Micro review sketch

Micro I

Consumer Theory

1. Utility maximization and expenditure minimization, Marshallian demand and hicksian demand.

   • 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$​

     Suppose that $u(\mathbf{x})$$u(\mathbf{x})$ is continuous and quasiconcave on ${\mathbb{R}}_{+}^n$$\mathbb{R}_{+}^n$, and that $(\mathbf{p},y)\gg \mathbf{0}$$(\mathbf{p}, y) \gg \mathbf{0}$. If $u$$u$ is differentiable at ${\mathbf{x}}^{\ast }$$\mathbf{x}^*$, and $({\mathbf{x}}^{\ast },{\lambda }^{\ast })\gg \mathbf{0}$$\left(\mathbf{x}^*, \lambda^*\right) \gg \mathbf{0}$ solves (1), then ${\mathbf{x}}^{\ast }$$\mathbf{x}^*$ solves the consumer's maximisation problem at prices $\mathbf{p}$$\mathbf{p}$ and income $y$$y$.

   • Marshallian demand function:

     By solving the UMP problem,

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

   • Other properties on Marshallian demand, see my note on Topic 2 on website

2. Indirect utility

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

   $$\begin{array}{}\text{(3)}& v(\mathbf{p},w)=u({\mathbf{x}}^{\ast }(\mathbf{p},w))\end{array}$$

   • Some properties, see my note on Topic 2 on website.

3. Walras law, Cournot aggregation, engel aggregation

   • With strictly monotonic preference, we have Walras' law

     $$\begin{array}{}\text{(4)}& {\mathbf{p}}^T{\mathbf{x}}^{\ast }(\mathbf{p},w)=w\end{array}$$

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

       $$\begin{array}{}\text{(5)}& {\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial w}|}_{{\mathbf{x}}^{\ast }}=1\end{array}$$

       • 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:

         $$\begin{array}{}\text{(6)}& \begin{array}{rl}& {\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial w}|}_{{\mathbf{x}}^{\ast }}=1\\ \Rightarrow & \sum _{\ell =1}^L{\eta }_{\ell }{s}_{\ell }=1\end{array}\end{array}$$

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

         $$\begin{array}{}\text{(7)}& {\eta }_{\ell }=\frac{\partial x_{\ell }/x_{\ell }}{\partial w/w}=\frac{\partial x_{\ell }}{\partial w}\frac{w}{x_{\ell }}\end{array}$$

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

         $$\begin{array}{}\text{(8)}& s_l=\frac{p_l{x}_l}{w}\end{array}$$

       •  

        

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

       $$\begin{array}{}\text{(9)}& x_j^{\ast }(\mathbf{p},w)+{\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial p_j}|}_{{\mathbf{x}}^{\ast }}=0\text{ for all }j=1,\dots ,n\end{array}$$

       • We can derive the Cournot aggregation

         $$\begin{array}{}\text{(10)}& \begin{array}{l}{x}_j^{\ast }(\mathrm{p},w)+{\sum _{\ell =1}^L{p}_{\ell }\frac{\partial x_{\ell }}{\partial p_j}|}_{{\mathrm{x}}^{\ast }}=0\\ \Rightarrow \sum _{\ell =1}^L{s}_{\ell }{ϵ}_{\ell j}=-s_j\end{array}\end{array}$$

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

         $$\begin{array}{}\text{(11)}& {ϵ}_{ij}=\frac{\partial x_i}{\partial p_j}\frac{p_j}{x_i}\end{array}$$

4. Expenditure minimization and Hicksian demand

   The cross substitution of Hicksian demand is symmatric:

   $$\begin{array}{}\text{(12)}& \frac{\partial x_i^h(\mathbf{p},U)}{\partial p_j}=\frac{\partial x_j^h(\mathbf{p},U)}{\partial p_i}i,j=1,\dots ,n\end{array}$$

    

5. Roy's identity

   $$\begin{array}{}\text{(13)}& x_j^{\ast }(\mathbf{p},w)=-\frac{\frac{\partial v(\mathbf{p},w)}{\partial p_j}}{\frac{\partial v(\mathbf{p},w)}{\partial w}}\end{array}$$

6. Duality

   $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)$​

   • Duality between the direct and indirect utility (we can recover the direct utility from the indirect utility)

   $$\begin{array}{}\text{(14)}& u(\mathbf{x})=\underset{\mathbf{p}\in {\mathbb{R}}_{++}^n}{min}v(\mathbf{p},1)\text{ s.t. }\mathbf{p}\cdot \mathbf{x}=1\text{. }\end{array}$$

   • Specifically, if we want to recover the direct utility from an indirect utility function, we can do the following procedures.

