11
votes
Accepted
Why are cost functions often assumed to be convex in microeconomics?
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 ...
- 57.7k
10
votes
Accepted
Fixed cost of a firm
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 ...
- 29.6k
9
votes
Accepted
Cost Minimization and Karush-Kuhn-Tucker
The term $\lambda_2(x_1-1)$ in your Lagrangian is incorrect; it treats the second constraint as an equality rather than an inequality. To allow for the constraint being an inequality you can include a ...
- 8,584
8
votes
CobbDouglas: Constant marginal costs and constant returns to scale
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 ...
- 81
7
votes
Long Term Economic Profit for Perfectly Competitive market
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 ...
- 33.9k
7
votes
Long Term Economic Profit for Perfectly Competitive market
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 ...
- 57.7k
7
votes
Under what condition is a cost function strictly concave in prices?
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 ...
- 3,481
6
votes
Cost function from CES production function
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
$$C(w_1,w_2,y) = \...
- 3,798
6
votes
Cost Minimization and Karush-Kuhn-Tucker
Here is the cost minimisation problem that we need to solve:
\begin{eqnarray*} \min_{x_1,x_2} & w_1x_1+w_2x_2 \\ \text{s.t. } & \sqrt{x_1x_2}=\overline{y} \\ \text{and } & x_1\geq 1, x_2\...
- 9,411
5
votes
Accepted
Prove all cost functions are concave in input prices and demand for inputs is downward
Let $x(w, q)$ denote the solution to the cost minimization problem :
\begin{eqnarray*} \min_{x} & \ w\cdot x \\ \text{s.t.} & \ \ f(x) \geq q \end{eqnarray*}
where $f$ is the production ...
- 9,411
5
votes
Accepted
Under what condition is a cost function strictly concave in prices?
As Bertrand pointed out, strict-concavity will necessarily fail along any rays through the origin. But one can have strict concavity for normalized price systems.
So let $f:\mathbb{R}^n_+\to\mathbb{R}...
- 13.5k
5
votes
Cost Minimization and Karush-Kuhn-Tucker
Let's setup the optimization problem first,
$$min_{\{x_1,x_2\}} \omega_1x_1+\omega_2x_2 $$
$$ s.t \hspace{5 mm} (\bar{y}=x_1^{\frac{1}{2}}x_2^{\frac{1}{2}}) \wedge(x_1 \ge 1)\wedge (x_2 \ge 0)$$
...
- 181
4
votes
Deriving long-run cost function
You're right. Divide Eq (1) by Eq (2):
$$
\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 = \...
- 1,226
4
votes
Is it possible to have constant marginal cost and decreasing average cost simultaneously?
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 $C(q)/q$). ...
- 125
4
votes
Is it possible to have constant marginal cost and decreasing average cost simultaneously?
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.
- 8,168
4
votes
Decision over "max" production function:
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 $x_2^*$ ...
- 15.8k
4
votes
Accepted
What is the returns to scale of the production function q = min {K, L^(1/2)}?
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 ...
- 5,291
4
votes
Marginal cost given (Cobb-Douglas) production
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 $A:= \prod_i \alpha_i^{\alpha_i}$ ...
- 3,798
4
votes
What is the returns to scale of the production function q = min {K, L^(1/2)}?
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 &...
- 33.9k
4
votes
Accepted
Trigonometric Cost Function
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 ...
- 12.5k
4
votes
Why are cost functions often assumed to be convex in microeconomics?
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 ...
- 5,291
4
votes
Why are cost functions often assumed to be convex in microeconomics?
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 ...
- 3,481
3
votes
Accepted
How to calculate the minimun of Average Fixed Cost knowing the Average Fixed Cost
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 ...
- 8,584
3
votes
What does "r" represent in the total cost function?
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 ...
- 3,798
3
votes
what results can be derived from the average cost curve?
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.
...
- 8,584
3
votes
Accepted
what results can be derived from the average cost curve?
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 ...
- 1,249
3
votes
Why min AC = min SRAC at the minima of AC curve?
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 $x$. ...
- 29.6k
3
votes
Why are cost functions often assumed to be convex in microeconomics?
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 ...
- 7,119
3
votes
Accepted
Finding production given total cost (shephard's lemma)
To find the production function, you can solve for $\frac{v}{w}$ in $ {l}_{c}$ and ${k}_{c}$ and set $\frac{v}{w}$ = $\frac{v}{w}$ then solve for $q$.
This will yield
$$ \frac{v}{w} = (\frac{4{l}_{c}}{...
- 366
3
votes
Accepted
Why is MC = ATC the same point for both the breakeven point and an investor maximizing return?
You write about two separate optimization problems,
$$
1. \max_y \pi(y) \\
2. \max_y r(y)
$$
The first problem's optimum yields $MC(y_1) = p_y$, while the second problem's optimum yields $MC(y_2) = ...
- 29.6k
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