11
votes
Accepted
Why are elasticities defined as logarithmic derivatives?
If I understand your question, first the elasticity haven't units. The problem with $\partial Y/\partial X$ is that if you change the measure units the result is different. Is less problematic to ...
- 156
10
votes
Proof coefficient in log-log model is equal to coefficient of elasticity
Because $\Bbb E[\varepsilon \mid x]= 0$ is one of the key assumptions for the estimation.
- 1,307
9
votes
Accepted
Elasticity of demand equals -1 but income decreases!
We discourage numeric questions as they are unlikely to be useful for future visitors but this is a very good example of why using non-marginal quantities can be misleading.
The exact definition of ...
- 28.4k
8
votes
Accepted
Elasticity and logarithms
Because $a$ is a parameter, and so
$$
\eta = \frac{ d \log y}{d \log x} = \frac{ d \log a + d \ b \log x}{d \log x} = 0 + b.
$$
- 28.4k
7
votes
Accepted
What determines the outcome of a price war, and why isn't that outcome reached instantaneously?
Answer to question
If we take your assumptions literally, Jim will decide not to enter the widget business. For suppose he did incur the cost of entry and that Mary is selling at price $p_m$. Jim can ...
- 16.9k
7
votes
Accepted
Constant Elasticity of Substitution: Special Cases
We know that if $u$ represents $\succeq$ on $X$, then for any strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$, then $v(x) = f(u(x))$ represents $\succeq$ on $X$
($X$ in this case ...
- 6,556
7
votes
How exactly does elasticity relate to slope?
The two demand functions $D_1(p),D_2(p)$ cross at the point $(Q,p)$. Their respective elasticities at price $p$ are
\begin{align*}
\epsilon_1(p) & = \frac{\text{d}D_1(p)}{\text{d}p}\frac{p}{D_1(p)}...
- 28.4k
7
votes
Elasticity and logarithms
Differentiating both sides of the equation with respect to $x$, using the chain rule for the left hand side and noting that, since $a$ is a parameter, $da/dx=0$:
$$\frac{1}{y}\frac{dy}{dx}=b\frac{1}{x}...
- 8,251
7
votes
Does Modern Monetary Theory (MMT) provide a useful insight into how to manage the economy?
Does Modern Monetary Theory (MMT) provide a useful insight into how to manage the economy?
That depends on your definition of MMT, because it is not generally agreed on what it even is. You will find ...
- 53k
6
votes
Accepted
If the Engel Curve of a Cobb-Douglas utility function is positive and linear, than does that mean it is neither a necessity nor a luxury good?
Recall the following equivalent definitions for luxury goods and necessities:
A good $x$ is considered a necessity if $e_{(x,I)}<1$.
A good $x$ is considered a luxury good if $e_{(x,I)}>1$.
As ...
- 231
6
votes
How do I calculate price elasticity of demand using historical price and quantity data?
You've fallen into a really common pitfall -- the spurious regression. The parameters you chose to include can't be chosen 'willy nilly' by throwing data into a regress command. Ultimately this can't ...
- 254
6
votes
Accepted
Intuition for the CES consumption index in New-Keynesian DSGE models
Heuristically, you can think of the integral as just a sum:
$$ \bar{C} = \left( \sum_{i=1}^n C_i^{1-\frac{1}{\epsilon}} \right)^{\frac{\epsilon}{\epsilon - 1}} $$
where $\bar{C}$ is an index of ...
- 2,156
6
votes
Accepted
First order condition of log functions in general and interpretation
It just comes from the derivative of profit function. I assume that $a_i$ is the choice variable here so the derivative of $\pi$ wrt $a_i$ is (step by step):
$$\frac{\partial \pi}{ \partial a_i} = \...
- 53k
6
votes
First order condition of log functions in general and interpretation
The $\gamma$ on the RHS comes from applying chain rule when differentiating the second term with respect to $a_i$.
Regarding elasticity, note that with a differentiable function $f$, the ratio $f'(x)/...
- 15.3k
6
votes
Microeconomics question on elasticity
The answer by Herr K is correct so far as it goes and is probably what your teacher is looking for, but it's worth adding the following.
Walt and Jessie place their orders before looking at the price. ...
- 8,251
6
votes
Accepted
Monopolies on Giffen Goods
If a good exhibits Giffen behavior at a certain price level, it implies that a (slightly) higher price will result in greater demand.
The Giffen property is local, a good cannot behave as a Giffen ...
