New answers tagged

0

In economics, as opposed to finance, people often use discount factor and interest rate sometimes interchangeably. This is specially when defining inter-temporal preferences. The idea stems from the fact that discount rate and interest rate measures a similar kind of trade-off: current utlity/wealth vs future utility/wealth. Interest rate, simply put, is the ...


0

Overview: interest rate is a broad term, discount rate is a specific type of interest An interest rate is a very broad term that applies to a host of assets. Interest rates are paid on mortgages, bonds, CDs, savings account, etc. The discount rate is a specific type of interest rate that applies to the rate of interest charged to banks borrowing from a ...


0

The sequence of bundles $\big\{\frac{1}{n}\frac{1}{n}\big\}_{n\in\mathbb{N}}$ converge to $(0,0)$ and each bundle in the seuqence is strictly preferred to $(0,0)$.


0

I think the answer above needs a little more clarification. The Envelope theorem is slightly more subtle than simple chain rule for derivatives. The idea is as follows: $\theta$ is a parameter that is held fixed in the optimization problem. So the optimal solution for $q,r$ depends on $\theta$ and is hence written as $q(\theta)$ and $r(\theta)$ and the ...


0

I'm guessing your inquiry is concerning the chain rule of total derivatives. In the special case that $B(q-r)$ is for some reason omitted out of the expression, the derivation sequence would be this. $$\frac{\partial{u}}{{\partial\theta}}= \frac{\partial{u}}{{\partial{C}}}\frac{\partial{C}}{{\partial\theta}} = -1*\frac{\partial{C}}{{\partial\theta}}=-\frac{\...


-2

Both the technologies are CRS technologies. Cost of one unit of output in case of complements is $w_1 + 2w_2$ and in case of substitutes it is $\min(w_1, 2w_2)$. So cost functions are respectively, $c_1(y, w_1, w_2) = (w_1 + 2w_2)y$ and $c_2(y, w_1, w_2)=\min(w_1, 2w_2)y$


4

As Michael Greinecker noted, the stag hunt is the leading example of a symmetric 2x2-game with a payoff-dominated but risk-dominant NE. In symmetric 2x2 coordination games, a pure NE is risk dominant iff it is the unique best reply to the mixture $(\frac12,\frac12)$. Since Level-0 types are usually assumed to mix uniformly over pure strategies, all higher-...


5

Level-k reasoning in the stag hunt game is analyzed in Gracia-Lázaro, Carlos, Luis Mario Floría, and Yamir Moreno. "Cognitive hierarchy theory and two-person games." Games 8.1 (2017): 1. The idea that playing $s$ guarantees its payoff is discussed in Aumann, Robert "Nash equilibria are not self-enforcing, in ‘‘Economic Decision-Making: Games, ...


2

I think what you are looking for is known as overselling.


3

This idea is known as the Fisher separation theorem. Without the investment opportunity to transfer $h$ units of present day value into $w(h)$ units of future value, the perfect credit market gives us the intertemporal budget constraint of $$ c_1 + \frac{c_2}{1+r} = y, $$ which can be represented by a straight line. Without knowledge of the consumer's ...


0

Well i just edited my answer a lot. I made a fundamentally mistake, from $ L_ {h} $ arises an unique value of h, even when $ h $ is in other first order conditions, that doesn't change anything. Once that i have clear that there is no relationship between $ \beta $ and $ h $, I think I know what is happening. The reason why it is invested in $ h $ although $ ...


0

I would recommend that you read what Harvey S. Rosen and Ted Gayer have to say about taxation and tax burdens on their book, Public Finance. Your hypotheses for perfectly inelastic demand might be flawed, to the extent that (1) the house market is competing with Airbnb-type companies and (2) -for inelastic supply- there is an affluence in housing available (...


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.


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 ...


2

If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem, $$ x c'(x) = k c(x).$$ That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is ...


3

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 ...


3

A possible approach is to find a compact set $Z$ of inputs and show that the PMP has an optimal solution if and only if the PMP has an optimal solution in $Z$. If so, we can replace the PMP by the following problem. $$max_{z \in Z} \,\,p f(z) - w z.$$ If $f$ is continuous and if $Z$ is compact, the existence of a solution follows from the Weierstrass theorem....


2

Intuitively, you'd want the profit function to "peak" at some finite vector $\mathbf z^*$. To ensure this, it's sufficient to require that the profit function $\pi(\mathbf z)=pf(\mathbf z)-\mathbf w\cdot\mathbf z$ be concave in $\mathbf z$, the production function $f$ be increasing and continuously differentiable in $\mathbf z$, and the ...


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(...


1

I’ll add just one caveat to the answers and comments already posted: Vast majority of economists do no treat commodities as vectors as we define in physics. It is simply because there is no way to interpret the “direction” of a commodity bundle! So you should rather imagine elements of the commodity space to be simply columns of numbers denoting the quantity ...


