Topic 2: Consumer theory

1. Utility maximization problem (UMP)

• Utility maximization

  $$\begin{array}{}\text{(1)}& \begin{array}{rl}& \underset{x}{max}U(x)\\ & s.t.px\le w\\ & x\ge 0\end{array}\end{array}$$

  Then we can set up the langrage function and use KKT conditions to find possible solutions.

• Are K-T conditions sufficient?

  Yes, when

  • utility function is twice differentiable and quasi-concave

  • all constraints are linear

  • $U_l({\mathbf{x}}^{\mathbf{\ast }})\ne 0$$U_l(\mathbf{x^*})\neq 0$ for some $l=1,...,L$$l = 1,...,L$

• Marshallian demand function:

  By solving the UMP problem,

  $$\begin{array}{}\text{(2)}& {\mathbf{x}}^{\ast }(\mathbf{p},w)=(x_1^{\ast }(\mathbf{p},w),\dots ,{\mathbf{x}}_L^{\ast }(\mathbf{p},w))\end{array}$$

• Properties of Marshallian demand function:

  • If the preference is strictly convex, then only one solution bundle remains under budget constraints (if utility function is strictly quasi-concave, then consumption bundle is unique).

    Proof:

    Suppose there are two consumption bundles $x_1^{\ast }$$x_1^*$ and $x_2^{\ast }$$x_2^*$ both satisfies the budget constraints,

    Since the preference is strictly convex, then let $t\in [0,1]$$t \in\left[0,1\right]$, such that $t{x}_1^{\ast }+(1-t)x_2^{\ast }\succsim x_1^{\ast },x_2^{\ast }$$tx_1^*+(1-t)x_2^* \succsim x_1^*,x_2^*$.

    Which means that $x_1^{\ast }$$x_1^*$ and $x_2^{\ast }$$x_2^*$ are not optimal bundle. Contradict!

  • If preference is convex, but not strictly convex, then there might be multiple ${\mathbf{x}}^{\ast }$$\mathbf{x}^*$ that are not unique.

    Proof:

    Suppose we have two optimal consumption bundle $x_1^{\ast }$$x_1^*$ and $x_2^{\ast }$$x_2^*$, but we have set of the optimal consumption bundles are not convex.

    which means $t{x}_1^{\ast }+(1-t)x_2^{\ast }\prec x_1^{\ast }$$tx_1^* + (1-t)x_2^* \prec x_1^*$,

    This means the upper contour set of utility function is not convex. Contradict!

    The set ${\mathbf{x}}^{\ast }(\mathbf{p},w)$$\mathbf{x}^*(\mathbf{p}, w)$ has to be convex.

    

  • ${\mathbf{x}}^{\ast }(\mathbf{p},w)$$\mathbf{x}^*(\mathbf{p}, w)$ is homogeneous degree of 0 with respect to $(\mathbf{p},w)$$(\mathbf{p}, w)$.

    This is because $\underset{x}{max}U(x),s.t.px\le w$$\max_x U(x),\ s.t.\ px \leq w$ is equivalent to $s.t.tpx\le tw$$s.t.\ tpx \leq tw$.

     

  • When utility is strictly quasi-concave (preference is strictly convex, indifference curve is strictly convex), and monotone, then our Lagrange function with equality constraint could be a sufficient method for solving UMP.

• Indirect utility function: substituting the ${\mathbf{x}}^{\ast }$$\mathbf{x}^*$ into the objective function

• Properties of indirect utility functions

  • $v(\mathbf{p},w)$$v(\mathbf{p}, w)$ is strictly increasing in $w$$w$.

