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6 votes
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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 ...
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3 votes
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Constrained optimization to find utility maximizing allocation

I think you are trying to find a feasible allocation that maximises the sum of the utilities of the two individuals. So we can write the objective function as: \begin{eqnarray*} \max_{x,y} & \ xy^...
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1 vote

Calculating price in a pure exchange economy

Let $\omega_1 = (\omega_{1}^X, \omega_{1}^Y)$, and $\omega_2 = (\omega_{2}^X, \omega_{2}^Y)$ be the endowment of the two consumers, respectively. Also, their utility functions are $u_1(x_1, y_1) = x_1$...
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2 votes

What is the assumption behind "indifference curve does not cross"?

Consider any utility function $u: \mathbb{R}^2_+ \rightarrow\mathbb{R}$. Indifference curve for satisfaction level $\mu$ is defined as: $\text{IC}(\mu) = \{(x, y)\in\mathbb{R}^2_+|u(x, y) = \mu\}$ I ...
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2 votes

Convex Preference but Convex Utility

Example given by Herr K. is perfect. Let me give another example of a dis-continuous utility function which is quasi-concave, but not concave. Consider $u:\mathbb{R}^2_+ \rightarrow \mathbb{R}$ ...
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2 votes

Demand correspondence is both upper and lower hemi-continuous; is the preference continuous?

If the commodity space is $\mathbb{R}^2_+$ and the preference is Lexicographic, then with the standard budget sets $B=B(p_X, p_Y, M) = \{(x, y) \in \mathbb{R}^2 | p_Xx + p_Yy \leq M\}$ where $(p_X, ...
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3 votes
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Constant relative risk aversion for wealth spanning from negative to positive

I'm afraid the answer is no. Let's take any continuous and strictly increasing utility function $U(W)$ which is twice differentiable almost everywhere on the reals. Constant relative risk aversion (...
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1 vote

What are some examples of goods/services whose utility functions have local maxima?

Such a point where utility is a local maxima is called a satiating point. Think of a good that turns into a bad beyond a certain amount. Most non-tradable consumables goods are of such kind, where ...
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0 votes

Reservation utility

It easier to understand this intuitively if one looks at μ as relative bargaining strength. Higher the bargaining strength, higher is the weight attached to their utility in the joint maximization ...
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5 votes
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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 $...
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2 votes
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risk aversion and the law of diminishing marginal utility

If your preferences over lotteries satisfy the vNM-axioms, i.e. if you are an expected utility maximizer, then you cannot be risk averse and at the same time not have diminishing marginal utility of ...
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0 votes

risk aversion and the law of diminishing marginal utility

I used to ask my students if they preferred getting \$4 with probability 1 or \$10 with probability 1/2 and \$0 otherwise. Most students went for the gamble. When I asked for their reasoning they said ...
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2 votes

risk aversion and the law of diminishing marginal utility

Risk aversion means that, when faced with a risky alternative and a sure alternative whose value is equal to the expected value of the risky one, the sure alternative is weakly preferred to the risky ...
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