2

You should read the fundamentals. I strongly suggest two textbooks for microeconomics: Hal Varian - Microeconomics Snyder, Nicholson - Microeconomic Theory If your read any of the two and find it confusing, I suggest you go for: Mankiw, Taylor - Economics


1

One of the simplest specification I can think of (and for which the first order condition can be solved analytically in $L$) is: $$ y=\left\{ \begin{array}{ccc} L^{\alpha} & & L\leq L_{e} \\ L_{e}^{\alpha}+g\left( L-L_{e}\right) & & L>L_{e}% \end{array}% \right. $$ with $g\left( L-L_{e}\right) =(L-L_{e})^\beta$ and $\alpha\geq1$ and ...


1

I'm not sure but it seems to me that the logistic function $\frac{e^{x}}{1+e^{x}}$ could serve your purpose. You may need to scale it as its output falls between 0 and 1, but it does have an analytical derivative that you can then use to solve for the labour demand function.


4

Here is an answer that is probably confusing, but adds some advanced perspective. The first general proof of the first welfare theorem (due to Kenneth Arrow) that did not rely on calculus used the assumption of strict convexity. Tjalling Koopmans later introduced the assumption of local-nonsatiation, which has become the standard assumption in textbooks for ...


2

I would add to Michael's answer that convexity is important because it is needed to prove that there exists at least one competitive equilibria (and also to prove the second welfare theorem).


4

Are convex preferences needed for the first welfare theorem? No, convexity of preferences is imposed for other reasons. A general sufficient condition is local non-satiation, which says the agent can be made better off by an arbitrary small perturbation of his consumption bundle. This can hold without the preference being convex. It would see so. For ...


1

No one would predict players to play the mixed NE in this game. In general, any sensible prediction of the choices of rational and experienced players must be a NE, but not the other way round. In this game the mixed NE is unstable under every payoff monotonic learning dynamics. In my opinion the more interesting question is why it is so hard to rule out the ...


4

Take the situation where $u_1(x,y)=x+y$, $u_2(x,y)=x+y$. Then the contract curve would be the entire box.


2

First, important thing to note here is that $K/Y$ is not the capital income to labor income ratio, but ratio of capital to total income (or output which is macro-economically equivalent to income). $K$ is not an income derived from capital it is the stock of capital. Wars destroy the stock of capital so even though returns to capital $r$ indeed increase ...


7

One of the most important consideration for using CARA utility is tractability. CARA utility and Gaussian errors yield a certainty equivalent completely described by a simple function of the mean and variance of the Gaussian distribution.. It also turns out that maximizing expected utility in this setup is equivalent to maximizing the certainty equivalent -...


3

Oligopolistic competition through both quantity and price can be regarded as Stackelberg competition, provided that there is a leader firm that moves first (commits) and other players move subsequently with some information on what the leader has committed herself to. It thus describes the dynamic form of the classic Cournot and Bertrand competition. Some ...


0

The Answer can be found in the comments by user brunosalcedo: "Remember that elasticity measures percentual rates of change. How does q change per unit of percentual change of p? If $p=0$, then any percentual change of p results in no change at all because 0 multiplied by any number is 0. The elasticity is not just 0 at p=0, it is also continuous at $p=0$. ...


6

There is a series of papers that address precisely this question. The most famous ones are probably Walker and Wooders (2001) and Chiappori, Levitt, and Groseclose (2002) that deal with penalty kicks and tennis serves. Both papers conclude that the behavior of professional athletes is consistent with them playing g a mixed strategy equilibrium. A more recent ...


0

If you just have one good in your utility function, then you really dont need an indifference curve. The reason we use IC is because we cant draw 3-D graphs properly without a computer (Remember $U(x,y)$ requires 3 axes to plot: X-axis, Y-axis, and a Z-axis to plot the values of U). So indifference curve (which basically draws the contours of the utility ...


-1

@brunosalcedo: Is this because the discontinuity in firm B’s payoff whenever firm A sets a price less than (or equal to) firm B ‘s monopoly price? (Firm B will always try to undercut firm A by an arbitrarily small amount rather than share the market) If this is the case, I think you’re correct. Any SPNE involves firm B choosing its monopoly price, and firm ...


-1

Your logic is correct. You need one more step. Using the same logic you can show that, in any SPNE, firm $A$ must set a price greater than the price that maximizes $B$'s profits. Firm $B$ then chooses such price.


1

Consumers buy airline tickets but don't hire pilots, so I wouldn't call those two complementary in the usual sense of "complementary goods".


2

Let $f(U)=U^{6/5}$. This is a positive monotone transformation of $U$ on $\mathbb{R}_0^+$. So the preferences represented by $U$ are also represented by $V(x_1,x_2):=f(U(x_1,x_2))=x_1^{3/5}x_2^{2/5}$. The utility function $V$ has Cobb-Douglas form and you can use the formula for the Hicksian demand for Cobb-Douglas utilities: $$x_1^*=\left(\frac{3p_2}{2p_1}\...


4

There are multiple ways how the discount factor can be estimated. I dont think its possible to make exhaustive review of all of them (within format of this site at least), but one that nicely matches your question would be through estimating the Euler equations. Following Attanasio & Browning (2009) an Euler equation for general asset would be given by: ...


2

This is one possible interpretation. Good 2 being removed from the market can simply be interpreted as $x_2 = 0$. In an economic interpretation the good does not simply disappear from the utility function in the sense that preferences do not change, it is just the availability of the good that changes. This is an external condition, so you can simply think ...


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