     Step 1: Normalize the $w=1$$w = 1$.

     Step 2: $\underset{\mathbf{p}\in {\mathbb{R}}_{++}^n}{min}v(\mathbf{p},1)\text{ s.t. }\mathbf{p}\cdot \mathbf{x}=1\text{. }$$\min _{\mathbf{p} \in \mathbb{R}_{++}^n} v(\mathbf{p}, 1) \quad \text { s.t. } \quad \mathbf{p} \cdot \mathbf{x}=1 \text {. }$ and obtain the inverse demand function $p(\mathbf{x},1)$$p(\bold x,1)$.

     Step 3: plug the inverse demand function into the indirect utility function to obtain the direct utility function.

   • Indirect utility and expenditure function

     $$\begin{array}{}\text{(15)}& \begin{array}{rl}u& \equiv v(\mathbf{p},e(\mathbf{p},u))\\ w& \equiv e(\mathbf{p},v(\mathbf{p},w))\end{array}\end{array}$$

   • Slutsky equation

     $$\begin{array}{}\text{(16)}& \frac{\partial x_j(\mathbf{p},w)}{\partial p_i}=\frac{\partial h_j(\mathbf{p},u)}{\partial p_i}-\frac{\partial x_j(\mathbf{p},w)}{\partial w}{x}_i\end{array}$$

   • Slutsky equation in elasticity form

     $$\begin{array}{}\text{(17)}& \begin{array}{c}\frac{\partial x_i^M}{\partial p_j}\frac{p_j}{x_i}=\frac{\partial x_i^h}{\partial p_j}\frac{p_j}{x_i}-\frac{p_j}{x_i}{x}_j^M\frac{\partial x_i^M}{\partial M}\frac{M}{M}\\ {ϵ}_{ij}^M={ϵ}_{ij}^h-\underset{{\alpha }_j}{\underbrace{\frac{p_j{x}_j^M}{M}}}{ϵ}_{iM}\end{array}\end{array}$$

   • Hotelling-Wold Identity (Derive the inverse demand function from the direct utility)

     Let $u(\mathbf{x})$$u(\mathbf{x})$ be the consumer's direct utility function. Then the inverse demand function for good $i$$i$ associated with income $y=1$$y=1$ is given by

     $$\begin{array}{}\text{(18)}& p_i(\mathbf{x})=\frac{\partial u(\mathbf{x})/\partial x_i}{\sum _{j=1}^n{x}_j(\partial u(\mathbf{x})/\partial x_j)}.\end{array}$$

7. Welfare - CV and EV

   CV: compensation variation, EV: equivalence variation.

   Consider a change in price vector from $p^0$$p^0$ to $p^1$$p^1$ :

   • $CV(p^0,p^1,y^0)$$C V\left(p^0, p^1, y^0\right)$ : holding nominal income constant at $y^0,CV$$y^0, C V$ is the minimum amount of money required to keep the consumer as well of as $\mathrm{s}$$\mathrm{s}$ was in the initial state $(p^0,y^0)$$\left(p^0, y^0\right)$ as she is in the new state $(p^1,y^0+CV)$$\left(p^1, y^0+C V\right)$. In terms of the expenditure function:

   $$\begin{array}{}\text{(19)}& \begin{array}{rl}CV(p_1^0,p_1^1,y^0)& =e(p^1,U^0)-e(p^0,U^0)\\ & =e(p^1,U^0)-y^0\end{array}\end{array}$$

   where $U^0=V(p^0,y^0)$$U^0=V\left(p^0, y^0\right)$ ​ In terms of the indirect utility function:

   $$\begin{array}{}\text{(20)}& V(p^1,y^0+CV)=V(p^0,y^0)\end{array}$$

   • $EV(p^0,p^1,y^0)$$E V\left(p^0, p^1, y^0\right)$ : uses utility after the price change as the basis for comparison:

   $$\begin{array}{}\text{(21)}& EV(p^0,p^1,y^0)=e(p^1,U^1)-e(p^0,U^1)\end{array}$$

8. CV, EV and CS (Path dependence)

   • Consider first a case when only one price changes from $p_1^0$$p_1^0$ to $p_1^1$$p_1^1$, all other prices are held constant: (a) Integrating between the two prices gives:

     $$\begin{array}{}\text{(22)}& \begin{array}{rl}CV(p^0,p^1,y^0)& =e(p^1,U^0)-e(p^0,U^0)\\ & ={\int }_{p_1^0}^{p_1^1}{h}_1(p,U^0)d{p}_1={\int }_{p_1^0}^{p_1^1}\frac{\partial e(p,U^0)}{\partial p_1}d{p}_1\end{array}\end{array}$$

     (b) Alternatively:

     $$\begin{array}{}\text{(23)}& \begin{array}{rl}EV(p^0,p^1,y^0)& =e(p^1,U^1)-e(p^0,U^1)\\ & ={\int }_{p_1^0}^{p_1^1}{h}_1(p,U^1)d{p}_1={\int }_{p_1^0}^{p_1^1}\frac{\partial e(p,U^1)}{\partial p_1}d{p}_1\end{array}\end{array}$$

   • How do we interpret the consumer surplus that derived from the Marshallian demand?

