7

Because "adjustment costs are linear and there is no aggregate uncertainty", the FOC for $N_t$ is $$f'\theta g_N(N_{t}, I_{t}) - w_N = \phi \lambda C_N$$. Notice that this is exactly the same form for each period. The same is for $I_t$. This means that a firm will choose the same labor inputs in all periods. In other words, the firm gets into the ...


5

As alluded to by Bertrand in his +1 comments this is because FOCs do not tell you where maximum or minimum occurs. This is common misconception among some students but it simply does not hold. FOCs give you location of stationary points. They give you points where $df(x,y)/dx = df(x,y)/dy= 0$. A maximum or minimum will occur at points where $df(x,y)/dx = df(...


5

It is possible to have $$g(x^*) = b\; {\rm and}\; \lambda^* = 0$$. When the multiplier is zero and the constraint is equal to zero, then a) The constraint does not really "bind" b) That's why the multiplier is zero. What does it mean "the constraint does not really bind"? It means that the solution $x^*$, that makes $g(x^*) = b$, would be ...


5

Your Lagrangian would be $$L = (ax+by)+\lambda (I−p_x x−p_y y) +\mu_x(x−0)+\mu_y(y-0),$$ where the final two terms represent the restriction that $x,y\geq0$. You then arrive at conditions $$\frac{\partial L}{\partial x}= a -\lambda p_x +\mu_x=0$$ $$\frac{\partial L}{\partial y}= b -\lambda p_y +\mu_y=0$$ $$I=p_x x+p_y y$$ and complementary slackness ...


4

Optimal transport methods are very much still in use in economics. The show up in two-sided matching with side-payments, contract theory, hedonic pricing, partial identification in econometrics, and a couple of other areas. You can find an excellent overview of economic applications in the 2016 book Optimal Transport Methods in Economics by Alfred Galichon.


4

I think the easiest way is to notice that since the problem is increasing in both arguments, we can assume $px_1 + x_2 = 1$ the budget binds (at least for $p \not =0$). Substituting our constraint into the objective function, we have: $$ \max_{x_1} x_1 + 5(1 - p x_1) = x_1(1 - 5p) + 5 $$ If $1 - 5p<0$, we choose the smallest possible $x_1$, so $x_1 = 0$ ...


4

Because you are talking about constraints, it appears you do not consider the case of inserting such "access costs" (because this is what they are) in the utility function. It implies that they do not create disutility directly, only direct or indirect monetary costs. Let $d$ be the distance in some units. Let $C_d = c_dd$ be a linear (for ...


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 $U(50,20)>\max\{U(0,120),\,U(60,0)\}$.


4

Converting my comments into an answer: At the bottom of p.268, the authors say: The set of consumption vectors $X_i$ available to individual $i$ $(=1,\cdots,m)$ is a closed convex subset of $R^l$ which is bounded from below. [Emphasis added.] Since the Heine-Borel Theorem establishes that $S\subset R^n$ is compact if and only if $S$ is both closed and ...


4

It's a matter of choice how one writes the Lagrangian in the context of Lagrange/KKT. Depending on how it's written, the gradients of the objective and constraint functions are either parallel or anti-parallel at a (suitable) optimum, and the Lagrange multiplier is neither negative or positive. At the end of the day, it is the same (subset of) optima that ...


3

Only two or three papers spontaneously come to my mind, they are not very recent (but quite seminal papers, useful to find further newest contributions): Lee, Lung-Fei, and Mark M. Pitt, 1986, "Microeconometric Demand System with Binding Nonnegativity Constraints: The Dual Approach," Econometrica, 54, 1237–1242. Lee, Lung-Fei and Mark M. Pitt, 1987,...


3

The constraint $g(x)\geq 0$ is binding if in optimum $g(x) = 0$ rather than the constraint being slack $g(x)>0$.


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 ...


2

The problem is that you are ignoring the division $ \frac {0} {0} $, which is in $ \frac {\partial L} {\partial v} $. Before looking at the solution, and seeing that indeed when $ \theta_ {2} = 0 \ \Rightarrow \ v ^ {*} = 0 $, I want to note that the first constraint $ c_ {1} = w - (1 -v) \theta_ {1} h_ {1} ^ {a1} $ can be more realistic and logical. It is ...


2

Take $\Omega_1=\Omega_2=[0,0.5]$. Let $p(x)=x^4$, so that $p''(x)=12x^2>0$ on $(0,0.5)$. Let $q(x)=0.5x$, which is linear in $x$, and $K=0.2>0$. For $\bar\gamma=1$, both objective functions attain their respective maximum at $x=0.5$. As the following figure shows, objective function $(1)$ (blue curve) has a higher maximum than that of objective ...


