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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 }...


4

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=...


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 ...


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

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 ...


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

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,...


1

The Income Offer Curve (which is the same as the Income Expansion Path) shows us the effect of a change in nominal money income on the consumption of both goods (in a 2 good model) in the real 2 good indifference curve space. Thus, to derive the income offer curve, one shifts the budget constraint by varying money income, and joins all the points of ...


1

Both of these curves describe the same phenomenon, the change in consumer choice as income changes. The difference is in the displayed variables. The Engel curve is a relationship between the consumption of a good $x$ and income $I$, whereas the income expansion path (IEP) shows the relationship between the consumption of a good $x$ and a good $y$ as income ...


1

As mentioned by @Henry: investment is itself a demand for capital goods (physical, human, etc.). If I decide to buy trucks this manifests itself as an increase in demand to the truck factory.


1

It seems that your confusion is more about classifying goods into necessities vs. luxuries. In economics, the criterion for such classification is to look at whether the income (not price) elasticity of demand is "high" or not. If a good has a high (and positive) income elasticity, then it's a luxury; if it has a low (but still positive) income elasticity, ...


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