     CS is an uncompensated measurement.

     $$\begin{array}{}\text{(24)}& \mathrm{CS}(p^0,p^1,y^0)={\int }_{p_1^0}^{p_1^1}{x}_i(p,y^0)d{p}_1=-{\int }_{p_1^0}^{p_1^1}\frac{\frac{\partial V(p,y^0)}{\partial p_1}}{\frac{\partial V(p,y^0)}{\partial y}}d{p}_1\end{array}$$

     To keep the individual at the same indifference curve, income level must be constantly adjusted along the path of price change. Since in CS, the income is fixed, this produce the difference between compensated and uncompensated welfare measures.

9. Homothetic functions

   Let $y=f(x_1,\dots ,x_n)$$y=f\left(x_1, \ldots, x_n\right)$ be homogenous of degree $k$$k$, and let $z=F(y)$$z=F(y)$, where $F^{\prime }(y)>0$$F^{\prime}(y)>0$ (then $F$$F$ is a monotonic transformation of $y$$y$ ), the function $z(x_1,x_2,\dots ,x_n)$$z\left(x_1, x_2, \ldots, x_n\right)$ is homothetic.

   • Homothetic functions have MRS homogeneous of degree 0 :

     $$\begin{array}{}\text{(25)}& \frac{\frac{\partial U}{\partial x_i}(tx)}{\frac{\partial U}{\partial x_j}(tx)}=\frac{\frac{\partial U}{\partial x_i}(x)}{\frac{\partial U}{\partial x_j}(x)}\text{ for all }t>0\end{array}$$

10. Quasi-linear functions and Gorman form,

    $$\begin{array}{}\text{(26)}& \begin{array}{r}U=x_i+V\left(x_{-i}\right)\\ V^{\prime }(x)>0\\ V^{\prime \prime }(x)\le 0\end{array}\end{array}$$

    for example: $U=x_2+\log x_1$$U=x_2+\log x_1$.

    • This situation produces vertically parallel indifference curve.

    • Demands for all other goods that do not enter linearly are independent of income and so is $\lambda$$\lambda$ (marginal utility of money).

    • Similarly for any general quasilinear utility function

    • $u(x_1,x_2)=v\left(x_1\right)+\beta x_2$$u\left(x_1, x_2\right)=v\left(x_1\right)+\beta x_2$.

    • Hence, $M{U}_1=v^{\prime }\left(x_1\right)$$M U_1=v^{\prime}\left(x_1\right)$, and $M{U}_2=\beta$$M U_2=\beta$, thus implying $MRS=\frac{v^{\prime }\left(x_1\right)}{\beta }$$M R S=\frac{v^{\prime}\left(x_1\right)}{\beta}$, which is independent of $x_2$$x_2$.

    • For any given level of $x_1$$x_1$ the increase in $x_2$$x_2$ does not affect the slope of the indifference curve.

      

Revealed Preference

WARP

If:

Then:

If $X^0$$X^0$ is revealed preferred to $X^1$$X^1$, then $X^1$$X^1$ can never be revealed preferred to $X^0$$X^0$​.

SARP

One property of the system of consumer demands that WARP does not imply is symmetry. For symmetry we need SARP.

The Strong Axiom of Revealed Preference (SARP) is satisfied if, for every sequence of distinct bundles ${\mathbf{x}}^0,{\mathbf{x}}^1,\dots ,{\mathbf{x}}^k$$\mathbf{x}^0, \mathbf{x}^1, \ldots, \mathbf{x}^k$, where ${\mathbf{x}}^0$$\mathbf{x}^0$ is revealed preferred to ${\mathbf{x}}^1$$\mathbf{x}^1$, and ${\mathbf{x}}^1$$\mathbf{x}^1$ is revealed preferred to ${\mathbf{x}}^2,\dots$$\mathbf{x}^2, \ldots$, and ${\mathbf{x}}^{k-1}$$\mathbf{x}^{k-1}$ is revealed preferred to ${\mathbf{x}}^k$$\mathbf{x}^k$, it is not the case that ${\mathbf{x}}^k$$\mathbf{x}^k$ is revealed preferred to ${\mathbf{x}}^0$$\mathbf{x}^0$. SARP rules out intransitive revealed preferences and therefore can be used to induce a complete and transitive preference relation, $\succsim$$\succsim$, for which there will then exist a utility function that rationalizes the observed behaviour.

