7

Not really, you're right in that (loosely speaking) the MRS is the amount of one good someone is willing to give up in order to get an additional unit of another good. However, the slope of the budget line measures the amount of one good someone has to give up in order to get an additional unit of another good. In the first case, only preferences matter, ...


7

The first order stochastic dominance relation is convex. An easy way to prove this is to use the property that a cdf $F$ FOSD another cdf $G$ if and only if $F(x)\le G(x)$ for all $x$. That is, $F$ FOSD $G$ if and only if the graph of $F$ is never above the graph of $G$. It is then easy to show that $F$ is never above any convex combination $H(x)=\alpha F(...


7

Sure you can, just that your interpretation of your variables in your analysis changes however. In this case you are analyzing how investment in differing factors of production affect output. I'd recommend that you may want to estimate a more flexible functional form like the Translog Production Function to check if your function is CES instead of just a ...


7

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.


7

This looks like constant relative risk aversion (CRRA) utility . Usually CRRA is written like $U = \frac{C^{1-\rho}-1}{1-\rho} $ (I omitted second part for brevity) in your case $a=1-\rho$. $\rho$ is the relative risk aversion. By extension $a$ is the function of $\rho$ so as $a$ increases (due to smaller $\rho$) the person should become less risk averse.


6

The indifference curves are not "parallel", as they are not straight lines. They are however shifted, that is they are supposed to maintain vertical distance regardless of the value of $x$. The curves you map maintain horizontal distance regardless of $y$. That is because the non-linear variable here is $y$, not $x$. The curves are still shifted, but along ...


6

Not all cdf’s have a density function, (for example if $F$ is not differentiable). However, when they do have a density, the notation $dF(z)$ is equivalent to $f(z)dz$. When performing integrals. However, even if the density does not exists, you can still write the expectation using the notation $dF(z)$. The details of what it actually means and the subtle ...


6

Gains and losses presuppose a reference point, which is not a feature in standard expected utility theory. In this theory, the only argument in the utility over wealth is $w$, the absolute level of wealth. A common form of utility function is the constant relative risk aversion (CRRA) form: \begin{equation} u(w)=\frac{w^{1-\rho}}{1-\rho}, \end{equation} ...


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 ...


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 ...


5

Ask your teacher what he meant, because there is either a misunderstanding or he is mistaken. Producer's surplus will come down to $y \cdot p - VC(y)$. In the first graph, by definition of $AVC$ we have $$ y \cdot (p - AVC(y)) = y \cdot p - VC(y). $$ In the second graph, using $\int MC = VC$ and $VC(0) = 0$, we have $$ \int_0^y p - MC(x) \text{d} x = \...


5

As many have stated before me, you can't just throw electricity away if you produce too much—it has to go somewhere. If you put more power into the system than the resistant (or consumption) in the system, it is like spinning your bike faster while you bike downhill, the generators start spinning faster, and the frequency increase above 50 Hz. The second ...


5

If only weak-ordering and continuity is assumed, ICs can definitely intersect. This is not true. First, if you're speaking of indifference curves, you'd already be assuming either local non-satiation or monotonicity. Let's speak of indifference sets instead. The analogue of two sets, $I_1$ and $I_2$, "crossing" each other can be formalized as $I_1\ne I_2$ ...


5

You are correct that an uniformly increasing cost of production cannot exist alongside economies of scale. A constant returns to scale production function is homogeneous of degree one: $$ f(\lambda x, \lambda y) = \lambda \cdot f(x, y)$$ That is, when you double all the inputs you also double the output. An increasing returns to scale production function ...


5

I feel that your question might be a little broad, but there are certainly many areas of economics where non-functional relations are used. Two simple examples (there are many others): One of the most fundamental models of behavior in economics relies on the idea that choices can be represented by "preferences" which mathematically are binary relations that ...


5

$AVC<AR$ means, without considering fixed cost, the firm is making a profit of $AR-AVC>0$ per unit of output. Compare the two options: keep producing vs shutdown: Keep producing: $\text{Avg Profit}=\underbrace{AR-AVC}_{>0}-AFC$ Shutdown: $\text{Avg Profit}=0-AFC$ Since $AVC<AR$, staying in production is better since the revenues can be used ...


