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7

For the 2x2 case being considered, write $$\mathbf{B}=\left[\begin{array}{cc} b_{1,1} & b_{1,2}\\ b_{2,1} & b_{2,2} \end{array}\right].\quad$$ It follows that the element (1,1) in $B^{-1}$ is given by $\frac{b_{2,2}}{b_{1,1}b_{2,2}-b_{1,2}b_{2,1}}$. Notice that $$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 }... 6 The supply of masks can't keep up with soaring demand because mask producers have capacity constraints in the short run. See this recent article. 5 I don't believe it is lower semicontinous. Let w = (0,\dots,0), p \in \mathbb{R}^n_+ be any vector such that p_1 = 0 (the first coordinate being 0). The allocation x=(1,0,\dots,0) \in B(p,w). Define the sequence p_n = p + (\frac{1}{n},0,\dots,0) and w_n = (\frac{1}{n},0,\dots,0). w_n \rightarrow w and p_n \rightarrow p. For any x^n \in B(... 4 It's true that given the utility function the y-good is a normal good, so the question is quite odd. Ignoring this, your calculations are correct, but you could simplify to y(p_x,p_y,b)=\frac{b}{2p_y-p_x}. Your curve contains a mistake, since x(1,3,48)=19.2. Indeed by substituting your expressions for x and y you can show that 2x+y=\frac{b}{p_x} ... 4 The price elasticity of demand is defined as:$$E_P=\frac{dQ}{dP} \frac{P}{Q}$$Although generally elasticity depends on price there is a special type of functions (isoelastic functions) for which elasticity remains the same along the whole function. For example consider demand given by:$$P=AQ^{1/e}$$This demand function will always have the same ... 4 To provide some real world(ish) interpretation, you could consider the following: Wallace enjoys eating cheese on its own. He doesn't much care for crackers on their own, but he especially loves eating crackers and cheese together, he makes nice little cracker n cheese sandwiches. In this example, we can think of cheese (x) and crackers (y) as perfect ... 4 I assume you know how does \min\{x,y\} look like? In order to draw utility function of interest, you need to consider cases: u(x,y)=x+\min\{x,y\}=\begin{cases}2x, \;\; \mathrm{for} \;\; x \leq y \\ x+y, \;\; \mathrm{for} \;\; x > y\end{cases} With x on horizontal and y on vertical axis: Not sure about the "usual" perfect complements. It is more ... 4 I will denote the demand function by Q(p) and the inverse demand function by P(q). Then$$ \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 ...

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: $$\ln D_t = ... 3 When we say that AD=AE we don’t mean that aggregate demand function is equal to aggregate expenditure - we mean by that the actual total quantity (Q) demanded on the market given the aggregate prices at a specific period of time is equal to aggregate expenditure. Aggregate expenditure is equal to the sum of all spending on output (Q) sold in the economy at ... 3 The main reason why long run aggregate supply is vertical is that in the end the production capacity of every country is limited. In the end there is always some maximum number of number of stuff we can produce (of course, there can be economic growth which expands our production possibilities but the LRAS is basically given by the production possibility ... 3 This is a common problem in many domains. For example, there's massive new demand of Airbus planes after the Boeing 737 Max 8 disaster, and their manufacturing capacity is maxed out. I'm familiar with it from the domain of software development, so I'll use that as an example. Suppose some website gets mentioned on Reddit and the post happens to hit the ... 3 If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over n goods, x_1,\dots,x_n) takes the form u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho}, \end{... 3 As for your first question: income elasticity of demand is just a percentage change in quantity demanded divided by a percentage change in demand. If you divide two things that are equal you get one: \frac{a}{b}=1 \iff a=b (as long as b \neq 0). Same thing goes for income elasticity of demand, 1 is not just some random value that was chosen to separate ... 3 From (1) and (2) you get$$\frac{x_j}{x_i}=\frac{a_j p_i}{a_i p_j},$$or equivalently,$$x_j =\frac{a_j p_i}{a_i p_j} x_i.$$Substituting this into equation 3 for j=2,...,n and i=1 (solving for the demand function for good 1) we get$$M=p_1x_1 + \sum_{j=2}^n p_j \frac{a_j p_1}{a_1 p_j} x_1M=p_1x_1 + \sum_{j=2}^n \frac{a_j p_1}{a_1} x_1M=...

