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How to derive elasticity of substitution The first step is to recall the definition of a differential. If you have a function $f: \Bbb R^n \to \Bbb R$, say, $f(x_1,\cdots,x_n)$, then: $${\rm d}f = \frac{\partial f}{\partial x_1}{\rm d}x_1 + \cdots + \frac{\partial f}{\partial x_n}\,{\rm d}x_n.$$ For example, $$d\log v = \frac{1}{v}dv$$ Now suppose $v = \... 13 Think of it this way. A yacht costs$\$0.01$. How much do you demand? Then double the price. Now how many would you demand? Then consider something closer to a market price. Something in 6 or 7 figures I guess. If you've got some money, you might buy one at the market price. But then if that price doubles, wouldn't that affect your demand a lot more than ...
The answer to the question is yes, it is indeed meaningful (at least mathematically speaking). If you estimate the linear equation $$W = \beta_0 + \beta_1 PTR,$$ then $\beta_1=\frac{\partial W }{\partial PTR}$, meaning that $\beta_1$ represents the marginal change of $PTR$ over $W$. Now, if you estimate $$log(W) = \beta_0 + \beta_1 log(PTR),$$ then $\... 11 In many practical instances, price elasticity of demand (PED) is calculated in a back of the envelope fashion, just as taught in the textbooks! Firms can adjust their price by some small amount and observe the demand response. For relatively small changes in price and quantity, little accuracy is lost by assuming that the demand function is locally linear, ... 10 Consider the Slutsky equation, $$\frac{\partial x}{\partial p} = \frac{\partial x^c}{\partial p} - \frac{\partial x}{\partial I} x.$$ A giffen good is the case where the income effect$\frac{\partial x}{\partial I} x$is negative and large (in magnitude) enough so that$\frac{\partial x}{\partial p} > 0$. From Wikipedia: There are three necessary ... 10 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 express it in percentages: $$El_X(Y)=\frac{\Delta Y/Y}{\Delta X/X}$$ if$\Delta Y \to 0$when$\Delta X \to 0$then you obtain $$El_X(Y)=\frac{dY}{dX}\frac{X}{Y}$$ ... 10 Because$\Bbb E[\varepsilon \mid x]= 0$is one of the key assumptions for the estimation. 9 There is rather low probability for demand of a good to exhibit the Giffen property at market level, where averaging over heterogeneous preferences, different income levels and consequent differentiated behavior, will usually offset Giffen phenomena. Looking at @jmbejara answer, goods that are likely to satisfy all three necessary conditions are drugs ... 9 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 the price elasticity of demand is $$\epsilon(p) = \frac{d D(p)}{d p} \cdot \frac{p}{D(p)}.$$ (In your notation$D(p) = Q$.) By straightforward calculation you ... 8 Like people have said in the comments, log-log is commonly used. It amounts to estimating a constant elasticity model$Y = \alpha X^\beta$, which is a commonly used functional form within economics. Once you take logs, this becomes$\ln Y = \ln \alpha + \beta \ln X$. You can read more about this here. I guess your question is whether or not using this ... 8 The truth is that it's unclear if firms use the concept at all. Alan Blinder wrote this wonderful little book called "Asking about prices". A survey of firms asking them how they set prices. And it's full of very puzzling finds. Elasticity is one of them(Page 99). So they ask firms what is their price elasticity of demand: This was a difficult question ... 8 Because$a$is a parameter, and so $$\eta = \frac{ d \log y}{d \log x} = \frac{ d \log a + d \ b \log x}{d \log x} = 0 + b.$$ 7 Alright, the other respondents have covered the logic behind a log-log regression pretty well, so I'm just going to add some practical tips. If you want to check whether your specification is reasonable, and your problem is the assumption of a constant elasticity, try splitting the sample into groups based on percentiles of$x$and recalculating$\alpha$and ... 7 I think there are pedagogical advantages to discussing both the raw numbers and the absolute values and I think the benefits of both explain why they both show up (sometimes in the same text, even). Each elasticity number gives two bits of information. First, the absolute value with respect to 1 and second, the sign. Now, clearly, if you had a negative ... 