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## Hot answers tagged consumer-theory

6 votes
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### How do I measure well-being without Utility function?

First, the consumer cannot be worse of. If the consumer would want to buy the same consumption bundle, he could. Here, he does not. Second, if you assume that there is a unique optimal consumption ...
• 13.4k
5 votes

### Does the Marshallian demand function always include prices and income?

Given that $u(x_1,x_2, x_3)=x_1x_2+x_3$, demand is the solution to the following problem: \begin{eqnarray*} \max_{x_1,x_2,x_3} & x_1x_2+x_3 \\ \text{s.t. } & p_1x_1+p_2x_2+p_3x_3 \leq I \\ \...
• 9,316
4 votes
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### Prove a preference preserved under limits if and only if its upper and lower contour is closed

Without any more assumptions, this need not be true. Here is an economically nonsensical but mathematically fine example: Consider the following relation on the real line: $x\succeq y$ holds if and ...
• 13.4k
3 votes
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### Does utility representation theorem need locally-nonsatiated as a condition?

The result holds without any nonsatiation assumption. However, proofs of the result in full generality tend to be very long and often use mathematical methods outside of the scope of MWG. For ...
• 13.4k
3 votes
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### Prove: The law of demand holds if WA, Walras' law, homogeneity of degree 0, and homogeneity of degree 1 in wealth hold for Walrasian demand functions

We need to show that: $$(p - p')\cdot(x(p,1) - x(p', 1)) \le 0.$$ Note that if $x(p,1) = x(p', 1)$ then this is obviously satisfied, so assume that $x(p,1) \ne x(p', 1)$. Note that the condition is ...
• 12.5k
3 votes

### Does the Marshallian demand function always include prices and income?

In response to a correct comment left by user @Amit below my 8-years-old answer, I am adding another answer here to provide intuition and some simple analytical steps that support user's @Amit full ...
• 33.9k
3 votes

