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Syllabus Outline: Economic History (Undergraduate)

Course objectives The objective of this course is to introduce students to the economic approach to historical change, with a focus on institutional evolution. Students will be exposed to a wide array of economic tools, mostly developed to study of markets, to be applied to historical instances of market and non-market behavior.ReadingsLopez, R. S. (1976). The commercial revolution of the middle ages. Cambridge University Press.Lane, F. C. (1979). Profits from power. SUNY Press.North, D. C. (1981). Structure and change in economic history. Norton.Allen, D. W. (2011). The institutional revolution: Measurement and the economic emergence of the modern world. University of Chicago Press.

Econ 340: Introduction to Mathematical Economics (2017)

General informationThursday, 4:30-7:10 PM, Buchanan Hall (formerly, Mason Hall), D005, Fairfax campus.

Econ 340: Week 7

- How do we differentiate a function that is expressed
*implicitly*?

\(xy=5\)[Note that this corresponds to the function \(y=f\left(x\right)=5x^{-1}\)]Rewrite this as\(xf\left(x\right)=5,\ \forall\ x\ne0\)We take the derivative wrt to x of both sides\(\frac{d}{dx}\left(xf\left(x\right)\right)=\frac{d}{dx}5\)Using the product rule yields\(f\left(x\right)+xf'\left(x\right)=0\)Solving for f'(x)\(f'\left(x\right)=-\frac{f\left(x\right)}{x}\)

\(y^3+3x^2y=13\)We must use the chain ruleandthe product rule\(3y^2y'+6xy+3xy'=0\)Once again, we solve for y'\(y'=-2\frac{xy}{y^2+x^2}\)

- Differentiate
*each side*of the implicit function wrt x, while considering y as a function of x - Solve the resulting equation for y'

\(x^2y^3+\left(y+1\right)e^{-x}=x+2\)Rewrite\(x^2f\left(x\right)^3+\left(f\left(x\right)+1\right)e^{-x}=x+2\)We can now differentiate both sides\(2xf\left(x\right)^{3}+3x^{2}f\left(x\right)^{2}f^{'}\left(x\right)+f^{'}\left(x\right)e^{-x}-\left(f\left(x\right)+1\right)e^{-x}=1\)Solving for f'(x)\(f^{'}\left(x\right)=\frac{1-2xf\left(x\right)^{3}+\left[f\left(x\right)+1\right]e^{-x}}{3x^{2}f\left(x\right)^{2}+e^{-x}}\)

Use the same exact method to find the *second* implicit derivative

\(D:\ Q_d=a-b(P+T)\)\(S:\ Q_s=\alpha+\beta(P)\)In equilibrium\(a-b(P+T)=\alpha+\beta P\)Taking the implicit derivative wrt T\(-b(\frac{dP}{dT}+1)=\alpha+\beta(\frac{dP}{dT})\)Solving for \(\frac{dP}{dT}\)\(\frac{dP}{dT}=-\frac{b}{\beta+b}\)Thus, taxes imposed on the consumer sidereducethe equilibrium price

- What is the relationship between the first derivative of a function and that of its inverse?
- In other words, what is the relationship between f' and g'?

\(f(x)=ax+b\)\(g(x)=\frac{1}{a}x-\frac{b}{a}\)Where g is the inverse of f\(f'(x)=a\)\(g^{'}(x)=\frac{1}{a}\)In general, when g(x) is the inverse of f(x)\(g(f(x))=x,\ \forall x\ in\ I\)Implicitly differentiating wrt x we get\(\)\(g^{'}(f(x))f^{'}(x)=1\)Solving for g'\(g^{'}(f(x))=\frac{1}{f^{'}(x)}\)Thus, the first derivative of the inverse of f(x) is equal to one over the first derivative of f(x)

\(f(x)=x^{5}+3x^{3}+6x-3\)\(f^{'}(x)=5x^{4}+9x^{2}+6\)Thus\(g^{'}(x)=\frac{1}{5x^{4}+9x^{2}+6}\)

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- Elasticity measures the
*percentage change*in the dependent variable due to a*"unit"*(actually, an infinitesimal) change in the independent variable - The single most important application of the notion of elasticity in economics is the
*price elasticity of demand*

\(Q=D(P)\)\(\epsilon_P=\frac{\Delta Q}{Q}=\frac{D(P+\Delta P)-D(P)}{D(P)}\)Multiply both sides by \(\frac{P}{\Delta P}\)\(\frac{\Delta Q}{Q}\frac{P}{\Delta P}=\frac{D(P+\Delta P)-D(P)}{D(P)}\frac{P}{\Delta P}=\frac{D(P+\Delta P)-D(P)}{\Delta P}\frac{P}{D(P)}\)

Find \(\epsilon_x\) for \(f(x)=Ax^b\)\(\epsilon_x=\frac{d f(x)}{d x}\frac{x}{f(x)}=f^{'}(x)\frac{x}{f(x)}\)Finding f'(x) and substituting in f'(x) and f(x) yields\(\epsilon_x=bAx^{b-1}\frac{x}{Ax^b}=b\)

