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Consider two situations: (i) a price so high that no consumer buys a good, (ii) the market price at which supply and demand are equal. CS=PS=0 in situation (i) and CS>0 and PS>0 in situation (ii). If you move from (i) to (ii) CS increases and PS as well. If you move from (ii) to (i) PS decreases and so does CS. If consumers cannot be forced to buy, the ...

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In my interpretation of the model, competitive firms imply that the wage always equals the expected productivity, which depends on the beliefs. Clearly, in any pooling equilibrium the on-path beliefs are equal to the prior such that the wage is simply $E[\theta]$. If, in your example, a worker sends the off-path messags $e=1$, the wage must be $w(e=1)=w_1=... 4 Your steps look okay, and the solutions are correct. You can rule out corner solutions by evaluating the utility function at the "corners" and compare those values to the optimum you found. You should be able to verify that$U(50,20)>\max\{U(0,120),\,U(60,0)\}$. 1 Elasticity of demand is normally considered in relation to market demand for a good, that is, the sum of individual customer demands. Different customers will probably respond in different ways to a price increase. If the increase is 5%, especially on a low value item, very likely many customers will not notice, some will notice but not change their buying ... 2 The standard elasticity still makes sense. Elasticity is not necessarily constant. Also, I am not sure what sort of formula you learned at your college but elasticity is rigorously defined (and also taught at college level for econ majors) as$EL = \frac{df(x)}{dx}\frac{x}{f(x)}$for some function$f(x)$(see Essential Mathematics for Economic Analysis by ... 0 That is not correct conclusion and there are several problems with the formulation you use.$A$is not a shock - it is a technology parameter of production function (see Woodford Interest and Prices pp 148). Heck,$A$cannot even be zero. Following Woodford:$A_t> 0$is a time-varying exogenous technology factor there can be shocks to$A_t$that ... 1 Let$\delta_i=N_i\beta_i$. I think what you want is for generation$i$'s utility to be something like: $$U_i=u(x_i)+\delta_{i+1}U_{i+1}+\cdots+\delta_{i+n}U_{i+n},$$ where$i$derives utility from his own consumption$u(x_i)$as well as from his descendants' utilities,$U_{i+t}$for$t=1,\dots,n, over their own consumption and ... 2 Here are two nice papers on recursive utility functions, a generalization of additive utility functions and compatible with "utility of the dynastic head to be partly a function of the utility of his children and grandchildren's utility, but where his children's utility is again partly a function of his grandchildren's and great-grandchildren's utility, ... 4 Note that, from your SOC: \begin{align} P''(t) &= V''(t)e^{-rt} - rV'(t)e^{-rt}- rV'(t)e^{-rt} + r^2V(t)e^{-rt} + rme^{-rt}\\ &=e^{-rt}\Bigl[\underbrace{\color{red}{V''(t)-rV'(t)}}_{=D}-r\underbrace{\bigl[V'(t)-rV(t)-m\bigr]}_{=0\text{ from FOC}}\Bigr] \end{align} 3 Local non-satiation means that you always want a little bit more. There are no sweet spots where you are content and wouldn't accept any more ofx_i$. What this means in practice is that if you are optimising your allocation of goods$x_i$, you will exhaust all of your resources ($w$). If you don't have (2), it's possible that your optimal bundle will ... 4 The most common version of Pareto dominance says that an allocation is Pareto-dominated if there is another feasible allocation in which at least one agent is better off and everyone else is at least as well off. In particular, the latter allocation can include consumers that are just as well off as before. It follows from the very definition of a ... 2 I suggest that the graph could be improved in two ways (apart from the point, which you seem to recognise, that the upward-sloping line labelled MSB should be labelled MPC (marginal private cost)). Firstly, the inclusion of the vertical arrow labelled subsidy rather suggests that the vertical distance between the two marginal cost curves equals the amount of ... 0 There could be many answers, but the one that comes to my mind immediately is product differentiation. If you're selling exactly the same thing as everyone else, buyers are going to flock to the lowest price buyer, so you everyone will have to follow a price cut. But say you are selling ice cream cones on the boardwalk at the beach. Any little thing, such as ... 0 Even if you have an unused resource, you would have an opportunity cost it that item had value to someone else. If you could sell, or rent a resource, this that lost income would be opportunity cost. Like wise if keeping an unused asset (my vehicle I no longer drive due to covid) costs you money (plates, insurance) you have an opportunity cost if you don't ... 7 the equilibria of the game in which the strategies of users who face the same reward and costs (i.e. same type) are the same. This sounds like an ex ante (or ex interim or ex post) symmetric equilibrium, depending on the timing of the realization of the types. If both dimensions of the types are realized at the beginning of the game, then I'd go with the ... 2 Without loss of generality, suppose$\mathbf x=(x_1,0)$where$x_1>0$. Consider the following sequence $$\mathbf y^n=\left(x_1\Bigl(1-\frac1n\Bigr),\,\frac1n\right).$$ Clearly,$\mathbf y^n\gg\mathbf 0$for all$n\in\mathbb N$, and thus$\mathbf y^n\succsim\mathbf 0$by monotonicity. It is also the case that$\lim_{n\to\infty}\...

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Hint: Simply apply integration by parts to the integral on the LHS. Simplify and you should arrive at the following expression: $$(1-R)-\int_{R-k}^1G(\theta)\mathrm d\theta.$$ Add and subtract $k$ to obtain: (1-R+k-k)-\int_{R-k}^1G(\theta)\mathrm d\theta = (1-(R-k))-k-\int_{R-k}^1G(\theta)\mathrm d\theta. \end{...

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