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}{...


8

Since the exponents add to one the production function has constant returns to scale, which means that, given factor prices, total cost is linear, which means that it's derivative (= marginal cost) is contant. If you change the exponent 1-alpha to beta where alpha+beta < 1, there will be decreasing returns to scale (but still homotheticity) and you will ...


7

To elaborate a bit on the answer by user 1muflon1, in economics the word "profit" is the surplus accrued to the firm after we have subtracted from revenues all compensation of production inputs, irrespective of whether these compensations have been recorded by Accounting as expenses or not. Two examples: Suppose you run your own business, and you ...


7

This is because we are talking about economic profit not accounting profit. An economic profit takes into account opportunity cost. If you are skilled programmer that can earn $\\\$100000$ per year being employed at Google then doing something else like operating your own business incurs an opportunity cost of $\\\$100000$ per year. Hence in this situation ...


7

No function that is homogeneous of degree one, is at the same time strictly concave in its arguments. If the function is differentiable (or non-differentiable at a finite number of points), then the Hessian of a linear homogeneous function is singular. So if you want to end up with a unit cost function that is strictly concave, you have to drop at the same ...


6

If you are interested in the case where $\rho \geq 1$ then look at the post CES $\ \ \rho \geq 1$. For the standard case where $0 < \rho < 1$ you should get a result like this $$C(w_1,w_2,y) = \left(w_1^{\frac{\rho}{\rho -1}} +w_2^{\frac{\rho}{\rho -1}}\right)^{\frac{\rho - 1}{\rho}} y.$$ To see this you should start by setting up the cost ...


5

Let $x(w, q)$ denote the solution to the cost minimization problem : \begin{eqnarray*} \min_{x} & \ w\cdot x \\ \text{s.t.} & \ \ f(x) \geq q \end{eqnarray*} where $f$ is the production function. Since $x(w, q)$ minimizes cost at $(w, q)$, following holds for all $w$ and for all $q$ : \begin{eqnarray*} w\cdot x(w, q) \leq w\cdot x(w', q) \ \ \ \...


5

As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems. So let $f:\mathbb{R}^n_+\to\mathbb{R}_+$ be a production function. We let $\Delta^n_{++}$ be the set of all points in $\mathbb{R}^n$ with all coordinates being strictly positive and summing to one. ...


4

You're right. Divide Eq (1) by Eq (2): $$ \frac{a L^{a-1}K^b}{bL^aK^{b-1}} = \frac{aK}{bL} = \frac{w}{r} ~~~\Rightarrow~~~ L = \frac{ar}{bw}K \tag{4} $$ Now use this in Eq. (3) $$ C = wL + rK = \left(\frac{a}{b} + 1\right)rK ~~~\Rightarrow~~~ K = \frac{C}{r(a/b + 1)} \tag{5} $$ Replace this in Eq. (4) to get $L$


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

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

Hint For profit maximization, either $x_1$ or $x_2$ (but not both) must be zero. If not, say $x_1^*>x_2^*>0$ at the optimum, then one could increase profit by lowering cost by reducing $x_2^*$ without affecting output and thus revenue. Let $z=\max\{x_1,8x_2\}$. The profit function can be written as \begin{equation} p[3(x_3)^{1/3}(z)^{1/3}]-w_3x_3-...


4

You have a Leontief production function and in optimum you will always have $K=\sqrt{L}=q_1$. Now increase both inputs by factor $k>1$ and you arrive at $k*K > \sqrt{kL}=q_2$ where the first inequality follows from $k>\sqrt{k}$ for $k>1$ and the second ineqality from the fact that only the minimum matters in your production function. Therefore ...


4

To understand what is the issue here, try dutifully to examine all possible sub-cases in the production function. The production function is $$Q_0 = \min\{K_0, L_0^{1/2}\}$$. Consider cases A. $K_0 < L_0^{1/2}$ Here $Q_0 = K_0$. Consider $Q_{\lambda }\equiv \min\{\lambda K_0, \lambda^{1/2}L_0^{1/2}\},\;\;\; \lambda>1.$ Subcase A1. If $\lambda K_0\leq \...