Uncertainty

1. vNM Utility

   If $(L,⪰)$$(L, \succeq)$ satisfy above axioms, there is a utility function $U:L\to \mathbb{R}$$U: L \rightarrow \mathbb{R}$ that satisfy the expected utility property: $U(p\circ x\otimes (1-p)\circ y)=pU(x)+(1-p)U(y)$$U(p \circ x \otimes(1-p) \circ y)=p U(x)+(1-p) U(y)$. Such a utility function is called von Neumann-Morgenstern (VNM) expected utility function.

2. Certainty equivalent

   The amont of money $c(F,U)$$c(F,U)$ for which the individual is indifferent between the lottery and the certain amont.

   $$\begin{array}{}\text{(29)}& U[c(F,U)]=\int U(x)dF(x)\end{array}$$

3. Risk premium ($\pi$$\pi$)

   The maximum amount of money that an agent is willing to pay to receive the expected value of the lottery instead of receiving the lottery itself.

4. Arrow-Reatt coefficient of absolute risk aversion

   $$\begin{array}{}\text{(32)}& r_A(x)=-\frac{U^{\prime \prime }(x)}{U^{\prime }(x)}\end{array}$$

   The larger this ratio is, the larger risk averse is the agent.

   $R_a(W)$$R_a(W)$ is only a local measure of risk aversion, so it need not be the same at every level of wealth. Indeed, one expects that attitudes toward risk, and so the Arrow-Pratt measure, will ordinarily vary with wealth, and vary in 'sensible' ways. Arrow has proposed a simple classification of VNM utility functions (or utility function segments) according to how $R_a(w)$$R_a(w)$ varies with wealth. Quite straightforwardly, we say that a VNM utility function displays constant, decreasing, or increasing absolute risk aversion over some domain of wealth if, over that interval, $R_a(w)$$R_a(w)$ remains constant, decreases, or increases with an increase in wealth, respectively.

   Decreasing absolute risk aversion (DARA) is generally a sensible restriction to impose. Under constant absolute risk aversion, there would be no greater willingness to accept a small gamble at higher levels of wealth, and under increasing absolute risk aversion, we have rather perverse behavior: the greater the wealth, the more averse one becomes to accepting the same small gamble. DARA imposes the more plausible restriction that the individual be less averse to taking small risks at higher levels of wealth.

5. Relative risk aversion

   $$\begin{array}{}\text{(33)}& r_R=-x\frac{U^{\prime \prime }(x)}{U^{\prime }(x)}\end{array}$$

Producer Theory

Production function

1. MRTS

   Measures the rate at which one input can be substituted for another without changing the amount of output produced.

   $$\begin{array}{}\text{(34)}& {\mathrm{MRTS}}_{ij}(\mathbf{x})=\frac{\partial f(\mathbf{x})/\partial x_i}{\partial f(\mathbf{x})/\partial x_j}=\frac{M{P}_i}{M{P}_j}\end{array}$$

   In the two-input case, as depicted in Fig. 3.1, ${\mathrm{MRTS}}_{12}\left({\mathbf{x}}^1\right)$$\operatorname{MRTS}_{12}\left(\mathbf{x}^1\right)$ is the absolute value of the slope of the isoquant through ${\mathbf{x}}^1$$\mathbf{x}^1$ at the point ${\mathbf{x}}^1$$\mathbf{x}^1$.

2. Elasticity of substitution

   $$\begin{array}{}\text{(35)}& \sigma =\frac{\mathrm{\%}\text{ change in }\frac{x_2}{x_1}}{\mathrm{\%}{\text{ change in MRTS }}_{21}}=\frac{\mathrm{d}\left(\frac{x_2}{x_1}\right)}{\mathrm{d}\left(\frac{f_1}{f_2}\right)}\cdot \frac{\left(\frac{f_1}{f_2}\right)}{\left(\frac{x_2}{x_1}\right)}=\frac{\mathrm{d}\ln \left(\frac{x_2}{x_1}\right)}{\mathrm{d}\ln \left(\frac{f_1}{f_2}\right)}\end{array}$$