    Proof

    $$\begin{array}{}\text{(4)}& V(p_1,p_2,w)=U(x_1^w(p_1,p_2,w),x_2^w(p_1,p_2,w))\end{array}$$

    对$w$$w$求偏导数 Diffrentiate w.r.t. $w$$w$:：

    $$\begin{array}{}\text{(5)}& \frac{\partial V}{\partial w}\equiv \frac{\partial U}{\partial x_1}\frac{\partial x_1^w}{\partial w}+\frac{\partial U}{\partial x_2}\frac{\partial x_2^w}{\partial w}\end{array}$$

    使用一阶条件 FOC：$U_1=\lambda p_1$$U_1 = \lambda p_1$ 和 $U_2=\lambda p_2$$U_2 = \lambda p_2$：

    $$\begin{array}{}\text{(6)}& \frac{\partial V}{\partial w}\equiv \lambda (p_1\frac{\partial x_1^w}{\partial w}+p_2\frac{\partial x_2^w}{\partial w}).\end{array}$$

    现在，当将需求曲线重新插入预算约束中时，我们得到一个恒等式：

    $$\begin{array}{}\text{(7)}& p_1{x}_1^w+p_2{x}_2^w\equiv w\end{array}$$

    对其求导，我们得到：

    $$\begin{array}{}\text{(8)}& p_1\frac{\partial x_1^w}{\partial w}+p_2\frac{\partial x_2^w}{\partial w}\equiv 1.\end{array}$$

    这给出了：$\frac{\partial V}{\partial w}\equiv \lambda >0$$\frac{\partial V}{\partial w} \equiv \lambda > 0$。

  • $v(\mathbf{p},w)$$v(\mathbf{p}, w)$ is homogeneous of degree 0 in $(\mathbf{p},w)$$(\mathbf{p}, w)$.

    This is because ${\mathbf{x}}^{\mathbf{\ast }}(\mathbf{p},w)$$\mathbf{x^*}(\mathbf{p}, w)$ is H.D.0, ${\mathbf{x}}^{\mathbf{\ast }}(\mathbf{tp},tw)={\mathbf{x}}^{\mathbf{\ast }}(\mathbf{p},w)$$\mathbf{x^*}(\mathbf{tp}, tw) = \mathbf{x^*}(\mathbf{p}, w)$

  • $v(\mathbf{p},w)$$v(\mathbf{p}, w)$ is quasi-convex in $(\mathbf{p},w)$$(\mathbf{p}, w)$

    Proof

    for $\mathrm{\forall }({\mathbf{p}}^{\mathbf{1}},w^1),({\mathbf{p}}^{\mathbf{2}},w^2)$$\forall (\mathbf{p^1},w^1), (\mathbf{p^2},w^2)$,

    Let (for $i=1,2$$i = 1,2$)

    $$\begin{array}{}\text{(9)}& \begin{array}{c}{v}^i=v({\mathbf{p}}^{\mathbf{i}},w^i)\\ {\mathbf{x}}^i={\mathbf{x}}^{\ast }({\mathbf{p}}^i,w^i)\end{array}\end{array}$$

    Define

    $$\begin{array}{}\text{(10)}& \begin{array}{c}{\mathbf{p}}^t=t{\mathbf{p}}^1+(1-t){\mathbf{p}}^2\\ w^t=t{w}^1+(1-t)w^2\\ v^t=v({\mathbf{p}}^t,w^t)\\ {\mathbf{x}}^t={\mathbf{x}}^{\ast }({\mathbf{p}}^t,w^t)\end{array}\end{array}$$

    , where $t\in (0,1)$$t\in(0,1)$

    If we want show $v^t=v({\mathbf{p}}^t,w^t)$$v^t = v(\mathbf{p}^t,w^t)$ is quasi-convex, we have to show

    $$\begin{array}{}\text{(11)}& v^t<max\{v^1,v^2\}\end{array}$$

    Let's prove by contradiction,

    Assume $v^t>v^1,v^2$$v^t > v^1, v^2$, then we have

    $$\begin{array}{}\text{(12)}& \begin{array}{c}{\mathbf{p}}^1{x}^t>w^1\\ {\mathbf{p}}^2{x}^t>w^2\end{array}\end{array}$$

    , because $p^1{x}^1=w^1$$p^1x^1 = w^1$, and to achieve a better utility consumption bundle, the expenditure has to be increased.

    This will result in ${\mathbf{p}}^t{x}^t>w^t$$\mathbf{p}^tx^t > w^t$ since $t{\mathbf{p}}^1+(1-t){\mathbf{p}}^2>t{w}^1+(1-t)w^2$$t \mathbf{p}^1+(1-t) \mathbf{p}^2 > t w^{1}+(1-t) w^2$. Contradiction!

