6
Completeness: given any pair, I have a preference, I can make a choice.
If had to choose between marrying Rachel and Monica, I would go for Rachel. Good looks, fun, etc.
If had to choose between marrying Chandler and Rachel, I would go for Chandler. Corny sense of humor but aware of it, even temperament, etc.
If had to choose between marrying Monica and ...
6
I'm quite surprised nobody has picked the obvious one:
I prefer rock over scissors.
I prefer scissors over paper.
I prefer paper over rock.
Complete, definitely not transitive.
6
I used to think that microeconomics was relatively more scientific than macroeconomics. Early macroeconomics, in particular, suffered from a choosing models of agent behavior that made modeling easier but bore little relationship (or at least a highly disputed relationship) with reasonable behavior of individual agents. This led to the microfoundations ...
5
Does quasi-concave utility function imply convex indifference curve?
No that is not true. Consider $u(x, y) = -x^2 - y^2$ defined on $\mathbb{R}^2_+$. Since $u$ is concave it is quasiconcave. Observing the graph of the indifference curves, we see that ICs of $u$ are not "convex".
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
One way to think about this utility function is "I can only use one of these two goods, and I will use the one that I have more of"
Some examples:
Choosing between two sets of equally-valued but non-matching dishware. Once you have one set, the other one is useless to you.
If your budget constraint is such that you can only have one kind of platform of ...
3
What you're asking for is equivalent to finding an injective function $f:\mathbb R^n\to\mathbb R$ that is monotone in the sense that if $x$ is coordinate-wise at most as big as $y$, then $f(x)\le f(y)$.
As a first step, notice that this is equivalent to finding such a function from $(0,1)^n\to\mathbb (0,1)$.
And this is easy by interleaving the digits of the ...
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
Those who decry the status of macroeconomics (to a certain extent), e.g. Krugman, do so because the macro field being closer to high-level political decision making is often involved in controversies, even if only a small part of the macro field itself is actually controversial (according to Krugman, 2013).
OK, here’s how I read the Gordon and Dahl ...
3
Mathematically, most neoclassical models assume that the profits are concave. This guaranties the uniqueness of the maximum. In economic terms, the Neoclassical theory usually assumes that the law of diminishing returns holds. Thus, the more you hire workers, the more you produce, but at a diminishing rate.
I am not so sure what your professor has in mind ...
3
What you're doing when you calculate $dQ/dP$ is assuming that all forms change their process by $dP$. This is the same as saying that the market price changes by $dP$ and is why you get the same number.
The main difference here is that when the book says price elasticity of demand is higher for an individual firm, it means that, holding the prices of other ...
3
Publicly funded healthcare and social welfare schemes typically serve redistributive functions and therefore can be considered as promoting equity (among the well-off and the poor). At the same time, these programs must be financed through taxation such as income tax that's usually distortive, thereby weighing down on efficiency.
3
For some intuition, rewrite the problem as the consumption problem $\max_{(c, l)\in\mathbb{R}_{+}^{2}} (cl)^{0.5}$ subject to the constraint $c/w + l \leq 1$.
The optimal level of $l$ is then given by $l = 1/2$ as per the usual solution for Cobb-Douglas preferences.
The intuition for why $l$ does not depend on $w$ in this example is the same intuition for ...
2
For the Nash equilibrium of any simultaneous-play game, you are looking for the point where each player is playing a best response to all other players at the same time.
So your steps to solving this game should be:
Determine player 1's best response function ($g_1$ as a function of $g_2$)
Determine player 2's best response function ($g_2$ as a function of ...
2
It depends on the function. A non-monotonic function with satiation:
$$
U(x_1,x_2) = 0.
$$
A non-monotonic function without (global) satiation:
$$
U(x_1,x_2) = \left \lfloor{x_1}\right \rfloor + \left \lfloor{x_2}\right \rfloor .
$$
2
The Wikipedia article shows it graphically. But here, you're leaving out the important bit: "The property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is preferred to it." This is the statement that you want to connect with the math.
So how does this relate?...