- 28.4k
5
votes
Constant Elasticity of Substitution: Special Cases
These are standard mathematical results for generalized means. For example,for the $\rho \rightarrow 0$ result, write (setting without loss of generality $\sum_{i=1}^na_i =1$),
$$U = \left[\sum^n_{i=...
- 33.2k
5
votes
Accepted
Does unit elasticity has to be at exactly the middle of the demand curve?
When the demand function is linear, $q = a-bp$, the only point were elasticity is unity is located in the midpoint of the demand curve (straight line). This is geometrical.
The demand line will cross ...
- 33.2k
5
votes
Does unit elasticity has to be at exactly the middle of the demand curve?
No, that is only true in the linear case. For a simple counterexample consider
$$
D(p) = 1 - \sqrt{p}.
$$
$$
\epsilon(p) = \frac{d D(p)}{dp} \cdot \frac{p}{D(p)} = -\frac{1}{2 \cdot \sqrt{p}} \cdot \...
- 28.4k
5
votes
Help with Income Elasticity Exercise in Becker's Economic Theory
The precise question the OP asked was what does $K_j N_j$ represent. As Alecos' response says, the statement $\sum_j K_j N_j = 1$, i.e. the weighted average of the income elasticities equals one, ...
- 81
5
votes
Accepted
Monopoly equilibrium with a completely inelastic demand
Perfectly inelastic demand means quantity demanded is $q$ irrespective of the price. If producing quantity $q$ costs $c$ then the monopolist's problem is
$$\max_p \{pq-c\}.$$
This problem is not ...
- 16.9k
5
votes
Accepted
Calculating Price Elasticity of Demand
The answer will vary slightly depending on which notion of elasticity you're using.
Arc elasticity (or midpoint elasticity) uses the formula
\begin{equation}
\epsilon^\text{arc}=\frac{Q_1-Q_0}{P_1-...
- 15.3k
5
votes
Accepted
If $X$ is a Giffen good then $Y$ must be a normal good
Reason: Both goods cannot be inferior.
Let's say originally you consume $x$ and $y$. So your budget constraint looks like
$$p_x x + p_y y = I.$$
If both X and Y are inferior, when income goes down ...
- 2,784
5
votes
Accepted
What is the empirical price elasticity of demand for insulin?
You are right! Textbooks are probably not careful enough.
However, there is evidence that the most inelastic drugs are those indicated for the treatment of chronic diseases with price-elasticity ...
- 6,885
5
votes
Accepted
Elasticity of demand
When people talk about 'elasticity of demand' without further qualification they normally mean 'price elasticity of demand':
$$\text{E}_p=\frac{\partial Q_t}{\partial P_t}\frac{P_t}{Q_t}$$
However, ...
- 53k
5
votes
Accepted
Calculating elasticity for $y^2e^{x+\frac{1}{y}}=3$
$$y'\frac{x}{y}= -\frac{f_1'(x,y)}{f_2'(x,y)}\frac{x}{y} = - \frac{y^2\exp(x+1/y)}{2y\exp(x+1/y)+y^2\exp(x+1/y)(-y^{-2})}\,\frac{x}{y}=\\=\frac{y^2}{1-2y}\,\frac{x}{y}=\frac{xy}{1-2y}.$$
- 6,355
5
votes
Accepted
Relation of Engel-curve to income elasticity of demand; is the slope of the Engel-curve equal to the elasticity of income?
Income elasticity of demand
Let $q(y)$ be the Engel curve for a good, i.e. it gives the demanded quantity for a given level of income $y$ (keeping prices fixed). The income elasticity of demand is ...
- 9,802
5
votes
Accepted
Finding the elasticity of a function with respect to a variable from logarithm
Elasticity of any differentiable function $f(x)$ wrt $x$ by definition is:
$$\epsilon = \frac{df(x)}{dx}\frac{x}{f(x)}$$
For a particular type of function given by:
$$f(x) = Ax^e$$
$$\epsilon = \frac{...
- 53k
5
votes
Accepted
Elasticity calculation
The elasticity, which we denote by $\eta$, is
$$\eta = \frac{\partial h}{\partial w} \frac{w}{h}.$$
Since
$$\frac{\partial h}{\partial w} = \frac{I A}{w^2(1+A)},$$
we have
$$\eta = \frac{I A}{w^2(1+A)}...
- 292
5
votes
Microeconomics question on elasticity
Since Jessie's expenditure on gas is fixed at $\\\$10$, it follows that a $1\%$ increase in price must lead to a $1\%$ decrease in quantity demanded, which in other words means Jessie has unit-elastic ...
- 15.3k
Only top scored, non community-wiki answers of a minimum length are eligible