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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 }... 7 You can solve it sequentially by noting the nesting structure of the utility function U. So first note that the utility function combines utility functions you are probably already familiar with U=\min\{u_1,u_2\} of complements and u_1=\sqrt{x+y} and u_2 = z+w both of which are perfect substitutes. Where u_1 and u_2 are nested within U. The ... 6 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 ... 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 ...

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

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

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 ...

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 ...

4

In general the demand for a certain good (say from a consumer) can be written as a function of the prices of all available goods and the total amount of money that the consumer has available. Take the setting of two goods, $q_1$ and $q_2$ with prices $p_1$ and $p_2$ and total income $y$. Then the demands can be written as: $$q_1 = d_1(p_1, p_2, y)\\ q_2 = ... 4 Nuance matters: In the comments under 1muflon1's answer the quote given is The demand curve represents marginal benefit. The vertical distance at each quantity shows the mount consumers are willing to pay for that unit. Willingness to pay reflects the benefit derived from each unit. So the actual claim is not that the demand curve is the same as the ... 4 To provide another answer with less equations: Consider first that the inner utility function u_1 = \sqrt{x+y} and u_2 = z + w are perfect substitutes implying that consumer only buy the cheapest of x and y and similarly only the cheapest of z and w. Because prices are given we know that y is cheaper than x and w is cheaper than z. Using ... 4 Assuming you mean unit elasticity of demand with respect to price the answer is yes. From the information you have, we can deduce that the demand function is as follows:$$Q^d(p)=10/p$$(I always spend \\\ 10 on coffee, which buys me exactly 10/p units of coffee, where p is the price of coffee). Now recall the formula for computing elasticity of ... 4 Below is a graph of the price offer curve of good y when income is 48 units, p_x = 8 and the utility function is$$ U(x,y) = \min\left(2x+2y,x+10\right). $$(Based on "Simple Utility Functions with Giffen Demand" by Sørensen). Good y exhibits Giffen behavior when 0 < p_y < 8. A gif of the optimal choice changing as p_y changes ... 3 This is a common problem in many domains. For example, there's massive new demand of Airbus planes after the Boeing 737 Max 8 disaster, and their manufacturing capacity is maxed out. I'm familiar with it from the domain of software development, so I'll use that as an example. Suppose some website gets mentioned on Reddit and the post happens to hit the ... 3 If you take the general class of CES utility functions, of which Cobb-Douglas is a special case, you do indeed get a demand function that depends on other prices. Specifically, the CES utility function (over n goods, x_1,\dots,x_n) takes the form u(x_1,\dots,x_n)=\bigl[\alpha_1x_1^\rho+\cdots+\alpha_nx_n^\rho\bigr]^{1/\rho}, \end{... 3 Assuming certain regularity conditions, the first order conditions for$$ \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 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 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

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.

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