11

There are several reasons: Didactic Reasons: Other users seem to have missed it but in your question you specify you are talking about "(introductory) microeconomics" [emphasis mine]. Well the most prosaic answer is simply that it is much easier to solve cost minimization, or various other models when costs are assumed to be convex. This in itself ...


10

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, ...


5

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

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 ...


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

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

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.


4

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....


4

Theoretically, the cost function is a result of a cost minimization problem with a given production technology. Convex/linear/concave costs are a result of decreasing/constant/increasing returns to scale. The thinking behind convex costs is the idea of decreasing marginal product of your input goods for production. As an example for the kind of thinking ...


4

If the cost function is globally concave in output $y$, then the profit function is convex in $y$ and the optimal (profit maximizing) output is not characterized by the equality between price and marginal cost, so price taker firms have an optimal output level that is either 0 or tends to infinity the profit is negative at least for low levels of output (if ...


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 ...


3

Increasing and convex costs are a result of decreasing returns to scale. These are mainly due to the limited availability of (local) input factors. Other contributing factors are the decline of management efficiency of large-scale production, the imperfection of internal supervision and control mechanisms, and more complex information transmission.


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 ...


3

You can do this in various ways for a value of output (e.g. goods and services) firm produces but not really very reliably for value of firm itself. Value of Output When it comes to value of output then this boils down to estimating production function of the company. For example, assuming that company has Cobb-Douglas production function: $$Y = AK^\beta L^{(...


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 ...


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