3

Assuming certain regularity conditions, the first order conditions for $$\max_{x, \lambda} U(x) - \lambda (p \cdot x - m)$$ are \begin{align*} &D_{x}U(x(p, m)) - \lambda p = 0 \\ \text{and} \quad & p \cdot x(p, m) - m = 0. \end{align*} Moreover $x(p, m)$ will be differentiable with respect to $m$ at $(p, m)$, and this fact together with the ...

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 ...

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.

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, $$\frac{p}{D(p)}\frac{d D(p)}{dp}=\frac{p}{p^\beta}(p^\beta)'=p^{1-\beta}\beta p^{\beta-1}=\beta.$$ ...

3

Since there is no change in consumer preferences, the demand function will not change. The firms supplying skateboards face increased costs due to the wheels being more expensive. This will alter their supply function and thus the equilibrium price in the skateboard market may change, and hence the equilibrium quantity demanded may also change.

2

You just need to use the condition $$MRS_{q_{1j},q_{2j}} = \frac{p_{1}}{p_{2}}$$ to obtain $$3 \frac{q_{2j}}{q_{1j}} = \frac{p_{1}}{p_{2}} \;\;\;\; \text{(1)}$$ Then solving for $p_{1}$ and plugging this into the budget constraint you obtain: $$y = 3p_{2} \frac{q_{2j}}{q_{1j}}q_{1j} - p_{2}q_{2j}$$ $$\Rightarrow q_{2j} = \frac{y/4}{p_{2}}$$ Accordingly,...

2

You have to use the midpoint method to resolve this (if I recall correctly). https://quickonomics.com/how-to-calculate-price-elasticities-using-the-midpoint-formula/ The reason I may not recall correctly is because the second you introduce calculus into the study of economics, you discard these formulae entirely and (try to) forget they ever existed. You ...

2

Exacerbating the short-run supply matter, these kinds of masks aren't actually for healthy people to keep pathogens away. They're for sick people to keep from spreading pathogens. So not only is "appropriate" demand spiking due to the virus spreading, "mistaken" demand is spiking due to fear - and I am comfortable making the data-less claim that this ...

2

Because higher wages do not necessarily mean that national income increased. From macroeconomic perspective national income is output. Higher wages in these simple models do not necessarily translate into higher output. For example if these higher wages come from company’s profit then on one hand one (worker) household income increases but another (...

2

Demand side effects: Effects to the economy whenever the demand-side factors (the components of GDP [in the abstracts case, Government spending]) either increases or decreases. Demand AD (GDP) = C + I + G +X-M https://www.economicshelp.org/blog/2671/economics/factors-affecting-economic-growth/

2

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 $v \not = \alpha p$ for any scalar $\alpha$ (i.e. $S$ is negative definite for all vectors except those proportional to price), then the Weak Axiom holds.

2

This is one possible interpretation. Good 2 being removed from the market can simply be interpreted as $x_2 = 0$. In an economic interpretation the good does not simply disappear from the utility function in the sense that preferences do not change, it is just the availability of the good that changes. This is an external condition, so you can simply think ...

2

Argue that, given your assumptions on the utility function, $x^*$ is the essentially unique (and hence global) maximum. (You need this because there may be local maxima when the assumptions on the utility function is relaxed - this will violate the proposition you're trying to prove). Now simply use the definition of global optima: for any $x\leq x^*$, \$u(x)...

1

Consumers buy airline tickets but don't hire pilots, so I wouldn't call those two complementary in the usual sense of "complementary goods".

1

This may just be a semantic misunderstanding. The Law of Diminishing Marginal Utility (LDMU) does "affect" the price in the sense that it is responsible for the convexity of preferences, which in turn guarantees a well-defined price effect. The formulation "price effect = income effect + substitution effect" is just a short way of saying that the change in ...

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