\(D(P)=8000P^{-1.5}\)\(\epsilon_P=-12000P^{-2.5}\frac{P}{8000P^{-1.5}}=-1.5\)

Take the following revenue function\(R(P)=P\cdot Q(P)\)Find the elasticity of revenue wrt price\(\epsilon_R=R^{'}(P)\frac{P}{R(P)}\)\(R^{'}(P)=\frac{d (P\cdot Q(P))}{dP}=Q(P)+PQ^{'}(P)\)Thus, substituting in yields\(\epsilon_R=\frac{Q(P)}{Q(P)}+Q^{'}(P)\frac{P}{Q(P)}\)But \(Q^{'}(P)\frac{P}{Q(P)}\) is just \(\epsilon_P\)Thus,\(\epsilon_R=1+\epsilon_P\)

- L'Hopital's rule is used to determine the limit when \(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{0}{0}\)
- This limit is indeterminate
- Simple version of the rule:

\(\lim_{x\to a}\frac{f(x)}{g(x)}=\frac{f^{'}(x)}{g^{'}(x)}\)

- We can now use this rule to show

\(\lim_{x\to a}\frac{e^{x}-1}{x}=\frac{e^{x}}{1}=1\)

\(\lim_{\lambda\to 0}\frac{x^{\lambda}-y^{\lambda}}{\lambda}=\frac{x^{lambda}\ln x-y^{\lambda}\ln y}{1}=\frac{\ln x-\ln y}{1}=\ln x-\ln y=\ln\frac{x}{y}\)

\(F(K,\ L)=A(aK^{-\sigma}+(1-a)L^{-\sigma})^{-1/\sigma}\)Using l'Hopital's rule we can show that, as \(\sigma\) converges to zero, the function converges to the Cobb-Douglas functional form

Syllabus: ECON 309 (Spring 2018)

Tuesday, 4:30-7:10 PM, Robinson B111, Fairfax campus.

Econ 340: Week 13

We use this method when our optimization problem is constrained by external factors and it is either impossible or impractical to substitute the constraint into the objective function.

Max \(u\left(x,\ y\right)\) subject to (s.t.) \(m=p_{x}x+p_{y}y\)

We rewrite the problem as a Lagrangian function:

Max \(L=u(x, y)-\lambda(p_{x}x+p_{y}y-m)\)

FOCs

\(\frac{\partial L}{\partial x}=u^{'}_{x}-\lambda p_{x}=0\)

\(\frac{\partial L}{\partial y}=u^{'}_{y}-\lambda p_{y}=0\)

\(m=p_{x}x+p_{y}y\)

Solving the system of equations yields the equimarginal principle:

\(\frac{u^{'}_{x}}{u^{'}_{y}}=\frac{p_{y}}{p_{x}}\)

- Write down the Lagrangian function
- Differentiate it wrt the variables of interest and \(\lambda\)
- Derive the FOCs
- Solve the system of equations wrt the variables of interest and \(\lambda\)

Max \(f(x, y)=x^{2}+y^{2}\) s.t. \(x^{2}+xy+y^{2}=3\)

- Write the utility maximization problem
- Write the Lagrangian function
- Derive the FOCs
- Solve the system of equations for the two commodities as functions of the parameters

- Max \(u(x,y)=Ax^{a}y^{b}\) subject to \(p_{x}x+p_{y}y=m\)
- Max \(L(x, y, \lambda)=Ax^{a}y^{b}-\lambda(p_{x}x+p_{y}y-m)\)
- FOCs:

\(Aax^{a-1}y^{b}=\lambda p_{x}\)

\(Abx^{a}y^{b-1}=\lambda p_{y}\)

\(y=\frac{p_{x}}{p_{y}} x \frac{b}{a}\)

Constraint: \(m=p_{x}x+p_{y}y\)

4. \(m=p_{x}x+p_{y}\frac{p_{x}}{p_{y}} x \frac{b}{a}=p_{x}x+p_{x}x\frac{b}{a}\)

\(x^{*}=\frac{m}{p_x}\frac{a}{a+b}\)

\(y^{*}=\frac{m}{p_{y}}\frac{b}{a+b}\)

Draw a tridimensional graph to explain the constrained optimization method

- Min \(C=3L+5K\) subject to \(2000=K^{.7}L^{.3}\)
- Max \(L=3L+5K-\lambda(K^{.7}L^{.3}-2000)\)
- FOCs:
\(3-\lambda(.3L^{-.7}K^{.7})=0\)\(5-\lambda(.7L^{.3}K^{-.3})=0\)\(\frac{3}{5}=\frac{.3}{.7}\frac{K}{L}\implies L=\frac{.5}{.7}K\)Constraint: \(2000=K^{.7}L^{.3}\)4. \(\)\(2000=K^{.7}(\frac{.5}{.7}K)^{.3}\)\(K^{*}=2212.4\)\(L^{*}=6321.11\)

Consider an economy that employs two factors, \(x_{ 1, 2}\), and produces one commodity via the production process expressed by a Cobb-Douglas function. Efficiency requires that, for any given quantity of output, the cost of production be minimized:

