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It is the first one, $TC(0) = FC$. This is the definition. Also consider that it is not clear what is "transformed by $q$ in some way". In case of $$\frac{5q}{q+1} + \frac{5}{q+1}$$ are the two fractions transformed by $q$, or should I just sum them up to 5? With your function, one can rearrange it to $$TC(q) = \frac{5}{q+1} + 5 + 5q + q^2 = -\frac{5q}{... 7 Let z_1 and z_2 be \geq 0 and solution to$$\min_z \{w^\top z\lvert f(z)\geq q\}$$then clearly f(z_1)\geq q and f(z_2)\geq q and since \{z\geq 0\lvert f(z)\geq z \} is convex it then follows that z_3 := \lambda z_1 + (1-\lambda)z_2 must satisfy the constraint f(z_3)\geq q. Since z_1 and z_2 are both minimizers it cannot be the case that ... 6 Question: is the following correct ?$$\frac{dL}{dq}=1/\frac{\partial q}{\partial L},\;\frac{dK}{dq}=1/\frac{\partial q}{\partial K}$$In general, no. Since q= f(L,K) is a multivariable, single-valued function, then by the implicit function theorem applied on the implied equation H = f(L,K)-q=0, what we can say is only that$$\frac {\partial L}{\...

6

These papers could be interesting to you. First, a classical contribution: Singh Nirvikar and Xavier Vives, 1984. "Price and Quantity Competition in a Differentiated Duopoly," RAND Journal of Economics, vol. 15(4), pages 546-554. And these two interesting papers, using the concept of competition toughness to reconcile Cournot and Bertrand: d'Aspremont, ...

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It is valid but only in the short run with the assumption that the capital is fixed. By assuming that output depends on labor and capital you can write $$q=q(L,K)$$ Now taking the total derivative $$dq=\frac{\partial q}{\partial L}dL+\frac{\partial q}{\partial K}dK$$ In the short run capital is fixed such as $dK=0$ $$\longrightarrow\quad dq=\frac{\... 5 Partial answer: for simplicity let P_c =1. The budget constraint: c= wn + \Pi Simplify (plug in \Pi): c= F(n)- fc Goods clearing: c = F(n) The household's budget constraint is inconsistent w/ goods market clearing. The firm pays a fixed cost that doesn't go to anyone. In a "true GE model" all payments have to go to someone in the economy. ... 4 The problem with \rho>1 is that it means the marginal product of factors is not decreasing (\rho<1) or constant (\rho=1) but increasing, which is an odd assumption. Such functions yield isoquants that are concave, and might lead to only one factor being used (as BKay said). As in any generic CES, the marginal product of factor x_i is$$ MP_i =...

4

There is a large economic literature on intellectual property rights. However, the issue seems far from settled on what even the optimal duration for patents are. Note that open source is even a step further than a 0 day patent duration. A strong case for your view would probably be found in Boldrin/Levine: http://levine.sscnet.ucla.edu/general/intellectual/...

4

I believe the answer is true. Giffen goods are goods where the income effect overpowers the substitution effect. \begin{align} \max_{\vec x} \ \ \ & U(\vec x) \\ & \text{s.t.} \ \ \ \vec p \cdot \vec x \leq I \end{align} To start, if you think about the consumer's problem (for example utility maximization, here), a change in a good's price ...

4

The contract curve is the locus of Pareto optimal points in an Edgeworth box. What we get from that: To be P.O., an allocation must be feasible. So, the contract curve does not extend beyond the edges of the box (opposed to indifference curves, which we can draw as extending beyond the edges of the box) because points outside the Edgeworth box are not ...

4

Yes, if there are non-zero fixed costs, and constant marginal cost, then average cost decreases strictly monotonically with quantity, asymptotic to the marginal cost.

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Short answer: Yes, it is possible. Decreasing average cost implies that marginal cost is less than average cost ($MC<AC$, which can be proved by simply taking the first derivative of $C(q)/q$). With constant marginal cost, there exists a simple linear cost function $C(q)=F+a\times q$ that satisfies the constant $MC$ condition, where the constant $F$ is ...

