14
votes
Why are utility functions typically assumed to be concave?
I disagree with @bbecon. I agree with @bbecon that concave utility functions present nice mathematical properties which help theorists develop analytical models.
If the OP's question was why utility ...
- 281
13
votes
Is the market price objective?
The market price is the current price at which something may be bought or sold. If a good is not sold or bought at a particular price, then that is not the market price. Whether or not any particular ...
- 683
9
votes
Is the market price objective?
The vast majority of economists subscribe today to the subjective theory of value that was in economics introduced by Jevons, Walras, and Menger. Subjective theory of value posits that value is ...
- 42.1k
7
votes
Accepted
Why are utility functions typically assumed to be concave?
More or less, yes.
Making the right assumption on the shape of the utility function allows you to prove existence or uniqueness of the equilibrium. The exact assumption you need depends on what ...
- 658
6
votes
Accepted
For any small perturbation dx, utility cannot change, or else, x* would not be optimal
It is basically a restatement of the first order condition - at an extrema (maxima or minima) of a well-behaved function its first derivative is equal to zero.
If you are at the point of maximization, ...
- 454
6
votes
Visualizing the expenditure minimization problem
I am not sure what you mean - the visualization is essentially the same, only the roles of the goal function and the constraint are switched. Given the appropriate utility and income levels the ...
- 26.2k
6
votes
Accepted
Does Debreu's representation theorem of ordinal utility require Hausdorff topology?
No. However, the problem can be reduced to representing preferences on a Hausdorff space. Instead of trying to represent a complete preorder on a set, one can try to represent linear orders on the ...
- 9,705
6
votes
Accepted
Proving Pareto-efficiency with MRS
One way to check that this allocation is Pareto efficient is using the first welfare theorem. Consider the exchange economy with utility functions
\begin{eqnarray*} u_i(x_i, y_i) = \sqrt{x_iy_i}\end{...
- 5,092
5
votes
Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?
(1) Satisfying completeness, independence, and continuity but not transitivity:
Take two outcomes, $\{0,1\}$, and the associated lottery space $[0,1]$. Consider the preference relation $\succsim$ ...
- 198
5
votes
Help solving utility optimization problem from Connelly 1992 (Economics)
I guess the text considers the problem
$$\max_{x,Q,t_L,t_Q,t_{cc},t_m,t_L,q} u(x,Q,t_L),$$
subject to
$$Q(t_Q,t_{cc}q)- Q=0$$
$$1-t_Q-t_{cc}\geq0$$
$$1-t_m-t_L-t_Q\geq 0$$
$$t_mW+V-x-p_{cc}t_{cc}\geq ...
- 3,177
5
votes
Accepted
How can I show convexity of this value function?
Suppose that $u(C,l)=\sqrt{C}-l^2$ and $f(l,A)=\big(l+g(A)\big)^2$, where $g$ is any function of $A$ that is not convex.
Then $$u\big(f(l,A),l\big)=l+g(A)-l^2.$$
The optimal labor supply is given by $...
- 9,705
5
votes
Is the linear probability model consistent with utility maximization?
This is a very interesting question. Let me do a little algebra.
Consider the latent variable model $y = I(a+xb > u)$, where $I(\cdot)$ is the indicator function. Let $F(\cdot)$ be the CDF of $u$. ...
- 2,024
4
votes
Is addiction a case of increasing marginal utility?
To expand on @1muflon1's answer. The theory of rational addiction assumes that the utility of a consumer at time instance $t$ depends both on current consumption of the addicitve good, say $c_t$, and ...
- 8,632
4
votes
Accepted
What utility function represents an agent with a time-discount factor?
Perhaps I misunderstand the question, because it seems trivial coming from such an established researcher.
As @HerrK. points out, utility functions that represent intertemporal discounting are ...
- 26.2k
4
votes
Is this a case of nonseparable utility (across states of nature)?
No, I don't think so. Let $a_i$ be your choice (e.g. ice tea) in situation $i$ (e.g. summer).
Separability over states says that if for some $b_j$, $a_i$ and $c_i$:
$$
u(a_i, b_j) \ge u(c_i, b_j),
$$
...
- 8,632
4
votes
Accepted
Is this a case of nonseparable utility (across states of nature)?
I think the core issue with this question (and the other related one the OP posted Nonseparable utility across states of nature: an intuitive example) is we need to clarify what is meant by "...