5

Just a model that can be used to state how (in)complete a particular contract is, whatever the reason. I remember a debate at the end of the 90's on Incomplete Contracts: Where do We Stand? by Jean Tirole and Foundations of Incomplete Contracts by Hart and Moore, where they develop a model that provides a rigorous foundation for the idea that contracts are ...


5

Let me pick up the discussion in the comments. Consider any diagonal through the origin. Suppose it intersects some IC more than once. Pick two of these points and call them A and B. Because they are on a straight increasing line, one of the bundles, say B, has more of both good 1 and good 2. By monotonicity, bundle B should be preferred over A, which ...


5

Reason: Both goods cannot be inferior. Let's say originally you consume $x$ and $y$. So your budget constraint looks like $$p_x x + p_y y = I.$$ If both X and Y are inferior, when income goes down from $I_0$ to $I'$, the quantity demanded for both has to go up (by definition) from $x$ to $x'$ and $y$ to $y'$. This implies $$p_x x' + p_y y' = I' < I = ...


5

If prices are constant then quantities are proportional to expenditures. Consider : $$ Y=AK^{\alpha}L^{\beta} = A(\frac{E_{K}}{r})^{\alpha}(\frac{E_{L}}{w})^{\beta} $$ $$ = (\frac{A}{r^\alpha w^\alpha})(E_{K})^{\alpha}(E_{L})^{\beta} $$ $$ = \tilde{A}(E_{K})^{\alpha}(E_{L})^{\beta} $$ If prices don't vary too much this may be an acceptable approximation. ...


5

Convexity can be a very important assumption as much of economic analysis is built on working with convex sets, which makes things easier. Very importantly, convexity of sets allows us to work with the Separating Hyperplane and Supporting Hyperplane theorems, which have applications to many results in economics, as partially discussed here and here. Here ...


5

If the utility is continuous and locally insatiable then the Hicksian demand equals the Walrasian demand. So you'll need to look for a utility function which violates atleast one of these. Lets try a simple violation of non-satiation $$ u(x,y) = \left\{ \begin{array}{ll} x+y & \quad x+y \leq 1 \\ 1 & \quad \text{ ...


5

Remember that utility function only matters ordinally... that is, you only care if $u(A) > u(B)$, and $u(A) = 100$ with $u(B) = 10$ is the same thing as $u(A) = -1000$ and $u(B) = -1000.0001$ (both have $u(A) > u(B)$.)


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".


5

Complete market is a market where every possible asset or good can be assigned a price and where you have perfect information, can make perfect contracts and zero transaction costs. Any market can be complete regardless of its market structure. So you can have complete market dominated by monopoly, or oligopoly or monopolistic competition etc. Perfectly ...


5

One interpretation I can offer. The demand function can be expressed as: $$Q_1 = Q_1(p_1,p_2)$$ Let us take the total differential: $$dQ_1 = \frac{\partial Q_1(p_1,p_2)}{\partial p_1}dp_1+\frac{\partial Q_1(p_1,p_2)}{\partial p_2}dp_2$$ Assume that $Q_1$ remains unchanged with respect to a change in prices. This implies that $dQ_1=0$. Solving the ...


4

Why not using the number of workers? And simply replacing sales (or output) by workers in the HHI or C4 indices. I saw this in the literature, but where? May be in a report of the German "Monopolkomission"... EDIT: I found an example for France. The share $C_{10}$ of the production of the 10 biggest firms is very correlated with their share in total ...


4

You're right that it's a bit counterintuitive that the shape of the indifference curves shouldn't change when you transform the utility function. The reason is that you are transforming along an axis that is perpendicular to the plane where the indifference curve lives. Let's imagine we have two goods, x and y, and let's say that the original utility ...


4

The idea is indeed to Taylor expand the production function. To justify it, you can start with the constant elasticity of substitution function, which in the two-factor case can be written as $$ Y = A[\alpha K^\gamma + (1 - \alpha)L^\gamma]^{1/\gamma} \tag{1} $$ in this case $X_1 = K$, $X_2 = L$. Now we expand $\ln Y$ around $\gamma = 0$ (recall the CES ...


4

Globally, there is Lakner and Milanovik (2015)'s elephant graph: Hellebrandt and Mauro (2015) Thus, the two previous distributions look like bimodal log-normal distributions. or CDFs, as in MacAskill's book Doing Good Better Did not find something strictly related to wages. For most of people, income may be a good proxy of wages.


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