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1

Let $b_i := \frac{a_i}{p_i}$, now choose the good that maximize $b_i$, let it be $b_j$ i.e. the $j$th good. Optimal choice is $x_j=m/p_j$, and $x_i = 0$ for $i\neq j$.


0

If you have $J$ consumers therefore $J$ demands for a good $X$. Denoting the individual demand of each consumer with $x_j^*$ as you have it, if $X$ is the aggregate demand, it is just the sum of every individual demand: $X=\sum_{j=1}^{J}x_j^*$ Then for your case it's: $x_1^*+x_2^*=\frac{(I_1+I_2)}{2P_X}$, and the same with $Y$.


3

The two papers you provide are explicit on how elasticities are computed. To take a simplified version of the specifications used in both papers, let $$\log D(p)=\beta\log p.$$ Now, $$D(p)=e^{\log D(p)}=e^{\beta\log p}=(e^{\log p})^\beta=p^\beta.$$ Therefore, $$\frac{p}{D(p)}\frac{d D(p)}{dp}=\frac{p}{p^\beta}(p^\beta)'=p^{1-\beta}\beta p^{\beta-1}=\beta.$$ ...


4

I would not call it average elasticity rather its elasticity at an average price. For example take the first paper about elasticity of oil you cite (that is Cooper, J. C. (2003). Price elasticity of demand for crude oil: estimates for 23 countries. OPEC review, 27(1), 1-8.). In that paper the Cooper estimates elasticity using the following model: $$\ln D_t = ...


3

Yes, for the standard case of a strictly decreasing demand function $Q(p)$ and price-elasticity of demand $\epsilon_p(Q)=Q'(p)\frac{p}{Q(p)}$ the inverse demand function $p(Q)$ exists and by the inverse function theorem $p'(Q)=\frac{1}{Q'(p)}$. This gives $p'(Q)=\frac{p(Q)}{\epsilon_p(Q)Q}$ wherever the derivatives exist.


4

I will denote the demand function by $Q(p)$ and the inverse demand function by $P(q)$. Then $$ \forall q: Q(P(q)) = q $$ so for any $h > 0$ and $q$ we have $$ \begin{align*} p & := P(q) \\ p_h & := P(q+h) \\ q & = Q(p) \\ q_h & := Q(p_h) = q+h \end{align*} $$ From the definition of derivatives $$ \frac{\text{d} P(q)}{\text{d} q} := \lim_{h ...


3

One approach could be the following. For a $(p_n,w_n)$ in the sequence and $x \in B(p,w)$ define: $$ \alpha_n = 1 \text{ if } p_n x \le w_n$$ and $$ \alpha_n = \frac{w_n}{p_n x} \text{ if } p_n x > w_n$$ Then define: $$ x_n = \alpha_n x$$ Here $x_n$ equals $x$ if $x$ is in the budget $B(p_n,w_n)$. If not, then $x_n$ is the radial projection of $x$ onto ...


5

I don't believe it is lower semicontinous. Let $w = (0,\dots,0)$, $p \in \mathbb{R}^n_+$ be any vector such that $p_1 = 0$ (the first coordinate being 0). The allocation $x=(1,0,\dots,0) \in B(p,w)$. Define the sequence $p_n = p + (\frac{1}{n},0,\dots,0)$ and $w_n = (\frac{1}{n},0,\dots,0)$. $w_n \rightarrow w$ and $p_n \rightarrow p$. For any $x^n \in B(...


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