     

• Walras' law

  With strictly monotonic preference, we have Walras' law

  $$\begin{array}{}\text{(13)}& {\mathbf{p}}^T{\mathbf{x}}^{\ast }(\mathbf{p},w)=w\end{array}$$

  • Taking derivative w.r.t. $w$$w$

    $$\begin{array}{}\text{(14)}& {\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial w}|}_{{\mathbf{x}}^{\ast }}=1\end{array}$$

    • This means the change in expenditure and the change in wealth must be the same.

    • Not all commodities can be inferior good. Because $\frac{\partial x_{\ell }}{\partial w}<0$$\frac{\partial x_{\ell}}{\partial w}<0$ for each consumption good will result in a violation of Walras' law.

    • We can derive Engel Aggregation from here:

      $$\begin{array}{}\text{(15)}& \begin{array}{rl}& {\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial w}|}_{{\mathbf{x}}^{\ast }}=1\\ \Rightarrow & \sum _{\ell =1}^L{\eta }_{\ell }{s}_{\ell }=1\end{array}\end{array}$$

      Note that $\eta$$\eta$ represents the income elasticity,

      $$\begin{array}{}\text{(16)}& {\eta }_{\ell }=\frac{\partial x_{\ell }/x_{\ell }}{\partial w/w}=\frac{\partial x_{\ell }}{\partial w}\frac{w}{x_{\ell }}\end{array}$$

      , and $s_l$$s_l$ represents the expenditure share on good $l$$l$

      $$\begin{array}{}\text{(17)}& s_l=\frac{p_l{x}_l}{w}\end{array}$$

    •  

     

  • Taking derivative w.r.t. $p_j$$p_j$

    $$\begin{array}{}\text{(18)}& x_j^{\ast }(\mathbf{p},w)+{\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial p_j}|}_{{\mathbf{x}}^{\ast }}=0\text{ for all }j=1,\dots ,n\end{array}$$

    • We can derive the Cournot aggregation

      $$\begin{array}{}\text{(19)}& \begin{array}{l}{x}_j^{\ast }(\mathrm{p},w)+{\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial p_j}|}_{{\mathrm{x}}^{\ast }}=0\\ \Rightarrow \sum _{\ell =1}^L{s}_{\ell }{ϵ}_{\ell j}=-s_j\end{array}\end{array}$$

      Note that ${ϵ}_{ij}$$\epsilon_{ij}$ is the price elasticity

      $$\begin{array}{}\text{(20)}& {ϵ}_{ij}=\frac{\partial x_i}{\partial p_j}\frac{p_j}{x_i}\end{array}$$

• Roy's identity

  $$\begin{array}{}\text{(21)}& x_j^{\ast }(\mathbf{p},w)=-\frac{\frac{\partial v(\mathbf{p},w)}{\partial p_j}}{\frac{\partial v(\mathbf{p},w)}{\partial w}}\end{array}$$

  Proof:

  By envelope theorem:

  $$\begin{array}{}\text{(22)}& \begin{array}{c}\frac{\partial v}{\partial p_l}=-\lambda x_l^{\star }\\ \frac{\partial v}{\partial w}=\lambda >0\end{array}\end{array}$$

  Then we have the Roy's identity.

  Intuition:

  • $\frac{\partial v}{\partial w}=\lambda >0$$\frac{\partial v}{\partial w } = \lambda >0$, income more results in more utility

  • $\frac{\partial v}{\partial p_l}=-\lambda x_l^{\star }$$\frac{\partial v}{\partial p_l} = -\lambda x_l^\star$. it is 0 if $x_l^{\ast }=0$$x^*_l = 0$. If we do not consume on $x_l$$x_l$, its price have no impact on our utility.

2. Expenditure minimization problem (EMP)

• Hicksian demand and expenditure function

  After solving an EMP, we have Hicksian demand $h(p,u)$$h(p,u)$ or $x^h(p,u)$$x^h(p,u)$. And expenditure function $e(p,u)=p{x}^h(p,u)$$e(p,u) = p x^h(p,u)$.

