9

A good is normal if its demand is increasing in income. So let $p_x$ and $p_y$ be the price of the goods with quantities $x$ and $y$ and let $m$ be income. Suppose $ax>by$. Then $\min\{ax,by\}=by$. By slightly reducing $x$ by and spending the saved money on $y$, one gets a better bundle. For an optimal bundle, this cannot be. Similarly, it cannot be ...


3

Suppose you have $M=20$ to spend on shoes. Left shoes cost $p_L=5$ and right shoes cost $p_R=5$. A bundle, $(x_L,x_R)$ consists of $x_L$ pairs of left shoes and $x_R$ pairs of right shoes. How would you figure out how many of each type of shoe to buy? Probably the reasoning would go like this: You only ever want to buy shoes in pairs (consisting of a left ...


3

I would suggest that ask yourself the following questions (hopefully, this should help you figure out how to solve the problem) : If good $z \in (x,y)$ was free, what would be the demand for both agents? Is the conjunction of these demands feasible given the endowments? (this should allow you to rule out one of the cases) If the demands are feasible, which ...


3

Air is a complement good for a lot of things. If you get no air at all then food, gold and iPhones give you no utility either. And air is free! Except in Spaceballs...


3

Set of Pareto Efficient Allocations consists of feasible allocations $((x_J, y_J), (x_D, y_D))$ satisfying $y_J=\displaystyle\frac{x_J}{2}$. Competitive Equilibrium is the price $(p_x, p_y=1)$ satisfying the following conditions: Budget Requirement: $p_xx_J+ y_J = 2p_x + 2$ and $p_xx_D+ y_D = 4p_x + 1$ Equilibrium Conditions: $\displaystyle\frac{2y_J}{x_J} ...


2

Sounds pretty good to me. I think it would be more coherent to change the way you say this: As demand for ice cream rises, the demand curve for it shifts to the right, and assuming the supply curve for it remains still, this shift will increase both the price and quantity of ice cream. In turn, as ice cream prices rise, producers make more of it, and thus ...


2

Perfect complements is equivalent to Leontief utility: $U(x,y) = min(x/a_x, y/a_y)$ The MRS is defined as: $MRS_x,y = MU_x / MU_y$ Since this utility function is not differentiable the concept of marginal substitution is not well defined for Derek. However, we don't need marginal arguments for Derek to solve the problem. A function doesn't have to be ...


2

Alex's preferences of sugar and lemons can be expressed in form of a utility function as: $U(x,y)=min(x/2,y) $ where $x$ is sugar and $y$ is lemons.This function tells us we need at least 2 spoons of sugar to consume 1 lemon. we need a minimum of 2 sugars initally so we can say $x=2$ for 1 lemon so $y=1$ Plugging those values into our budget constraint: ...


2

Since demand equals supply holds for every price $p$, this simply means that every $p$ is an equilibrium price. However, the equilibrium allocation that $p$ supports varies with $p$. To be precise, price $p$ supports the allocation in which 1 consumes $\left(\frac{1+3p}{1+p}, \frac{1+3p}{1+p}\right)$ and 2 consumes $\left(\frac{3+p}{1+p}, \frac{3+p}{1+p}\...


2

$u(x, y) = -\max(x, y) = \min(-x, -y)$ is a concave function. Since it is concave, it is also quasiconcave (or equivalently, it represents weakly convex preferences). Here is the indifference map of $u$ : $u$ does not represent the same preference as $v(x, y) =\min(x, y)$. Here is the indifference map of $v$ :


2

Instead of directly giving you the answers, I'm going to give you a series of hints to help you figure out the answers on your own. 1.What would be the shape of the indifference curve? Consider three different consumption bundles: $(3,10)$, $(10,10)$, and $(10,3)$. Verify that these three bundles yield the same utility to the consumer. In other words, ...


2

Public goods are goods which have a fixed cost and are sometimes free for businesses. Think of police. They patrol streets, protecting businesses premises. Businesses do not pay directly for these services, except in very general terms with taxation and land rates. But these are fixed costs in the sense that the provision of the service do not depend on the ...


1

If Sally only prefers A or B but not both, this could be an example of Independent Goods. These are goods that are neither complements nor substitutes, and changes with one good generally do not affect the other. However, with the information about Sally's preferences, it sounds to me like it could also be an example of Cross-Category Substitute Goods. ...


1

Although one can derive the Hicksian demands by solving the expenditure minimisation problem, but the Leontief function is not differentiable at the 'kink', or at the 'point of optimality'. Thus one has to resort to simpler algebraic manipulations to find out the Hicksian demands. Let $\textbf{p}\in\mathbb{R_{++}^2}$(that is both $p_1 \ \& \ p_2$ are $&...


1

Hicksian demand is the consumption bundle that minimizes the expenditure of the consumer subject to the constraint that he attains some target level of satisfaction in equilibrium. In the problem, the expenditure on any bundle $(x, y)$ is given by $p_Xx + p_Yy$ and the target level of satisfaction is $\mu$. Given that the utility function is $u(x, y) = \min(...


1

You can solve this question by breaking the Utility function into 2 parts. Use U(x,y) = i) 6x+y if 6x+y < x+2y ii) x+2y if x+2y < 6x+y This would simplify into the Utility Function U(x,y) = i) 6x+y if 5x < y ii) x+2y if 5x > y When you graph the function you'll get 5x=y as the line of kinks. When 5x>y, that is, to the right of the line of ...


1

If the prices $p_1$ and $p_2$ are positive than as you pointed out the equations \begin{eqnarray*} \frac{30p_1}{p_1+p_2} + \frac{20p_2}{p_1+4p_2} & = & 30 \\ \\ \frac{30p_1}{p_1+p_2} + \frac{4 \cdot 20p_2}{p_1+4p_2} & = & 20. \end{eqnarray*} hold. This is troublesome, because subtracting the first equation from the second yields \...


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