2

I would leave this as a comment but I cant. You are on the right track. Once you know $V_2(k)$ then you can plug that into to the first hjb and solve. To solve for $V_2$ you need to find the optimal $i$ as a function of $k$. Then plug $i(k)$ into the 2nd HJB. That will give you a second order ode. Solving that will give you $V_2(k)$ and you go to 1.


2

Almost correct. Setting $W(h)=0$ is wrong (but inconsequential for the solutions). Checking the SOC for completeness should be included, but is somewhat obvious here. Correctness of 3. holds only under the assumption that the consumer owns no shares of the firm, which seems to hold in this exercise.


1

The concepts of feasible set and utility should be thought of separately. Non-satiation just means that there are no "thick" indifference curves - this guarantees that an agent is not indifferent between two close, but different bundles of goods.


1

In the economy described above, set of efficient allocations is given by the red curve. Just do the slope of ICs comparisons at the boundaries and you will get that.


1

In order to have $x^*_j>0$, it is possible to impose either (i) a condition on the marginal utility: $\lim_{x_j\rightarrow0} \partial{U}/\partial{x_j}(x)=+\infty$ (ii) or an inequality $x_j \geq a_j>0$ where $a_j$ is interpreted as a subsistence level of $x_j$ Often the utility function is reparameterized and written $U(x-a)$ with the constraint $X:=x-...


1

The two ways of properly setting up Lagrangian are completely equivalent (although you have mistakes above) so it does not matter which way you set it up. You actually have mistake there resulting in the two ways being different: In the first equation you forgot minus sign since: $ \sum_i^l p_i w_i = \sum_i^l p_i x_i \implies 0 = \sum_i^l p_i w_i - \sum_i^l ...


1

I think this is due to how terminology surrounding ZLB changed/evolved over time. In most models ZLB is actually mathematically zero lower bound in a way that is closer to a boundary solution as you put it (see examples in Romer Advanced Macroeconomics chapter 11 and more importantly sources cited therein for the models including ZLB). Nonetheless, the ZLB ...


1

The equation $100\sqrt{x_1} = 8x_1^{3/2} - 5$ is cubic in $\sqrt{x_1}$. Unless you have been asked for an analytic solution (which is possible but complex), the approximate method I suggest is first to ignore the minus $5$, enabling division by $\sqrt{x_1}$ yielding: $$100 = 8x_1$$ The first approximation is then $x_1 = 100/8 = 12.5$. Since $100\sqrt{12.5} \...


1

To solve this with the exact constraint given in the question, one could set up the following Lagrangian: $\mathcal{L}(c_{1},\lambda(\cdot))=u\left(c_{1}\right)+\int u\left(c_{2}\left(z\right)\right)g(z)\mathrm{d}z-\int\lambda(z)\left\{ f\left(c_{1},z\right)-c_{2}\left(z\right)\right\} \mathrm{d}z,$ where $g(z)$ is the probability density function for the ...


1

I assume $x_i$ represents the quantity and belongs to $\mathbb R_{+}$. You can form the constraints as follows: $$ x_i \geq0 \quad\forall i \in [3] \\ \sum_{i=1}^3p_ix_i \leq I $$ You can simplify the objective by noting that for the utility to be the maximum, $x_2 =x_3$. Try to reason why this is true. Hence, the final problem becomes, $$\ \max_{x_1, x_2, ...


1

I think there is an approach problem in the constraint. (1) implies that: \begin{align} v_{n,t} - v_{n,t+1}=H_t - (\sum_{i=1}^{n-1} v_{it}-\sum_{i=1}^{n-1} v_{i,t+1}) \end{align} As the number of fish caught that are not the $ n $ species increases, the number of fish that belong to the $ n $ species decreases. This, i understand, may arise from a production ...


1

These are many questions. O.k., so let's go step by step: (Q1) What is a mapping actually? A map is just another term for a function. Here, every "law of motion", the actual one (ALM) and the perceived one (PLM), is characterized by its parameters $a$ and $b$. The ALM depends on the PLM, and the function mapping the PLM-parameters to the ALM-parameters is ...


1

Right now, the issue is not so much a dollar cost, rather the physical feasibility of either testing, or the availability of N95 masks. 1) To run a testing programme, a jurisdiction needs the testing capacity, as well as the organisational capacity to administer tests. In many jurisdictions, neither capacity exists at the time of writing (mid-April), ...


1

Thunstrom et al. at Wyoming have a working paper on this issue although they look at social distancing rather than masks or testing. The basic reasoning is the same though: look at costs of measures for the economy and benefits in terms of lives saved (using the value of a statistical life) and calculate the net benefits. The working paper is here. From the ...


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