• Properties of expenditure function

  • strictly increasing in $u$$u$

    $$\begin{array}{}\text{(23)}& \begin{array}{c}e(\mathbf{p},u)\equiv \underset{X\in R_{+}^n}{min}\mathbf{p}\mathbf{X}\text{ s.t. }U(\mathbf{X})=u\\ \mathcal{L}(\mathbf{X},\lambda )=\mathbf{p}\mathbf{X}+\lambda [U-U(X)]\\ \text{ FOCs: }\frac{\partial \mathcal{L}(X^{\ast },{\lambda }^{\ast })}{\partial X_i}=p_i-{\lambda }^{\ast }\frac{\partial U\left(X^{\ast }\right)}{\partial x_i}=0,i=1,2,\dots ,n.\end{array}\end{array}$$

    The solution is : $X^{\ast }=X^h(p,U)\gg 0$$X^*=X^h(p, U) \gg 0$ for $p\gg 0$$p \gg 0$ and $U>U(0)$$U>U(0)$, i.e. Hicksian demands. Because $p_i>0$$p_i>0$ and $\frac{\partial U\left(X^{\ast }\right)}{\partial X_i}>0$$\frac{\partial U\left(X^*\right)}{\partial X_i}>0$, so is ${\lambda }^{\ast }>0$$\lambda^*>0$, hence by envelope theorem:

    $$\begin{array}{}\text{(24)}& \frac{\partial e(p,U)}{\partial U}=\frac{\partial \mathcal{L}(X^{\ast },{\lambda }^{\ast })}{\partial U}={\lambda }^{\ast }>0\end{array}$$

    In this case $\lambda$$\lambda$ is marginal cost of utility (the inverse of the marginal utility of income). It shows by how much is the minimum expenditure going to increase if we increase utility by a small amount.

  • Non-decreasing in $p$$p$.

    By Shephard' lemma: ${\frac{\partial e}{\partial p_{\ell }}|}_{{\mathbf{x}}^{\ast }}=h_{\ell }^{\ast }(\mathbf{p},u)\ge 0$$\left.\frac{\partial e}{\partial p_{\ell}}\right|_{\mathbf{x}^*}=h_{\ell}^*(\mathbf{p}, u) \geq 0$

  • $e(p,u)$$e(p,u)$ is homogeneous of degree 1 in prices

    Intuition:

    No relative price change, all price goes up, to reach the same utility, Hicksian demand $x^h(tp,u)=x^h(p,u)$$x^h(tp,u) = x^h(p,u)$. Therefore, expenditure $e(tp,u)=x^h(p,u)\times tp$$e(tp,u) = x^h(p,u)\times tp$, which is H.D.1

  • Concave in $p$$p$

    Proof:

    Suppose ${\mathbf{h}}^1$$\mathbf{h}^1$ and ${\mathbf{h}}^2$$\mathbf{h}^2$ are solutions to the expenditure minimization problems with ${\mathbf{p}}^1$$\mathbf{p}^1$ and ${\mathbf{p}}^2$$\mathbf{p}^2$ respectively. For $\mathrm{\forall }t\in [0,1]$$\forall t \in[0,1]$, define ${\mathbf{p}}^t=t{\mathbf{p}}^1+(1-t){\mathbf{p}}^2$$\mathbf{p}^t=t \mathbf{p}^1+(1-t) \mathbf{p}^2$.

    $$\begin{array}{}\text{(25)}& \begin{array}{rl}e({\mathbf{p}}^t,u)& \equiv {\mathbf{p}}^t{\mathbf{h}}^t\\ & =t{\mathbf{p}}^1{\mathbf{h}}^t+(1-t){\mathbf{p}}^2{\mathbf{h}}^t\\ & \ge te({\mathbf{p}}^1,u)+(1-t)e({\mathbf{p}}^2,u)\end{array}\end{array}$$

    Intuition: EMP 解出来是最小的expenditure，任何在这个价格水平下其他达到这个utility的consumption bundle 对应的expenditures 都要更大

   