   Also, rigorously,

   $$\begin{array}{}\text{(36)}& {\sigma }_{ij}(\mathrm{x})=\frac{d\ln \left(x_j/x_i\right)}{d\ln MRT{S}_{ij}}=\frac{d\ln \left(x_j/x_i\right)}{d\ln \left(M{P}_i/M{P}_j\right)}=\frac{\frac{d\left(x_j/x_i\right)}{x_j/x_i}}{\frac{d\left(M{P}_i/M{P}_j\right)}{\left(M{P}_i/M{P}_j\right)}}\end{array}$$

• % change in MRTS vs. % change in factor ratio at $\mathrm{x}$$\mathrm{x}$

• $\sigma$$\sigma$ measures how easily input factors can be substituted for one another (holding other inputs and output constant). (The curvature of the isoquant curve at $x$$x$)

• If $\sigma$$\sigma$ is large, then it is easier to substitute between the input factors.

• Strong (weak) substitution: when increasing $x_1$$x_1$ and reducing $x_2$$x_2$ along one isoquant, MRTS ${}_{ij}$$_{i j}$ - the ability of $x_1$$x_1$ to substitute for $x_2$$x_2$ - drops a little(a lot)

• $\sigma \in [0,+\mathrm{\infty })$$\sigma \in[0,+\infty)$

3. Return to scale (global and local)

   A production function $f(\mathbf{x})$$f(\mathbf{x})$ has the property of (globally):

   1. Constant returns to scale if $f(t\mathbf{x})=tf(\mathbf{x})$$f(t \mathbf{x})=t f(\mathbf{x})$ for all $t>0$$t>0$ and all $\mathbf{x}$$\mathbf{x}$;

   2. Increasing returns to scale if $f(t\mathbf{x})>tf(\mathbf{x})$$f(t \mathbf{x})>t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$;

   3. Decreasing returns to scale if $f(t\mathbf{x})<tf(\mathbf{x})$$f(t \mathbf{x})<t f(\mathbf{x})$ for all $t>1$$t>1$ and all $\mathbf{x}$$\mathbf{x}$.

   (Local) Returns to Scale The elasticity of scale at the point $\mathbf{x}$$\mathbf{x}$ is defined as

   $$\begin{array}{}\text{(37)}& \mu (\mathbf{x})\equiv \underset{t\to 1}{lim}\frac{d\ln [f(t\mathbf{x})]}{d\ln (t)}=\frac{\sum _{l=1}^n{f}_l(\mathbf{x})x_l}{f(\mathbf{x})}.\end{array}$$

   Returns to scale are locally constant, increasing, or decreasing as $\mu (\mathbf{x})$$\mu(\mathbf{x})$ is equal to, greater than, or less than one. The elasticity of scale and the output elasticities of the inputs are related as follows:

   $$\begin{array}{}\text{(38)}& \mu (\mathbf{x})=\sum _{i=1}^n{\mu }_l(\mathbf{x})\end{array}$$

Cost Minimization

1. Cost function and conditional input demand

   If the object of the firm is to maximize profits, it will necessarily choose the least costly, or cost-minimizing, production plan for every level of output.

   The cost function, defined for all input prices $\mathbf{w}\gg \mathbf{0}$$\mathbf{w} \gg \mathbf{0}$ and all output levels $y\in f\left({\mathbb{R}}_{+}^n\right)$$y \in f\left(\mathbb{R}_{+}^n\right)$ is the minimum-value function,

   $$\begin{array}{}\text{(39)}& c(\mathbf{w},y)\equiv \underset{\mathbf{x}\in {\mathbb{R}}_{+}^n}{min}\mathbf{w}\cdot \mathbf{x}\text{ s.t. }f(\mathbf{x})\ge y.\end{array}$$

   If $\mathbf{x}(\mathbf{w},y)$$\mathbf{x}(\mathbf{w}, y)$ solves the cost-minimization problem, then

   $$\begin{array}{}\text{(40)}& c(\mathbf{w},y)=\mathbf{w}\cdot \mathbf{x}(\mathbf{w},y).\end{array}$$

   Marginal rate of technical substitution between two inputs is equal o the ratio of their prices

   $$\begin{array}{}\text{(41)}& \frac{\partial f\left({\mathbf{x}}^{\ast }\right)/\partial x_i}{\partial f\left({\mathbf{x}}^{\ast }\right)/\partial x_j}=\frac{w_i}{w_j}.\end{array}$$

   Solutions to this cost minimization problem are called conditional input demand ${\mathbf{x}}^{\ast }(w,y)$$\mathbf{x}^*(w,y)$.