- 5,167
4
votes
Is the market price objective?
The other two up-voted answers are both correct but unnecessarily convoluted for such simple quesiton.
Simple answer is that market price is objective. Market price is:
The market price is the ...
- 2,184
4
votes
Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function
Take a strictly increasing mapping $f:\mathbb{N} \to [0,1)$, such as
$$
f(y) = 1 - \frac{1}{y+1}.
$$
Then
$$
U(x,y) = x + f(y)
$$
represents the Lexicographic preference in the $\mathbb{N}^2$ choice ...
- 26.2k
4
votes
Accepted
How do I get to this demand function in the monocentric city model?
Let $K:=y-(f+fa)-(t+ta)x$, and so $c=K-pq$.
From condition $(\mathrm{A1})$,
\begin{align}
&&\frac{v_q(\cdot)}{v_c(\cdot)}&=p \\
\quad\Rightarrow\quad &&
\frac{\alpha[K-pq]^{1-\...
- 14.4k
4
votes
Accepted
What is a simple demand function that allows for different price and income elasticities than 1 and -1?
Among the simplest demand functions allowing for arbitrary price and income elasticities there is:
$$ x_1^M(p_1,p_2,y) = \alpha + \beta\ln(\frac{p_1}{p_2}) + \gamma\ln(\frac{y}{p_2}),
$$
or its log ...
- 2,552
4
votes
Mathematical definition of perfect substitutes
If $X$ and $Y$ are perfect substitutes such that a unit of $X$ can be replaced by $n$ units of $Y$ [...] I know [...] $U=b(nx+y)$ for this case
This is a possible representation but not the only one. ...
- 26.2k
4
votes
Accepted
Is it possible to get a demand function as function of income and utility from this log linear indirect utility?
First of all, I do not really understand how
$$v=-c\frac{p^{(-β+1)}}{(-\beta+1)}+\frac{y^{(-\gamma+1)}}{(-\gamma+1)},$$
can be an indirect utility function since it is a function of $c$ and the ...
- 3,177
4
votes
Accepted
Concave utility functions solution example
This is the problem we want to solve:
\begin{eqnarray*} \max_{x_1, x_2} & x_1 +\ln x_2 \\ \text{s.t.} & \ p_1 x_1 + p_2x_2 \leq w \\ \text{and} & \ x_1\geq 0, x_2>0 \end{eqnarray*}
Here ...
- 5,092
4
votes
Accepted
Does a neoclassical production with constant returns to scale implies type of Cobb-Douglas
$F(K, L) = K^{\frac{1}{4}}L^\frac{3}{4}+K^{\frac{3}{4}}L^\frac{1}{4}$
- 5,092
3
votes
CES utility maximization two goods two period
The problem can be solved using two stage budgetting. In stage 1 total income $m = \sum_t p_{at} c_{at} + \sum_t p_{bt} c_{bt}$ is allocated across periods. In stage two the optimal expenditure $E_t$ ...
- 8,632
3
votes
Accepted
Why is Deadweight Loss Bad for Society?
I understand the negative effect on the market specifically, but won't those who decide not to participate in the market being taxed simply spend their money in another market, thus not affecting ...
- 42.1k
3
votes
Accepted
Log Linearising CES demand
Here's my guess.
Let use the notation
$$
\tilde x_t \approx \ln(x_t) - \ln(x) \approx \dfrac{x_t - x}{x}.
$$
If we take logs on both sides we get:
$$
\ln(G_t) = \frac{1}{1 -\rho} \ln(p_t) + \ln(y_t) - ...
- 8,632
3
votes
Nonseparable utility across states of nature: an intuitive example
I'm not aware of any intuitive justification for the state-non-separability in Epstein-Zin preference. However, as both @MichaelGreinecker and @afreelunch alluded to, there are micro/behavioral ...
- 14.4k
3
votes
Accepted
How do you measure the MRS for more than two goods?
Pairwise MRS
The Marginal Rate of Substitution is usually defined as a pairwise thing. E.g., in case of three goods $x,y$ and $z$, you would have
$\text{MRS}_{xy}, \text{MRS}_{xz}$ and $\text{MRS}_{yz}...
- 26.2k
3
votes
Is addiction a case of increasing marginal utility?
It is possible for an addict to be rational. A famous work on this was done by Becker (1988) Theory of Rational addiction.
In order for agent to have rational preferences the preferences have to ...
- 42.1k
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