4

Suppose that, initially, crop A is the most profitable crop that can be grown on a particular farm, yielding an average annual profit of \$X, profit being measured after deduction of all costs (including a normal return to the farmer) other than the cost of the property (ie the land). In a competitive property market, the capital value of the property might ... 4 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.... 3 There are no Giffen inputs. Suppose there are$l$-goods, including all inputs and outputs. A price system is then a vector$p=(p_1,\ldots,p_l)\in\mathbb{R}^l$. One can give a firms production decision by a production plan$y=(y_1,\ldots,y_l)\in\mathbb{R}^l$. The idea is that$y_j$denotes the net output produced of good$j$. If it is an input, this entry is ... 3 By Roy's Identity we have that Marshallian (uncompensated) demand for good$x_i$is $$x_i^M = \frac {\partial U^*/\partial p_i}{\partial U^*/\partial B} \tag{1}$$ where$U^*$is optimized utility over goods vector$\mathbf x = (x_1,...,x_i,...x_n)$and$B$, which is the available budget, and given the price vector$\mathbf p = (p_1,...,p_i,...,p_n)$. To ... 3 Here is my attempt at this question, it's incomplete and/or incorrect so please help make suggestions and I will edit this. Cost Minimization Since$f(x_1,x_2)$is not quasi-concave, the corresponding isoquant curves are not going to be covex to the origin (i.e. their upper contour set will not be convex). In this case firm should employ corner solution ... 3 A homothetic function can be characterized as follows: Let$f(\mathbf x)$,$\mathbf x \in \mathbb R^n$be a function homogeneous of degree$r$. Let$g()$be a function with$g'\neq 0$. Then $$G(\mathbf x) = g[f(\mathbf x)]$$ is homothetic. Since$f(\mathbf x)$is homogeneous of degree$r$we have that $$f(\lambda \mathbf x) = \lambda ^ rf(\mathbf x)$$ ... 3 In Farrell, M. J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society: Series A (General), 120(3), 253-281. which is widely seen as the starting point of systematic Efficiency and Productivity Analysis, the author writes, referring to what he is about to present: Similarly, although the treatment of the efficient ... 2 The question "how are the first order conditions" seems very unclear to me, and I am providing a set-up for finding and writing them out, while explaining the Kuhn-Tucker conditions that are easy to struggle with. Though we try to avoid giving away basic study question answers, it's a positively voted question without an answer, and I still think these ... 2 A constant elasticity demand function has the form$q=p^{-\epsilon}$. Let's check this indeed gives us a constant elasticity... $$\frac{d q}{dp}=-\epsilon p^{-\epsilon-1}$$ so, as we'd hoped, the elasticity is constant: $$\frac{d q}{dp}\frac{p}{q}=-\epsilon p^{-\epsilon-1}\frac{p}{p^{-\epsilon}}=-\epsilon p^{-\epsilon-1+1+\epsilon}=-\epsilon p^{0}=-\epsilon.... 2 The classic reference on this topic (as suggested by my professor) is the book; Theory of Cost and Production Functions by Ronald Shepherd. 2 I have figured this as the answer to this question; As we know profit maximization problem is given as,$$ \pi(p,w) = \mathop{max}_{\textbf{y}}\quad p.y - C(\overrightarrow{w},y) $$When f(\overrightarrow{x}) is homothetic,$$ C(\overrightarrow{w},y)=h(y).C(\overrightarrow{w},{1}) $$Substituting in the profit function gives;$$ \pi(p,w) = \mathop{max}... 2 Sorry I don't have enough reputation yet in the Economics community to comment. But the answer provided by @Alecos Papadopoulos can be illustrated further. Please feel free to move my "answer" to comments of his answer. With the assumption of only$F' > 0$and$F > 0$and$h' > 0$, then there is a simple case where the minimization problem has no ... 2 Some economists (Oliver Hart and John Moore are among the most prominent) have developed a "theory of the firm" which could help address some of the issues involved. One of the considerations in that theory is that the different parties involved in production can 'hold-up' each other. For example, in this case, the brake manufacturer or the auto manufacturer ... 2 One way to look at this is through the idea of "fixed costs". It only makes sense to pay the fixed cost if the market is big enough. There is very likely a large a fixed cost of entering a market. You have to establish a network of retailers or at least distribution channels, a way for the customers to return the defective units, partners that will also ... 2 Given the production function \begin{eqnarray*} F(L) = 10 \sqrt{L} - 2L \end{eqnarray*} marginal product is \begin{eqnarray*} F'(L) = \frac{5}{ \sqrt{L}} - 2 \end{eqnarray*} Since there is no cost for the family for using its own labor and it cannot hire from outside, family will maximize its production subject to the constraint that it has 4 units of ... 2 We say that$x \le y$when$x_i \le y_i$for each$i=1,\ldots,n$. If the production set can be represented by a production function$F$, and$F$is homogeneous of degree$r < 1$($F(\lambda x) = \lambda^r F(x)$for$\lambda >0$), then the production set satisfies the non-increasing returns to scale property you state. Moreover, if$F(0)=0$, and$F$is ... 2 The statement "In the short run, only the minimum point on the SRAC curve is productively efficient" is false. Suppose producing 1 unit of a good costs 5 dollars. 2 unit costs 8, x>2 units costs 5x. Clearly AC is minimal if you produce 2 units. But perhaps I do not want two units of the good, only one. Why spend the extra 3 dollars on the second good? ... 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 ...

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