• Properties of Hicksian demand function

  • Homogeneous of degree 0 in $\mathbf{p}$$\mathbf{p}$

  • No excess utility: $u(\mathbf{h}(\mathbf{p},\underset{―}{u}))=\underset{―}{u}$$u(\mathbf{h}(\mathbf{p}, \underline{u}))=\underline{u}$, binding constraint and ${\gamma }^{\ast }>0$$\gamma^*>0$

  • Cross elasticity: $\frac{\partial h_i}{\partial p_j}=\frac{{\partial }^{2}e}{\partial p_i\partial p_j}$$\frac{\partial h_i}{\partial p_j}=\frac{\partial^2 e}{\partial p_i \partial p_j}$

  • The matrix of own and cross-partial derivatives w.r.t $\mathbf{p}$$\mathbf{p}$ is symmetric and negative semi-definite - Hicksian substitution matrix $e(p,u)$$e(p, u)$ is concave in $p$$p$.

  • Non-positive own-price elasticity: $\frac{\partial h_{\ell }(\mathbf{p},u)}{\partial p_{\ell }}=\frac{{\partial }^{2}e(\mathbf{p},u)}{\partial p_{\ell }^2}\le 0$$\frac{\partial h_{\ell}(\mathbf{p}, u)}{\partial p_{\ell}}=\frac{\partial^2 e(\mathbf{p}, u)}{\partial p_{\ell}^2} \leq 0$

3. Duality

When ${\mathbf{x}}^{\ast }\gg 0$$\mathbf{x}^* \gg 0$

• In expenditure minimization: ${\gamma }^{\ast }\mathrm{\nabla }u\left({\mathbf{x}}^{\ast }\right)=\mathbf{p}$$\gamma^* \nabla u\left(\mathbf{x}^*\right)=\mathbf{p}$

• In utility maximization: $\mathrm{\nabla }u\left({\mathbf{x}}^{\ast }\right)={\lambda }^{\ast }\mathbf{p}$$\nabla u\left(\mathbf{x}^*\right)=\lambda^* \mathbf{p}$

• Both give $MR{S}_{ij}=\frac{p_i}{p_j}$$M R S_{i j}=\frac{p_i}{p_j}$

• In terms of demand functions

• In terms of value functions

• In terms of Lagrange multipliers: ${\gamma }^{\ast }=\frac{1}{{\lambda }^{\ast }}$$\gamma^*=\frac{1}{\lambda^*}$

• They are dual problems in the sense that they contain the same information

3.1 Duality between $U(q)$$U(q)$ and $V(p)$$V(p)$

$U(q)$$U(q)$ $\to$$\rightarrow$ UMP $\to$$\rightarrow$ $x^{\ast }(p,w)$$x^*(p,w)$ $\to$$\rightarrow$ $V(p,w)=U(x(p,w))$$V(p,w) = U(x(p,w))$

$V(p)$$V(p)$ $\to$$\rightarrow$ Roy's identity: $x_j^{\ast }(\mathbf{p},w)=-\frac{\frac{\partial v(\mathbf{p},w)}{\partial p_j}}{\frac{\partial v(\mathbf{p},w)}{\partial w}}$$x_j^*(\mathbf{p},w) = -\frac{\frac{\partial v(\mathbf{p},w)}{\partial p_j}}{\frac{\partial v(\mathbf{p},w)}{\partial w}}$ $\to$$\rightarrow$ $U(q)$$U(q)$

3.2 Indirect utility and expenditure function

3.3 Marshallian and Hicksian demand

By duality, $h_j(\mathbf{p},u)=x_j(\mathbf{p},e(\mathbf{p},u))$$h_j(\mathbf{p}, u)=x_j(\mathbf{p}, e(\mathbf{p}, u))$, thus

• Slutsky equation links the observable $\mathbf{x}(\mathbf{p},w)$$\mathbf{x}(\mathbf{p}, w)$ to the "useful" $\mathbf{h}(\mathbf{p},u)$$\mathbf{h}(\mathbf{p}, u)$;