4

There is a long tradition of using flexible functional forms for cost, utility or production functions. This can be done using a (low) order series expansion. Usually, these are Taylor expansions. For example the translog is a second order polynomial in log prices. Of course, one could also use a Fourier expansion, which would give a function in terms of ...


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

The Cobb Douglas production function with constants returns to scale $$y = \prod_i x_i^{\alpha_i} = A \prod_i \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i} ,$$ where $A:= \prod_i \alpha_i^{\alpha_i}$ annoying constant. Cost minimization with perfect competition $$\min_x \ \ p^\top x\ \lvert \ y = \prod \left(\frac{x_i}{\alpha_i}\right)^{\alpha_i},$$ implies ...


3

As the quantity of output of a good increases, fixed cost (by definition) remains the same and therefore average fixed cost per unit of output continuously decreases. Therefore the average fixed cost curve does not have a minimum, except in the sense that it is asymptotic to zero. A much more important concept than average fixed cost is average total cost, ...


3

r is the user cost of capital. User cost refers to the expenses borne by the owner or renter of a capital asset resulting from the use of the asset for a given period of time. A standard metaphor is that of a machine being worn down over time as it is being used. During the time of use it creates a flow of goods through a production process. The capital gets ...


3

Assuming a firm is a perfect competitor in input markets, the long-run average cost curve, which traces out the minimums of short-run average cost curves, can be used to characterize economies and diseconomies of scale for a firm. This is definitely a practical concept in business and IO. In an industry where the long-run average cost curve is always ...


3

Short-run implies that some decision variable cannot be free set, it is fixed for a time (in the short-run). In the long-run all variables may be freely set. Let us denote the fixed variable by $x$. This can take different values, so there is a family of short run cost functions is $SC(x,q)$. Given a value $q$ let us denote the cost minimizing value of $x$ ...


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.


2

Short answer: whether a firm's maximum 'profit' at the point where marginal revenue equals marginal cost is positive or negative depends on whether average revenue exceeds average cost at that point. To give a simple numerical example, suppose for a firm in a perfectly competitive market: $$MR = AR = 1$$ $$TC = 2 + 0.4Q + 0.1Q^2$$ where $MR$ is marginal ...


2

I understand that the "bundle price" is cost to us. Then your table depicts a linear system of equations $$C_j = a_1x_1+a_2x_2+...+a_mx_m,\;\;\; j=1,...,n$$ with $C_j$ and the $x$'s being the known quantities, and we want to determine the unknown alphas. If it so happens that the bundles on offer are same in number as the $x$'s, ($n=m$), then the system,...


2

When you say non-linear cost function, I assume you aren't referring to the firm's production having non-linear costs, but judging by your example, you rather mean the firm's output prices having weird optimal bundle pricing. From the abstract of the linked article (Hanson and Martin, 1990): " Bundle pricing is a widespread phenomenon. However, despite ...


2

If your cost function is also homogeneous of degree $k$ (which is often assumed to model different types of returns to scale, whether constant, increasing, or decreasing), then by Euler's Homogeneous Function Theorem, $$ x c'(x) = k c(x).$$ That is, $x c'(x)$ is your cost itself, up to some scaling factor $k$ (for example, if $c(x) = ax$ so that $c(x)$ is ...


2

So to understand why the long run average cost curve and short run average cost curve have the same minimum in perfect competition, as well as some of the other stuff you ask, you have to understand the different assumptions that underlie the models you're working with. Let's start with what the short run and long run mean here. In this context, the short ...


2

In general the statement is wrong. Here is a counterexample: Suppose you have $f(k,l) = -k l^\beta$ with $\beta >0$ and $(k,l)\in\mathbb{R}^2_{++}$ (you can interpret $f$ as a production function for a "bad" commodity). Then you have: $$f(tk,tl) = - t^{1+\beta} k l^\beta = t^{1+\beta} f(k,l) < t f(k,l)$$ so that $f(k,l)$ is decreasing returns to scale....


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