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 ...
1muflon1's user avatar
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10 votes
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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 ...
Giskard's user avatar
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7 votes
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Properties on conditional demand correspondence from the textbook of Mas-Colell et al

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 ...
bomadsen's user avatar
  • 313
6 votes
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Intermediate Case of Bertrand and Cournot

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 ...
Bertrand's user avatar
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5 votes
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CES Production Function with $\rho>1$

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 ...
luchonacho's user avatar
  • 8,591
5 votes
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Walras Law in a production economy with fixed costs

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/...
Albert Zevelev's user avatar
5 votes

Prove that if a production function is such that f'>0 and f''<0, then f'<Average Product

Assuming $f''<0$ implies strict concavity and hence $$tf(y) + (1-t)f(x)<f(ty+(1-t)x) \Leftrightarrow f(y) - f(x)<\frac{f(ty+(1-t)x)-f(x)}{t}$$ $$f(y) - f(x)<\frac{f(x + t(y-x))-f(x)}{t(y-x)...
bomadsen's user avatar
  • 313
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 $C(q)/q$). ...
HarmlessEcon's user avatar
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.
410 gone's user avatar
  • 8,143
4 votes

Are There Giffen Inputs?

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 ...
Michael Greinecker's user avatar
4 votes
Accepted

Are There Giffen Inputs?

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 ...
Kitsune Cavalry's user avatar
  • 6,608
4 votes

Does the contract curve always have to connect the initial points on an edgeworth box? Why or why not?

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 ...
123's user avatar
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4 votes
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Do property values capture producer choice in agriculture?

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 (...
Adam Bailey's user avatar
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4 votes
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The existence of solution for profit maximization problem

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 ...
tdm's user avatar
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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 ...
Bayesian's user avatar
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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 ...
Bertrand's user avatar
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3 votes

The existence of solution for profit maximization problem

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)-\...
Herr K.'s user avatar
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3 votes
Accepted

Why does a homothetic function have constant ratio of marginal products along rays?

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(\...
Alecos Papadopoulos's user avatar
3 votes
Accepted

Why was activity analysis abanadoned as a field of research?

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 ...
Alecos Papadopoulos's user avatar
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 ...
VARulle's user avatar
  • 6,735
3 votes
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Micro: proving that cost minimizing input vector for producing y cannot produce more than y

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 ...
tdm's user avatar
  • 11.4k
3 votes
Accepted

How do you calculate the contribution of each factor of production to the value of the final product?

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 ...
1muflon1's user avatar
  • 55.6k
3 votes

How are returns to scale of a non homogeneous production function defined?

The standard definition is that a production function $F(\vec{x})$ has constant, increasing, or decreasing returns to scale if $F(a\vec{x})$ is equal to, greater than, or less than $a F(\vec{x})$ (...
tparker's user avatar
  • 770
3 votes
Accepted

Solve long run production function of a firm using technical rate of substitution

The optimization problem you solve is $\min w_1 x_1 + w_2 x_2$ s.t. $x_1^{0.5} x_2^{0.5} = q $ I assume you’re used to solving optimization problems with the Lagrangian. The Lagrangian is: $\mathcal{L}...
Nicolas Torres's user avatar
3 votes
Accepted

Derive cost function from production function

$z_2$ is included in the production function. Please see the production function $f:\mathbb{R}^2_+\rightarrow\mathbb{R}$: \begin{eqnarray*} f(z_1,z_2) = \left(\min(\lfloor z_2\rfloor, 1)\right)z_1^\...
Amit's user avatar
  • 8,411
2 votes
Accepted

Maximisation problem in a multiproduct firm

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 ...
Kitsune Cavalry's user avatar
  • 6,608
2 votes

Why are some goods without close substitutes not sold in some countries?

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. ...
Fix.B.'s user avatar
  • 2,648
2 votes

Are all points on the Long Run Average Cost (LRAC) curve productively efficient?

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. ...
Giskard's user avatar
  • 29.3k
2 votes
Accepted

Help with this microeconomics exercise

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 ...
Amit's user avatar
  • 8,411
2 votes
Accepted

Some doubts about netput vectors

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) = \...
Theoretical Economist's user avatar

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