### Reasons for why slutsky matrix may be non symmetric

The necessary and sufficient conditions on a demand system are homogeneity of degree 0, Slutsky symmetry and Slutsky negativity. From a theoretical point of view, symmetry of the Slutsky matrix is due ...
• 12.5k
3 votes
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Let $\succeq$ be a relation on $\mathbb{R}^l_+$ such that $x\gg y$ implies $x\succeq y$ for all $x,y\in\mathbb{R}^l_+$, and such that all upper contour sets are closed. Then $x\geq y$ implies $x\... • 13.4k 3 votes Accepted ### Do standard consumer theory axioms rule out corner solutions? A textbook example of a utility function with frequent corner solutions is perfect substitution, e.g.$U(x,y) = x + 2y$. This, however is not strictly convex. But if you think about it, the corner ... • 29.6k 3 votes ### Do standard consumer theory axioms rule out corner solutions? Here are some examples of preferences that satisfy (1) completeness, (2) transitivity, (3) continuity, (4) non-satiation, and (5) strict convexity of the indifference curves, along with the ... • 9,316 3 votes ### Infinite marginal utility for vanishing consumption: Is this always true? The reason for using a power utility function is typically not that it fits some intuition (and I also don't think this intuition is particularly convincing). It is rather that these utility functions ... • 7,109 2 votes ### Utility Maximization of a quasi-linear utility function This is the problem we want to solve: \begin{eqnarray*} \max_{x_1,x_2,x_3} & x_1^{0.5}x_2^{0.5}+cx_3 \\ \text{s.t.} &x_1+2x_2+px_3\leq w \\ \text{and }& x_1\geq 0, x_2\geq 0, x_3\geq0\end{... • 9,316 2 votes Accepted ### The sufficient condition for unique interior solution in utility maximization problem Consider a consumer with utility function$u:\mathbb{R}_{++}\times\mathbb{R}_+\rightarrow\mathbb{R}$defined as follows:$u(x,y)=\sqrt{x}+y$. It is continuous, differentiable, strictly increasing, ... • 9,316 2 votes Accepted ### Consumer theory with subproblem The subproblem gives: $$v(H,\theta) = \max_{h_i} \sum_{i = 1}^I \frac{(\theta_i h_i)^{1 - 1/\eta_i}}{1 - 1/\eta_i} \text{ s.t. } \sum_{i = 1}^I h_i \le H.$$ The first order condition for$h_i$gives,... • 12.5k 2 votes Accepted ###$H$is a constant? Maximizing:$\int _0^Te^{-t}f(x,u)dt$st$x_t=g(t,x,u)$and$g$is independent of$tConsider the problem: \begin{align*} \max_{u} \int_0^T f(x(t), u(t)) dt& \\ \text{ s.t. } &\dot x = g(x(t), u(t)),\\ &\text{ + boundary conditions} \end{align*} Assume thatg(x,u)$... • 12.5k 2 votes ### Exponential Income Consumption Curve One example of the utility function that can give$y=x^2$as income consumption curve is$u(x,y)=\min(x^2,y)$• 9,316 2 votes Accepted ### Example of a utility function which yields inelastic demand function If we could get the demand function$c(p) = 1 - p$(for$p\leq1$) that would be great, because then $$\epsilon(p) = \frac{\text{d}c(p)}{\text{d}p}\frac{p}{c(p)} = -\frac{p}{1-p}$$ would run the ... • 29.6k 2 votes Accepted ### Utility function for a combination of a normal good and necessary good From "Dynamic economics An online textbook with dynamic graphics for the introduction to economics" by Prof. Dr. Christian Bauer: A function$f:\mathbb{R} \to \mathbb{R},(x,y)\to f(x,y)$is ... • 29.6k 2 votes ### Showing UMP and EMP do not exhibit duality if the assumption of local non-satiation is absent A counterexample would work as well. Consider the utility function$U(x) = 0$, and the budget constraint$1 \cdot x \leq 1$. The solution$x^* = 1$is feasible and maximizes utility, but it does not ... • 29.6k 2 votes ### Marshallian demand for x^2+y^2 Observe that the points of tangency of the indifference curve and the budget line are not optimal, as these points lie on a lower indifference curve in all three cases compared to the indifference ... • 9,316 2 votes Accepted ### Proof for Marshallian Demand function Let$x(p,w)$be the demand at prices$p$and income$w$. Let$x_0 = x(p_0,w)$and$x_1 = x(p_1, w)$. Note that$p_0 x_0 = w = p_1 x_1$Assume that WARP is violated, so$x_1 \ne x_0$, $$p_0 x_0 \ge ... • 12.5k 2 votes ### Why is the marginal utility of money assumed to be constant in Marshallian Theory of Consumer Behaviour Unlike the ordinal analysis or the revealed preference approach, where there is no need for a measuring rod, the cardinal utility analysis requires a measuring rod. Money acts as this measuring rod. ... • 161 2 votes ### Finding Utility Function for Optimal Allocation in Consumer Choice Model I think your formula is still too general, so what you want will not be possible. Given -1 < \alpha < 0 and$$ m^* = A \left(\frac{p}{\omega}\right)^\alpha, $$we have$$ \frac{m^*p}{w} = A \... • 29.6k 2 votes ### When are marginal rates of substitution consistent with a utility function? This is a rather indirect way. For$\omega, z \in \mathbb{R}_{++}$, define the (demand) correspondence: $$D(\omega, z) = \left\{(x,y) \in \mathbb{R}^2_+| MRS(x,y) = \omega \text{ and } \omega x + y = ... • 12.5k 2 votes Accepted ### Shape of the contract "curve" If you are happy to restrict attention to the interior, then in a two-good-two-agent economy - given differentiable, strictly concave, and increasing utility functions - the Pareto curve must be ... 2 votes ### Two-period two-good optimal consumption problem I guess you can use a Bellman approach. Observe that optimal x_2 and y_2 are functions of C_2 and second-period prices - I suppress dependence on second-period prices for notational simplicity. ... • 3,728 2 votes Accepted ### Two-period two-good optimal consumption problem You can solve it in this way:$$\max_{\{(a_1,a_2)\in\mathbb{R}^2_+|(1+\rho)a_1+a_2=(1+\rho)w_1+w_2\}} \left[\left[\max_{\{(x_1,y_1)\in\mathbb{R}^2_+|p_{x,1}x_1+p_{y,1}y_1\leq a_1\}} u(x_1,y_1)\right]+\... • 9,316 1 vote Accepted ### Simultaneously a substitute and a complement (validity of a claim) First, you need a clear definition of what is a substitute and what is a complement (as there are various definitions out there). I do not know how most economists would answer, but I would model your ... • 12.5k 1 vote ### Negative marginal utility and negative marginal product Actually, even if in the textbooks in most cases the marginal product is always positive, it is not unusual to have a production function with negative marginal product, that is a total product ... • 4,122 1 vote ### Vertical income expansion path: explanation? If a consumer has the utility function$U(x,y) = x + y$, and$p_x > p_y$, then the optimal bundle is$(x,y) = (0,m/p_y)\$. This seems to result in a vertical income expansion path. Note that the ...
• 29.6k

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