2. Sheppard's lemma

   $c(\mathbf{w},y)$$c(\mathbf{w}, y)$ is differentiable in $\mathbf{w}$$\mathbf{w}$ at $({\mathbf{w}}^0,y^0)$$\left(\mathbf{w}^0, y^0\right)$ whenever ${\mathbf{w}}^0\gg 0$$\mathbf{w}^0 \gg 0$​, and

   $$\begin{array}{}\text{(42)}& \frac{\partial c({\mathbf{w}}^0,y^0)}{\partial w_i}=x_i({\mathbf{w}}^0,y^0),i=1,\dots ,n\end{array}$$

3. Symmetric substitution matrix for conditional input demands.

   The substitution matrix, defined and denoted

   $$\begin{array}{}\text{(43)}& \begin{array}{c}{\sigma }^{\ast }(\mathbf{w},y)\equiv \left(\begin{array}{ccc}\frac{\partial x_1(\mathbf{w},y)}{\partial W_1}& \cdots & \frac{\partial x_1(\mathbf{w},y)}{\partial W_n}\\ ⋮& \ddots & ⋮\\ \frac{\partial x_n(\mathbf{w},y)}{\partial w_1}& \cdots & \frac{\partial X_n(\mathbf{w},y)}{\partial w_n}\end{array}\right),\end{array}\end{array}$$

   is symmetric and negative semidefinite. In particular, the negative semi-definiteness property implies that $\partial x_i(\mathbf{w},y)/\partial w_i\le 0$$\partial x_i(\mathbf{w}, y) / \partial w_i \leq 0$ for all $i$$i$​.

   This is because (By the Young's theorem)

   $$\begin{array}{}\text{(44)}& \frac{{\partial }^2{C}^{\ast }}{\partial w_1\partial w_2}=\frac{\partial x_1^{\ast }}{\partial w_2}=\frac{{\partial }^2{C}^{\ast }}{\partial w_2\partial w_1}=\frac{\partial x_2^{\ast }}{\partial w_1}\end{array}$$

4. Hicksian Third Law (derived from the Euler's theorem)

   $$\begin{array}{}\text{(45)}& \sum _j\frac{\partial x_i^{\ast }(\mathbf{w},y)}{\partial w_j}{w}_j=0\end{array}$$

   Or in elasticity form:

   $$\begin{array}{}\text{(46)}& \sum _j{ϵ}_{ij}^{\ast }=0\end{array}$$

5. Recall:

   $$\begin{array}{}\text{(47)}& \frac{\partial x_i^{\ast }}{\partial w_j}\equiv \frac{\partial x_j^{\ast }}{\partial w_i}\end{array}$$

   and expand:

   $$\begin{array}{}\text{(48)}& \begin{array}{rl}\frac{x_i}{x_i}\frac{w_j}{w_j}\frac{\partial x_i^{\ast }}{\partial w_j}& \equiv \frac{x_j}{x_j}\frac{w_i}{w_i}\frac{\partial x_j^{\ast }}{\partial w_i}\\ \frac{x_i}{w_j}\left(\frac{\partial x_i^{\ast }}{\partial w_j}\frac{w_j}{x_i}\right)& \equiv \frac{x_j}{w_i}\left(\frac{\partial x_j^{\ast }}{\partial w_i}\frac{w_i}{x_j}\right)\\ \left(w_i{x}_i\right){ϵ}_{ij}& \equiv \left(w_j{x}_j\right){ϵ}_{ji}\end{array}\end{array}$$

   Dividing both sides by the cost: $C=\sum _i^n{w}_i{x}_i$$C=\sum_i^n w_i x_i$

   $$\begin{array}{}\text{(49)}& k_i{ϵ}_{ij}\equiv k_j{ϵ}_{ji};i,j=1,\dots n\end{array}$$

   where $k_i=\frac{w_i{x}_i}{c}$$k_i=\frac{w_i x_i}{c}$. From the above result, express ${ϵ}_{ij}\equiv \frac{k_j}{k_i}{ϵ}_{ji}$$\epsilon_{i j} \equiv \frac{k_j}{k_i} \epsilon_{j i}$ and then substitute it

   back into $\text{(46)}$$\eqref{eq46}$ :

   $$\begin{array}{}\text{(50)}& \sum _{j=1}^n{ϵ}_{ij}\equiv \sum _{j=1}^n\frac{k_j}{k_i}{ϵ}_{ji}\equiv 0\end{array}$$

   Then multiplying through by $k_i$$k_i$ yields:

   $$\begin{array}{}\text{(51)}& \sum _{j=1}^n{k}_j{ϵ}_{ji}\equiv k_1{ϵ}_{1i}+k_2{ϵ}_{2i}+\cdots +k_n{ϵ}_{ni}\equiv 0;i=1,2,\dots ,n\end{array}$$

   Notice the difference between $\text{(51)}$$\eqref{eq51}$ and $\text{(46)}$$\eqref{eq46}$: in $\text{(51)}$$\eqref{eq51}$ the elasticities being considered are those between the various factors and one particular factor price. In $\text{(46)}$$\eqref{eq46}$ the elasticities all pertain to one particular factor $x_i$$x_i$ and all factor prices.

