13 votes
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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 ...
Luis Salazar's user avatar
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.
E. Sommer's user avatar
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9 votes
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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 ...
Giskard's user avatar
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8 votes
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Elasticity and logarithms

Because $a$ is a parameter, and so $$ \eta = \frac{ d \log y}{d \log x} = \frac{ d \log a + d \ b \log x}{d \log x} = 0 + b. $$
Giskard's user avatar
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7 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 ...
aisync's user avatar
  • 264
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)}...
Giskard's user avatar
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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}...
Adam Bailey's user avatar
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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 ...
1muflon1's user avatar
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6 votes
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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 ...
galoosh33's user avatar
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6 votes
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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 ...
Theoretical Economist's user avatar
6 votes
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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} = \...
1muflon1's user avatar
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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)/...
Herr K.'s user avatar
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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. ...
Adam Bailey's user avatar
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6 votes
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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 ...
Giskard's user avatar
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5 votes
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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 ...
Ubiquitous's user avatar
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5 votes
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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-...
Herr K.'s user avatar
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5 votes
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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 ...
Art's user avatar
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5 votes
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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 ...
emeryville's user avatar
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5 votes
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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, ...
1muflon1's user avatar
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5 votes
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Calculating elasticity for $y^2e^{x+\frac{1}{y}}=3$

$$y'\frac{x}{y}= -\frac{f_1'(x,y)}{f_2'(x,y)}\frac{x}{y} = - \frac{y^2\exp(x+1/y)}{2y\exp(x+1/y)+y^2\exp(x+1/y)(-y^{-2})}\,\frac{x}{y}=\\=\frac{y^2}{1-2y}\,\frac{x}{y}=\frac{xy}{1-2y}.$$
VARulle's user avatar
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5 votes
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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 ...
tdm's user avatar
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5 votes
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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{...
1muflon1's user avatar
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5 votes
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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)}...
Green.H's user avatar
  • 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 ...
Herr K.'s user avatar
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4 votes

Question on significance of different ways of measuring Price Elasticity of Demand

Ideally the points $(x_1,y_1)$ and $(x_2,y_2)$ are very close. The assumption is that that they are so close that the elasticity of the demand function on the curve between them is nearly constant. ...
Giskard's user avatar
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4 votes
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Usefulness of the value for the slope of the demand curve

Price elasticity of demand is related (but not equal) to the inverse of the slope of the demand curve. This Wikipedia article defines PED as: Price elasticity of demand (PED or Ed) is a measure ...
FreeMarketUnicorn's user avatar
4 votes

Why are the formulas for price elasticity of supply and demand the same in intro microeconomics?

The elasticity is a very general concept. Consider an arbitrary function $$y=f(x_1,x_2, \cdots , x_n) $$ The elasticity of $y$ with respect to $x_i$ is $$ e_{y,x_i} = \frac{\partial y}{\partial x_i}...
luchonacho's user avatar
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4 votes
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Intuition behind Engel Aggregation and Cournot Aggregation

These expressions can be easily derived formally by taking the derivatives of Walras' law w.r.t $I$ (Engel aggregation) and $p_j$ (Cournot aggregation). The Engel aggregation means that not all goods ...
Egg's user avatar
  • 56
4 votes

Using an elasticity to evaluate changes in a dependent variable

why formula #3 may be more appropriate than formula #2? Actually, #2 is a linear approximation of #3. Note that $\partial \log Z$ can be written/seen as $\log Z_2 - \log Z_1 = \log\frac{Z_2}{Z_1} (\...
keepAlive's user avatar
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4 votes

How high of a carbon fee would reduce global emissions by 50% in 15 years?

No one knows No one knows how high a carbon fee would be required to reduce annual global emissions by 50% in 15 years: that would require a global restructuring that takes us outside the bounds of ...
410 gone's user avatar
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