2
It is possible to use a Lagrangian to obtain your Marshallian demands, provided you break each min function up into two different pieces and remain wary of boundary solutions. So for example, if you impose the condition $x_1<\frac{a}{2}, x_2<\frac{b}{2}$ you can simplify your utility function to $x_1 x_2 + x_3$, and then solve for Marshallian demands ...
2
This seems like a homework question, so I'll just give hints.
By definition, a quasilinear utility has the form $u(x, y) = x + v(y)$ where $y$ is a vector of all other goods and $v(\cdot)$ is strictly concave. In this case, $x$ is called the numeraire.
From utility maximization, what's the first order condition that relates $MU_x$, $MU_y$, $p_x$, and $p_y$?...
2
Here's an example of an incomplete but transitive preference.
Consider three fruits, an apple ($A$), a banana ($B$), and a coconut ($C$). I cannot choose between individual fruits, i.e. I don't have a preference over $A$, $B$, or $C$ --- not that I'm indifferent between them, I just can't compare them. However, I do prefer more variety to less, namely, I'd ...
2
First, assume risk aversion. By the definition of the certainty equivalent and Jensen's:
$$u(CE(u,F))=E(u(x))<u(E(x))$$
Now, from monotonicity:
$$CE<E(x)$$
Second, assume $CE<E(x)$. By monotonicity and the definition of $CE$:
$$u(E(x))>u(CE)=E(u(x))$$
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.
1
Several things:
It is not clear that labor productivity as a share of GDP tracks allocative efficiency. For example, it could rise simply because the returns to capital (capital's share of GDP) are falling, which would most obviously be a consequence of allocative inefficiency through over-investment in capital inputs. There are potentially many other ...
1
The idea that the long run average cost curve (LRAC) must pass through the minimum points of the short run average cost curves (SRAC) is a fallacy, but it seems to be a remarkably plausible one. It was the source of a famous error by the economist Jacob Viner, referred to in this paper by Silberberg. Underlying the fallacy is perhaps an assumption that the ...
1
Adam Bailey is correct.
Consider the production function $f(x_1,x_2) = x_1 + x_2/2$ where $(x_1,x_2)$ are inputs.
If the input costs are $w_1=w_2=1$ and all inputs are freely chosen, the solution to the cost minimization problem is
\begin{align*}
x_1 & = y \\
\\
x_2 & = 0.
\end{align*}
However, in the short run one or more of the input quantities ...
1
It may be helpful to consider a phenomenon from politics called the Condorcet paradox. This is a situation that can happen in votes, in which the overall population would vote for A over B, would vote for B over C, and would vote for C over A. It is not a purely theoretical possiblity: it is the present reality in the UK over the best resolution to the ...
1
I`d propose you to follow these steps:
Set up the minimization cost problem (i.e. for a given output quantity $y$ minimize costs):
\begin{align}
\min_{H,L,K}& \quad sH + wL + rK \tag{1} \label{1}\\
\text{such that} &\quad \min\{H,L\} + \min\{H, K\}\geq y \tag{2} \label{2}
\end{align}
In principle you have 3 cases, depending on price of factors $(s,...
1
In the original problem, for $L=2$, the consumption set was $(-\infty,\infty) \times \mathbb{R}_{+}$. Now, the consumer is restricted to $[0,\infty)\times \mathbb{R}_{+} = \mathbb{R}_{+}^2$. Fix a price $p = (1,p_2)$, and now see what happens to Walrasian demand (i.e. $x(p,w) = (x_1,x_2))$ as we vary $w$.
You saw from part (a) that a consumer with these ...
1
A shortcut to solve $$\min_{x,y} x^{\alpha} y^{\beta}$$ s.t. $$xp_x + yp_x \leq m$$ and $$0 \leq x,y$$ is to set $x = 0$ and/or $y = 0$, as this results in $U(x,y) = 0$, and the function does not map to negative values.
1
I don't think it is a case. On a page you linked we have a proof that the profit function is non-decreasing in $p$. If the output price decreases, $p \geq p'$ and factor prices remain constant, $w_i \leq w'_i$ for all inputs, then the profit would be less or equal to the previous one.
Then, in connection to your previous question: How to prove that a ...
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