• The process is called Hicksian decomposition

  • (Hicksian) Substitution effect

  • Income effect

• More on Hicksian substitution

  • The cross substitution of Hicksian demand is symmatric:

    $$\begin{array}{}\text{(30)}& \frac{\partial x_i^h(\mathbf{p},U)}{\partial p_j}=\frac{\partial x_j^h(\mathbf{p},U)}{\partial p_i}i,j=1,\dots ,n\end{array}$$

    Proof - By the Young's Theorem

    Since we have $\frac{\partial e(\mathbf{p},U)}{\partial p_i}=x_i^h(\mathbf{p},U)$$\frac{\partial e(\mathbf{p}, U)}{\partial p_i}=x_i^h(\mathbf{p}, U)$, we can derive

    $$\begin{array}{}\text{(31)}& \frac{{\partial }^{2}e(\mathbf{p},U)}{\partial p_i^2}=\frac{\partial x_i^h(\mathbf{p},U)}{\partial p_i}\le 0.\end{array}$$

    , then we have,

    $$\begin{array}{}\text{(32)}& \frac{\partial x_j^h}{\partial p_i}=\frac{{\partial }^{2}e(\mathbf{p},U)}{\partial p_j\partial p_i}=\frac{{\partial }^{2}e(\mathbf{p},U)}{\partial p_i\partial p_j}=\frac{\partial x_i^h}{\partial p_j}\end{array}$$

    That implies the substitutive terms are symmetric.

  • Combining the results, we can derive a matrix

    $$\begin{array}{}\text{(33)}& \begin{array}{c}\sigma (\mathbf{p},u)\equiv \left[\begin{array}{ccc}\frac{\partial x_1^h(\mathbf{p},u)}{\partial p_1}& \dots & \frac{\partial x_1^h(\mathbf{p},u)}{\partial p_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial x_n^h(\mathbf{p},u)}{\partial p_1}& \cdots & \frac{\partial x_n^h(\mathbf{p},u)}{\partial p_n}\end{array}\right]\end{array}\end{array}$$

    called (Hicksian) substitution matrix which has to be negative semidefinite.

• Slutsky Matrix

  Notice that Slutsky equation can be written as:

  $$\begin{array}{}\text{(34)}& \frac{\partial x_i^h(\mathbf{p},u)}{\partial p_j}=\frac{\partial x_i^M(\mathbf{p},M)}{\partial p_j}+x_j^M\frac{\partial x_i^M(\mathbf{p},M)}{\partial M}\end{array}$$

  So $\sigma (\mathbf{p},U)$$\sigma(\mathbf{p}, U)$ can be replaced by| the sum of 2 elements on the RHS to obtain Slutsky matrix $\mathbf{S}(\mathbf{p},M)$$\mathbf{S}(\mathbf{p}, M)$. Obviously:

  $$\begin{array}{}\text{(35)}& \sigma (\mathbf{p},U)=\mathbf{S}(\mathbf{p},M)\end{array}$$

  and hence $\mathbf{S}(\mathbf{p},M)$$\mathbf{S}(\mathbf{p}, M)$ has to be symmetric and negative semi-definite.

• Slutsky equation in elasticity

  Slutsky equation can be written in dimensionless elasticity form. Multiply by $\frac{p_j}{x_i}$$\frac{p_j}{x_i}$ throughout and the last term by $\frac{M}{M}=1$$\frac{M}{M}=1$, we can get

  $$\begin{array}{}\text{(36)}& \begin{array}{c}\frac{\partial x_i^M}{\partial p_j}\frac{p_j}{x_i}=\frac{\partial x_i^h}{\partial p_j}\frac{p_j}{x_i}-\frac{p_j}{x_i}{x}_j^M\frac{\partial x_i^M}{\partial M}\frac{M}{M}\\ {ϵ}_{ij}^M={ϵ}_{ij}^h-\underset{{\alpha }_j}{\underbrace{\frac{p_j{x}_j^M}{M}}}{ϵ}_{iM}\end{array}\end{array}$$

  The difference between the (cross) price elasticity of the uncompensated and compensated demand curves depends on income elasticity of the good $(i)$$(i)$ and the importance of the good whose price has changed $(j)$$(j)$ measured by its budget share.