7 In theory, the elasticity quantifies in proportional terms the "reaction" of the dependent variable to a change of the independent variable, where the two are related through a functional relationship. So the concept is universal and mathematical -it applies to any univariate function, and not only in the field of Economics (when we have a multivariate ... 7 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 only sell to consumers if his price$p_j\leq p_m$. The best price for Jim is$p_m-\epsilon$(where$\epsilon$is some very small, positive amount). But this ... 7 The two demand functions$D_1(p),D_2(p)$cross at the point$(Q,p)$. Their respective elasticities at price$pare \begin{align*} \epsilon_1(p) & = \frac{\text{d}D_1(p)}{\text{d}p}\frac{p}{D_1(p)} \\ \\ \epsilon_2(p) & = \frac{\text{d}D_2(p)}{\text{d}p}\frac{p}{D_2(p)}. \end{align*} However since both function cross at the point(Q,p)$we know that ... 7 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,$da/dx=0$: $$\frac{1}{y}\frac{dy}{dx}=b\frac{1}{x}$$ Rearranging: $$\frac{dy/y}{dx/x}=\eta=b$$ 7 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 some arguing it is just a macro/monetary theory (such as the Wikipedia page) but then I seen MMT proponents on this site arguing it is a whole new paradigm ... 6 The usual textbook example of a Giffen good (i.e. a good whose demand curve slopes upwards) is the Irish potato famine. The idea is that as potatoes (a staple food) became more expensive, people could no longer afford expensive foods such as meat and so ended up buying more potatoes! However, this example has come in for criticism, not least of all because a ... 6$\alpha$handles the conversion between the units in which labor is measured and the the units in which consumption is measured. Consider the leisure part of the utility function: $$- \alpha \frac{n^{1 + \frac{1}{\nu}}}{1 + \frac{1}{\nu}}$$ We can rewrite it like this: $$- (\alpha^{\frac{1}{1 + \frac{1}{\nu}}})^{1 + \frac{1}{\nu}} \frac{n^{1 + \frac{1}{\... 6 We know that if u represents \succeq on X, then for any strictly increasing function f: \mathbb{R} \rightarrow \mathbb{R}, then v(x) = f(u(x)) represents \succeq on X (X in this case is \mathbb{R^n}) Consider v(x, \rho) = \ln(u(x, \rho)) - \frac{\ln\left[\sum^n_{i=1}\alpha_i \right]}{\rho}, which is strictly increasing.$$v(x, \rho) = \... 6 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 you can see, these definition do not encompass all possible scenarios, so any specific good does not have to be either a luxury or a necessity. In the case of ... 6 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 aggregate consumption, and utility is given by$u \left( \bar{C} \right)$. It's easy to check that the marginal rate of substitution between goods$j$and$k$is ... 6 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} = \frac{\partial \pi}{ \partial a_i} [ \ln R(a_i) ] + \frac{\partial \pi}{ \partial a_i} [ \ln N_i(\gamma a_i, \gamma a_{-i}) ] \\ = \frac{1}{ R(a_i)} R'(a_i) ... 6 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$f'(x)/f(x)$can be interpreted as the percentage change in the value of$f$around$x$. So isolating$\gamma$from the FOC, you'll get an expression for elasticity, ... 5 Say elasticity (of demand) gives the percentage change in quantity demanded in response to a one percent change in price. Since the change is porcentual, if you are in a point of the demand where consumption is low, then a one percent decrease in price will result in a relatively big change in consumption, so elasticity is relatively high. As the quantity ... 5 Related to the MRS, this is a more general problem regarding negative slopes. I confess to continuously being confused for many-many years on the matter (and having to pose and think), until I constructed the following mental image in my mind, which I share here just in case somebody else may find helpful: The trick is to put minus and plus infinity side-by-... 5 These papers look relevant, to one degree or the other: Karlan, D., & Zinman, J. (2009). Observing unobservables: Identifying information asymmetries with a consumer credit field experiment. Econometrica, 77(6), 1993-2008. The authors write:"We estimate the presence and importance of hidden information and hidden action problems in a consumer credit ... 5 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, means that if total income goes up by one percent then total expenditures/consumption also increases by one percent. The total market expenditure for some goods (...