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1

In your example, the playstation is a public good to be used by both roommates. Note that $\theta_1,\theta_2 < \theta_3$, but $\theta_1+\theta_2>\theta_3$, that is, one roommate alone would not buy it, but together it is efficient to buy. The idea of your VCG mechanism is that people only pay when they are "pivotal," that is, when they have ...

4

There is another way to compute the symmetric BNE in increasing strategy. Let $U(v)$ denote the expected utility of a player in equilibrium when his type is $v$: Given that the bidding strategy is increasing, a player with type $0$ will get the good with probability zero. Thus he/she must bid zero and $U(0) = 0$. For any other $v > 0$, the probability ...

1

Because by definition of short-run it is not possible. In economics, short-run is defined as a period when (some) factors/variables are fixed and not flexible. Consequently, by definition firm cannot exit or enter in the short-run as it cannot change it's fixed costs - for example firm prepaid rent and can't get the money back, or it takes few days to rent ...

3

This looks like a contest. There is a large economic literature on contests. Have a look at this survey by Corchón and Serena. Often these papers use a Tullock contest success function or model the contest as an all-pay auction, see, e.g., papers by Ron Siegel. There are papers that analyze given contests (research contests, lobbying, etc) and papers that ...

4

To expand on @1muflon1's answer. The theory of rational addiction assumes that the utility of a consumer at time instance $t$ depends both on current consumption of the addicitve good, say $c_t$, and the consumption of the addictive good in the past. For simplicity say $c_{t-1}$. So at period $t$ the instantaneous utility looks something like: $$u(c_t, c_{t-... 3 It is possible for an addict to be rational. A famous work on this was done by Becker (1988) Theory of Rational addiction. In order for agent to have rational preferences the preferences have to satisfy the following definition (See MWG Microeconomic Theory pp 6): Definition 1.B.1: The preference relation \succeq is rational if it possesses the following ... 2 Let x = D(p) be the demand for a good if the price is equals to p. The inverse demand curve (as you would draw it) is then given by p = D^{-1}(x). It gives the price as a function of the quantity. If there is a rebate of r and if p is the price, then the consumer only pays p^\ast = p - r. Then the demand is given by:$$ x = D(p^\ast) = D(p - r). $... 0 An attempt with adding an extra assumption: Let$0 \in V(0)$and$0 \notin V(y), \forall y>0$Since$V(y)$is set of all$x$that can at least produce$y$, we have that$V(y’) \subseteq V(y), \forall y’>y$Now let$x^*$(as defined in question)$\in V(y’)$. From definition:$wx^* \leq wx, \forall x \in V(y)$. Therefore, we also have that $$wx^* \leq wx,... 3 I don't think you need convexity. However, I think you do need to assume some monotonicity condition. The following should work (but might not be the minimal set of assumptions that provides the result). Consider the production possibility set V(.).$$ V(y) = \{x \in \mathbb{R}^n_+| x \text{ can produce } y\}.$$We assume that$V(y)$is a closed non-empty ... 1 No, it is not. The verbatime citation is The slope of this curve represents the rate at which the individual is willing to trade$x$for$y$while remaining equally well off. To trade$x$for$y$here means to give up some$\Delta x$to receive$\Delta y$per unit of$\Delta x$. Letting$\Delta x\rightarrow 0$the rate is$-\frac{dy}{dx}=MRS_{xy}$. 3 Production functions are defined without specific values for parameters, so they all could if you impose that the logical parameter implies a negative return. For example, consider a Cobb-Douglas production function of capital and labor,$Y=\beta_0 K^{\beta_k}L^{\beta_l}\omega \varepsilon$where$\omega$denotes firm-observed productivity and$\varepsilon$... 7 To follow up on the answer of @VARulle let me give you some conditions for which the indifference curve is path connected. The argument can also be found in the book Mathematical Methods and Models for Economists by Angel de la Fuente. Preferences are monotone if$x > y$implies$x \succ y$and that preferences are continuous if$x_n \succeq y_n$,$x_n \...

7

Given your last comment above it seems that what you are really asking is whether the indifference sets of a continuous preference relation on $\mathbb R^n_+$ are path-connected. The answer is No. Let $n=1$ and let the preference relation be represented by $u(x)=(1-x)^2$. Then the indifference set e.g. for $u=1$ is $\{0\}\cup\{2\}$, which is not path-...

3

There is most likely an assumption in the background that utility is increasing in the amount of flour, independently of its packaging. Then you are willing to exchange two 1kg-bags of flour for one 2kg-bag of flour. Thus the MRS is -2 (or -0.5, depending on the direction of exchange).

1

It was certainly a large part of the DOJ's case against Microsoft at the turn of the millennium. Get your favorite internet search tool and search for "doj v microsoft monopoly network effects" (no quotes) and you'll find the original complaint (https://www.justice.gov/atr/complaint-us-v-microsoft-corp): Microsoft has maintained a monopoly share (...

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