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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 $$\frac{\partial q_1(p_1,p_2)}{\partial p_1}=(\frac{\partial p_1(q_1,q_2)}{\partial q_1 }... 6 The supply of masks can't keep up with soaring demand because mask producers have capacity constraints in the short run. See this recent article. 6 You don't need to use any fancy theorem, the trick is to disentangle the definitions. Everything follows directly from the definitions. x^0=x^*(p^0,w) means that p^0x^0\leq w and that u(x)\leq u(x^0) for all x such that p^0x\leq w. In words x_0 is affordable and at least as good as every other affordable bundle (in many cases, better and not ... 6 That was tricky. The idea is as follows: First, under standard assumptions, demand is continuous. If you change prices a little bit, demand will not change a lot. In particular, if your excess demand for the first good was initially strictly positive, it will still be strictly positive for small changes in price. Now, the new bundle after the price was ... 5 I don't believe it is lower semicontinous. Let w = (0,\dots,0), p \in \mathbb{R}^n_+ be any vector such that p_1 = 0 (the first coordinate being 0). The allocation x=(1,0,\dots,0) \in B(p,w). Define the sequence p_n = p + (\frac{1}{n},0,\dots,0) and w_n = (\frac{1}{n},0,\dots,0). w_n \rightarrow w and p_n \rightarrow p. For any x^n \in B(... 5 Your Lagrangian would be$$L = (ax+by)+\lambda (I−p_x x−p_y y) +\mu_x(x−0)+\mu_y(y-0),$$where the final two terms represent the restriction that x,y\geq0. You then arrive at conditions$$\frac{\partial L}{\partial x}= a -\lambda p_x +\mu_x=0\frac{\partial L}{\partial y}= b -\lambda p_y +\mu_y=0I=p_x x+p_y y$$and complementary slackness ... 5 To solve$$\max_{x \geq 0} \ (x_1+1)^\alpha(x_2 + 1)^\betas.t. \ \ I \geq p_1x_1 + p_2x_2,$$I would define y_1 = x_1+1 and y_2 = x_2 + 1 to get the problem$$\max_{y \geq 0} \ y_1^\alpha y_2^\betas.t. \ \ \bar I \geq p_1y_1 + p_2y_2,$$where \bar I := I + p_1 + p_2. For \alpha + \beta = 1 the solution is well known to be$$y_1^* = \frac{\...

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 ...

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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 ...

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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 ...

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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

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 = ...

4

The word monotonic means "always moving in the same direction", in our case, always going up. Monotonic preferences mean that the customer always prefers more of a good. This comes in two flavors: Strictly monotonic: More of one good is always preferred to less of that good. Weakly monotonic: More of one good is always equivalent or better than ...

4

Demand is positive, so $A>0$. If $p_1$ goes to $\infty$, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\alpha < 0$. If $p_2$ goes to 0, $x_1$ cannot go to $\infty$, since $p_1x_1$ is bounded by $M$. Thus $\beta\ge 0$. If $M$ goes to 0, $x_1$ has to go to 0, since $p_1x_1$ is bounded by $M$. Thus $\gamma > 0$. If both prices and ...

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{...

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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

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 ...

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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

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 ...

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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 $$M=p_1x_1 + \sum_{j=2}^n p_j \frac{a_j p_1}{a_1 p_j} x_1$$ $$M=p_1x_1 + \sum_{j=2}^n \frac{a_j p_1}{a_1} x_1$$ $$M=... 3 One approach could be the following. For a (p_n,w_n) in the sequence and x \in B(p,w) define:$$ \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.$$... 3 None if the axiom is to be on preferences, as any smooth utility representation can be monotone transformed into a non-smooth utility function. 3 Answer to the Question on Welfare The welfare analysis is not as simple as that. First, let us set aside for a second any inequality considerations (we can add them on later but there are some misconceptions about welfare analysis that have to be corrected first). Welfare in supply-demand analysis is conventionally measured by amount of consumer and ... 3 You can find demand functions like this in textbook and exam problems. Just a random example from internet:$$Q = \ln 4 - 0.5 \ln P$$which is just a special case of Q=−1/a \ln (P/b) where a is 1 and b is also 1 and there is some additional constant \ln 4. We just had an exam where I also saw one of the other examples. If I remember right it was Q = \... 3 Why don't you just plug in some values for x_1 and x_2? Start with something like x_1=1, x_2=1 and find utility u(1,1)=1+5*1. Then increase x_1 or x_2 and let x_1=2, x_2=1 and find utility u(2,1)=1+5*2=11, and x_1=1, x_2=2 and find utility u(1,2)=1+5*2=11. The utilities of both bundles are identical, and the goods seem to be substitutable.... 3 There are many functions where absolute value of elasticity is decreasing and which are continuous closed form. One example of such function would be:$$Q= a-\ln(p), p\geq1 \implies EL = - \frac{1}{p}\frac{p}{a-\ln p} = -\frac{1}{a - \ln p},  which is continuous closed form and decreasing in price in its absolute value. Original Answer: Originally I ...

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