1 - x_2
$$
subject to $x_1 + x_2 \leq \overline{x}$ where $\overline{x}$ denotes the budget. The ...
- 28.1k
2
votes
Accepted
Budget-feasible set in a portfolio choice problem
The agent owns nothing initially but the endowment. If their endowment would be $0$ in every state, then it should be clear that their initial wealth is zero, and they could only afford a portfolio ...
- 11.2k
2
votes
Accepted
how to derive marshallian demand functions from leontief preferences?
The utility function looks like this:
$(big)^2 - (small)^2$
Since $small$ is something that takes away utility, you want $small = 0$. Otherwise, you’re spending some money in getting unhappier, ...
- 2,008
2
votes
What is the Lagrangian a function of?
If you write $L(x,\theta,\lambda)$ this means that the unknowns of the lagrangian function that can be estimated are $x,\theta,\lambda$. This implies that your first order conditions will be 3, i.e., ...
- 665
2
votes
Accepted
Are other 'variables' in demand functions always fixed?
Generally you would write the hicksian demand $h(p_x,p_y,U)$. But when you graph it is easier to think of it as a single variable function.
What you see on a typical demand curve (assuming it is a ...
- 2,008
2
votes
Accepted
Remainder term in Linear Approximations going to 0
Question: How do I show that: $\lim \limits_{∆θ_i \to 0} \frac{R}{∆θ_i} = 0$
[...]
Question: How do we show that: $\lim \limits_{∆x_i \to 0} \frac{R_2(∆x)}{(∆x)^2} = 0$
As you can see from your ...
- 2,717
2
votes
Why do we need Complementary Slackness Condition for Karush-Kuhn-Tucker Conditions
Opportunity given, there can be cases where we have both $\lambda_j=0$ and $g_j(x^*) = c_j$. This happens when the unconstrained optimum $x^*_u$ equals the constrained one. In such a case, while the ...
- 33.1k
2
votes
Accepted
Solving first-order conditions to this social planner's problem
For two periods, the objective function is
$$...+\,\beta^{t} \left\{ \sigma[u(q_{t}) - q_{t}] + \theta U(C_{t}) + \bar{N} - N_{t} \right\} \\+ \beta^{t+1} \left\{ \sigma[u(q_{t+1}) - q_{t+1}] + \theta ...
- 33.1k
2
votes
Accepted
Second Order Condition - Always means second derivative?
Yes, this is terminology borrowed from mathematics. First order conditions relate to derivative of first order and second order conditions to derivative of second order. It has nothing to do with ...
- 51.3k
2
votes
Accepted
Mixed Partial Derivatives in Profit Function
In my classes, they’re called Marshallian or uncompensated demands, as in consumer theory.
The partial derivative on the right hand side means to directly differentiate the demand for $z$ (the ...
- 2,008
2
votes
Accepted
Inappropriate use of Calculus in estimating ΔCost?
If this is correct, is it unsurprising that it is wildly different
from the estimate given that the change in capital was a 50% increase,
and calculus is designed to approximate small changes? ...
- 2,717
2
votes
Accepted
Quadratic Form Single Summation notation
The sum $\sum_{i \le j}=a_{ij}x_ix_j$ goes over all indices $(i,j)$ with $i\leq j$. We can do that by either summing over all $i$ and then for each $i$ all $j\geq i$, which would give us
$$\sum_{i=1}^...
- 11.2k
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 ...
- 6,202
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