6. Homothetic production function

   • The cost function is multiplicatively separable in input prices and output and can be written $c(\mathbf{w},y)=h(y)c(\mathbf{w},1)$$c(\mathbf{w}, y)=h(y) c(\mathbf{w}, 1)$, where $h(y)$$h(y)$ is strictly increasing and $c(\mathbf{w},1)$$c(\mathbf{w}, 1)$ is the unit cost function, or the cost of 1 unit of output;

   • The conditional input function are also multiplicatively separable in input prices and output can be written $\mathbf{x}(\mathbf{w},y)=h(y)\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, y)=h(y) \mathbf{x}(\mathbf{w}, 1)$, where $h^{\prime }(y)>0$$h^{\prime}(y)>0$ and $\mathbf{x}(\mathbf{w},1)$$\mathbf{x}(\mathbf{w}, 1)$ is the conditional input demand for 1 unit of output.

Long run cost

1. Economies of scale

   With respect to cost function, we define economies of scale. The definition relies on elasticity of cost w.r.t. output:

   $$\begin{array}{}\text{(52)}& E_{c,y}=\frac{\partial \ln C(w,y)}{\partial \ln y}=\frac{\partial C(w,y)}{\partial y}\frac{y}{C}=\frac{LMC}{LAC}.\end{array}$$

   Cost function has economies of scale if $E_{c,y}<1$$E_{c, y}<1$ which means that $\frac{LMC}{LAC}<1$$\frac{L M C}{L A C}<1$ which implies $LMC<LAC$$L M C<L A C$ and decreasing LAC (left of the minimum LAC). Cost function has diseconomies of scale if $E_{c,y}>1$$E_{c, y}>1$ which means that $\frac{LMC}{LAC}>1$$\frac{L M C}{L A C}>1$ which implies $LMC>LAC$$L M C>L A C$ and increasing LAC (right of the minimum LAC).

   The relationship between returns to scale and economies of scale is given by:

   $$\begin{array}{}\text{(53)}& E_{c,y}=\frac{LMC}{LAC}⋚1⟺E⋛1.\end{array}$$

Short run cost

Profit maximization

Then the first-order conditions require that marginal cost equals marginal benefit of production.

the MRTS between any two inputs is equated to the ratio of their prices.

1. Hotelling Lemma, under strictly concave production function.

   $$\begin{array}{}\text{(57)}& \frac{\partial \pi (p,\mathbf{w})}{\partial p}=y(p,\mathbf{w}),\text{ and }\frac{\partial \pi (p,\mathbf{w})}{\partial w_i}=-x_i(p,\mathbf{w}),i=1,2,\dots ,n\text{. }\end{array}$$

   Note that there is a negative sign on $\frac{\partial \pi (p,\mathbf{w})}{\partial w_i}=-x_i(p,\mathbf{w}),i=1,2,\dots ,n$$\frac{\partial \pi(p, \mathbf{w})}{\partial w_i}=-x_i(p, \mathbf{w}), \quad i=1,2, \ldots, n$​.

2.  

Partial Equilibrium

Perfect competition

1. Supply in the short run

   Demand curve facing individual producer is horizontal (perfectly elastic):

   $$\begin{array}{}\text{(58)}& p=\text{ const. }\end{array}$$

   But the market demand is downward sloping.

   Firm's total revenue: $R=p\cdot q$$R=p \cdot q$, and $MR=\frac{\mathrm{d}R}{\mathrm{d}q}=p$$M R=\frac{\mathrm{d} R}{\mathrm{~d} q}=p$​.

   

   $S_i=S_i(p)$$S_i=S_i(p)$ for $p\ge minAVC$$p \geq \min A V C$ $S_i=0$$S_i=0$ for $p<minAVC$$p<\min A V C$

   $$\begin{array}{}\text{(59)}& S=\sum _{i=1}^n{S}_i(p)=S(p)\end{array}$$

   Because of SOC for $\mathrm{Max}\pi ,MC$$\operatorname{Max} \pi, M C$​ is increasing, hence firm's supply is monotonically increasing, for prices above minAVC. Therefore, SR aggregate supply (obtained by horizontal summation) is also increasing.

