5
votes
Accepted
Derivative of $x_1^S(p_1, p_2, \overline{x}_1, \overline{x}_2) \equiv x_1(p_1,p_2,p_1\overline{x}_1 + p_2\overline{x}_2)$ to derive Slutsky equation
This is more of a calculus question. Recall the total differential of a function $f(z_1,z_2,z_3)$:
\begin{equation}
\mathrm df(z_1,z_2,z_3)=\frac{\partial f(z_1,z_2,z_3)}{\partial z_1}\mathrm dz_1 + \...
- 15.8k
3
votes
Why does the Substitution effect always have to be negative?
Consider the following utility maximisation problem of the consumer with utility function $u:\mathbb{R}^n_+\rightarrow\mathbb{R}$:
\begin{eqnarray*} \max_{x\in\mathbb{R}^n_+} & u(x) \\ \text{s.t.} ...
- 9,431
3
votes
Accepted
is income effect in the given quasilinear function equal to 0?
First let's consider the following problem:
\begin{eqnarray*} \max_{(x, y) \in \mathbb{R}^2_+} & x + 4\sqrt{y} \\ \text{s.t.} \ & p_Xx + p_Yy \leq M\end{eqnarray*}
Solution to the above ...
- 9,431
3
votes
Accepted
Income and substitution effect for perfect substitutes
An indifference curve for perfect substitutes is a straight line. In fact it is the line defined by $y=const-x$, for a utility level of $const\in\Bbb R$. We maximize the utility when our budget line ...
- 286
3
votes
Accepted
Income effect $-\frac{\partial x_i}{\partial m} x_i$ or $\frac{\partial x_i}{\partial m}x_i$?
The income effect is defined as $\mathbf{-\frac{\partial x_i}{\partial m}x_i}$.
Let $x_i$ be a normal good; that is, a good whose Marshallian demand increases with an increase in income $\left(\frac{...
- 1,249
2
votes
Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition
You interpreted the passage you read as essentially saying that "with log preferences, the Marshallian demand for the good does not depend on its price."
That doesn't sound very plausible, don't you ...
- 33.9k
2
votes
Question about the relationship between Weak Axiom and Slutsky Matrix
It is almost true.
There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom.
However, if we ask that
$$v \cdot S(p,w) v <0 $$
whenever $v \not = \alpha ...
- 1,924
2
votes
Using Slutsky method to understand substitution and income effect for luxury goods
I think that you are confusing a luxury good with an inferior good.
First, a luxury good is a superior good, and a superior good is NORMAL (which means that the demand of that good increases as ...
- 1,262
2
votes
Accepted
A question on Intermediate Microeconomics 9ed by Hal Varian, Chapter 8 Slutsky Equation, Figure 8.8
The book does not claim that
$$
p' x_1 + x_2 = m + p'(x_1^* - x_1)
$$
is the equation describing the new budget line. The budget line with slope $-p'$ that passes through $(x_1^*,x_2^*)$ is
$$
p' ...
- 29.6k
1
vote
Explaining the relative share of the income & substitution effects of a price change
The income effect is given by:
$$
x^\ast \frac{\partial x}{\partial I} = \frac{0.1 I}{P_x} \frac{0.1}{P_x} = 0.1\left(0.1\frac{I}{(P_x)^2}\right).
$$
The Price effect is:
$$
\frac{\partial x}{\partial ...
- 12.5k
1
vote
Does an "optimal" MRS exist?
If yes, how does it make sense practically? I was forced to consume less but i am more happy?
You are not consuming less in general though, you are consuming less electricity but spending more money ...
- 29.6k
1
vote
Complements/substitutes estimation from data (Slutsky matrix)
If you neither observe the utility level nor the expenditure (or income) level, it seems not possible to identify separately Hicksian and Marshallian demands. So it is not advisable to impose symmetry....
- 3,481
1
vote
How to prove Slutsky matrix's symmetry for L=2
Let $c(p, u)$ be the expenditure function. The Hicksian demand for good $j$ is the derivative of $c$ with respect to $p_j$.
$$
\frac{\partial c(p,u)}{\partial p_j} = h_j(p,u).
$$
From this, it follows ...
- 12.5k
1
vote
Accepted
Calculate income and sustitution effect from utility funcion
First, the fact that $MRS=\frac{1}{4}$ does not tell you by itself that the consumer will only buy $x_2$.
We need to go back to the 2nd Gossen's law:
$\frac{Um{x_1}}{p_1}=\frac{Um{x_2}}{p_2}$ (this is ...
- 98
1
vote
Accepted
Marshalian and Hickisian Demands and Slutsky Equation
Although my teacher has yet to verify my solution for d) (which I believe is incorrect), he shared his answer and the condition $x_2<k$ derives from the positivity of $V(p_1,p_2,b)$ and from the ...
- 400
1
vote
Reason behind the decomposition of price effect into substitution and income effects
Why are we decomposing the effect of a change in price of one commodity into all these effects which are (as I have understood), all hypothetical.
They're hypothetical, but they are nevertheless ...
- 1,806
1
vote
Why do the income and substitution effects cancel for log preferences? Trouble reconciling Slutsky decomposition
Try taking derivative of x* w.r.t. py (which is 0)
And verify with Slutsky decomposition.
- 11
1
vote
Calculating income and substitution effects
First of all:
In general, for preferences that are quasi-linear in $x(y)$ it holds true that:
Overall effect = Substitution effect, if in the initial situation both goods or if only good $y(x)$ are ...
- 111
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