2. Supply in the long run

   the long-run equilibrium is determined by:

   • firms max profit: $p=MC$$p=M C$

   • D(p)=n S_i(p)

   • Zero profit condition (free entry):

     $$\begin{array}{}\text{(60)}& {\pi }_i=p{S}_i-\varphi \left(S_i\right)=0\end{array}$$

     where $\varphi \left(S_i\right)$$\phi\left(S_i\right)$ is LRTC of the $i^{\text{th }}$$i^{\text {th }}$ firm for an output $q_i=S_i=S/n$$q_i=S_i=S / n$ $\Rightarrow$$\Rightarrow$ Also requires $p=AC$$p=A C$ or $p=\frac{\varphi \left(S_i\right)}{S_i}$$p=\frac{\phi\left(S_i\right)}{S_i}$

Monopoly

1. Goods supply

   $$\begin{array}{}\text{(61)}& mr\left(q^{\ast }\right)=mc\left(q^{\ast }\right)\end{array}$$

2.  

   Since $r(q)=p(q)q$$r(q) = p(q)q$, differentiate on the both side,

   $$\begin{array}{}\text{(62)}& \begin{array}{rl}\mathrm{mr}(q)& =p(q)+q\frac{dp(q)}{dq}\\ & =p(q)[1+\frac{dp(q)}{dq}\frac{q}{p(q)}]\\ & =p(q)[1-\frac{1}{|\epsilon (q)|}]\end{array}\end{array}$$

   This implies that

   $$\begin{array}{}\text{(63)}& p\left(q^{\ast }\right)[1-\frac{1}{\mid \epsilon \left(q^{\ast }\right)}]=mc\left(q^{\ast }\right)\ge 0\end{array}$$

   Rearranging this:

   $$\begin{array}{}\text{(64)}& \frac{p\left(q^{\ast }\right)-mc\left(q^{\ast }\right)}{p\left(q^{\ast }\right)}=\frac{1}{\left|\epsilon \left(q^{\ast }\right)\right|}\end{array}$$

   This is because the marginal cost is always non-negative. Since the price is also non-negative, we must have $\left|\epsilon \left(q^{\ast }\right)\right|\ge 1$$\left|\varepsilon\left(q^*\right)\right| \geq 1$. Therefore, a monopoly will never produce in the inelastic portion of the demand curve, as shown in the figure.

   

   When market demand elasticity is not infinite, $\left|\epsilon \left(q^{\ast }\right)\right|$$\left|\varepsilon\left(q^*\right)\right|$ will be finite, and at equilibrium, the price set by a monopoly will be greater than marginal cost. Furthermore, under unchanged conditions, the more inelastic the market demand, the more the price will exceed marginal cost. The firm will possess greater market power.

Price discrimination

1. First-Degree (Perfect): Monopolist charges different price for each unit of a good such that $P=MaxWTP$$P=M a x W T P$​ for that unit. Monopolist extracts the entire CS from a heterogeneous set of consumers.

    

2. Second-Degree: When monopolist knows that consumers differ but cannot discriminate directly because it is unable to identify them. Prices differ depending on the number of units bought (quantity discounts) but not across consumers.

    

3. Third-Degree: Different groups of buyers are charged different prices, but each buyer pays the same amount for each unit bought. Market segmentation can be implemented when monopolist knows market demand for different groups and can prevent arbitrage.

   Consider 2 separate markets: let $p_i\left(q_i\right)$$p_i\left(q_i\right)$ be the inverse demand function and the constant marginal cost function is $C\left(q_i\right)=c{q}_i$$C\left(q_i\right)=c q_i$. Monopolist maximizes profits by solving:

   $$\begin{array}{}\text{(65)}& \underset{q_1,q_2}{max}{p}_1\left(q_1\right)q_1+p_2\left(q_2\right)q_2-c{q}_1-c{q}_2\end{array}$$

   FOCs:

   $$\begin{array}{}\text{(66)}& \begin{array}{rl}& p_1\left(q_1\right)+p_1^{\prime }\left(q_1\right)q_1=c\\ & p_2\left(q_2\right)+p_2^{\prime }\left(q_2\right)q_2=c\end{array}\end{array}$$

   or:

   $$\begin{array}{}\text{(67)}& \begin{array}{c}{p}_1\left(q_1\right)[1+\frac{1}{{ϵ}_1}]=c\\ p_2\left(q_2\right)[1+\frac{1}{{ϵ}_2}]=c\\ \frac{p_1}{p_2}=\frac{1+\frac{1}{{ϵ}_2}}{1+\frac{1}{{ϵ}_1}}\end{array}\end{array}$$

Principal-agent problem