Min \(p_{1}x_{1}+p_{2}x_{2}\) subject to \(\bar{y}=x_{1}^{\alpha}x_{2}^{\beta}\)

Using the Lagrangian method, we can write:

Min \(p_{1}x_{1}+p_{2}x_{2}-\lambda(x_{1}^{\alpha}x_{2}^{\beta}-\bar{y})\)

FOCs

\(p_{1}-\lambda \alpha x_{1}^{\alpha-1}x_{2}^{\beta}=0\)

\(p_{2}-\lambda \beta x_{1}^{\alpha}x_{2}^{\beta-1}=0\)

\(\bar{y}=x_{1}^{\alpha}x_{2}^{\beta}\)

Because \(f^{'}_{1}=\alpha x_{1}^{\alpha-1}x_{2}^{\beta}\) and \(f^{'}_{2}=\beta x_{1}^{\alpha}x_{2}^{\beta-1}\), we can rewrite the FOCs as

\(\lambda f^{'}_{1}=p_{1}\) and \(\lambda f^{'}_{2}=p_{2}\)

Multiplying both sides of each equation by, respectively, \(x_{1}\) and \(x_{2}\) yields:

\(\lambda f^{'}_{1}x_{1}=p_{1}x_{1}\)

\(\lambda f^{'}_{2}x_{2}=p_{2}x_{2}\)

Using the properties of equations, we can write:

\(\)\(\lambda f^{'}_{1}x_{1}+\lambda f^{'}_{2}x_{2}=p_{1}x_{1}+p_{2}x_{2}\)

\(\lambda(f^{'}_{1}x_{1}+f^{'}_{2}x_{2})=p_{1}x_{1}+p_{2}x_{2}\)

Under competitive conditions, \(\lambda=q\), where \(q\) is the price of the commodity produced in the economy. Using Euler's theorem, this becomes:

\(qf(x_{1},x_{2})=p_{1}x_{1}+p_{2}x_{2}\)

Thus, the entirety of the revenues from the production of the commodity is exhausted in the payment of the two factors. QED

A general equilibrium is defined by a price vector such that excess demand for all commodities is exactly zero. These implies two conditions for the general equilibrium model:

a. Every agent maximizes utility in equilibrium

b. Excess demand for all commodities is zero

The following the list of steps one must take to find the general equilibrium price vector:

- Derive the demand curve for all commodities for all consumers
- Set the sum of all demand curves equal to the total of initial endowments in the economy
- Solve for the ratio of prices for n-1 commodities in the economy

Econ 340: Week 12

Suppose \(z=f\left(x,\ y\right)\) is a differentiable function of two variables, then

\(z=f\left(x,\ y\right)\Rightarrow dz=f^{'}_{x}(\cdot)dx+f^{'}_{y}(\cdot)dy\)

Let \(Y=f(K,\ L)\) be a production function then

\(dY=f^{'}_{K}dK+f^{'}_{L}dL\)

- \(d(af+bg)=a\cdot df+b\cdot dg\)
- \(d(fg)=g\cdot df+ f\cdot dg \)
- \(d(f/g)=\frac{g\cdot df+f\cdot dg}{g^{2}},\ g\neq 0\)

\(z=Ax^{a}+By^{b}\)\(dz=aAx^{a-1}+bBy^{b-1}\)

\(z=e^{xu},\ u=u(x,\ y)\)\(dz=e^{xu}d(x,\ u)=e^{xu}\cdot (u\ dx+x\ du)=e^{xu}\cdot [u\ dx+x\ (u^{'}_{x}dx+u^{'}_{y}dy)]\)

The differential of a functionzhas the same form whetherxandyare free variables, as whether they depend on not on some other variablestands.

Consider the following system of equation\(5u+5v=2x-3y\)\(2u+4v=3x-2y\)We have 4 variables and 2 equations, thus we have 2 degrees of freedom (4-2=2)Taking the total differential of both equations yields\(5du+5dv=2dx-3dy\)\(2du+4dv=3dx-2dy\)Solving the first equation wrtdu:\(du=\frac{2}{5}dx-\frac{3}{5}dy-dv\)Substituting into the second equation and solving fordv\(dv=\frac{11}{10}dx-\frac{2}{5}dy\)Substituting back into the equation fordu\(du=-\frac{7}{10}dx-\frac{1}{5}dy\)

The optimization of a multivariate function works pretty much the same way as that of functions of one variable.

A differentiable function \(z=f(x,\ y)\) can only have a maximum or a minimum at an interior point \((x_{0},\ y_{0})\) of *S* if it is a **stationary point**, that is, the point must satisfy the following two equations:

\(f^{'}_{x}(\cdot)=0\)

\(f^{'}_{y}(\cdot)=0\)

These are the first order conditions (FOCs) of a multivariate optimization

\(f(x,\ y)=-2x^{2}-2xy-2y^{2}+36x+42y-158\)Find the FOCs

A firm produces two different kindsAandBof a commodity. The daily cost of producingxunits ofAandyunits ofBis\(C(x,\ y)=.04x^{2}+.01xy+.01y^{2}+4x+2y+500\)Suppose the firm can send a unit ofAfor 15 and one ofBfor 9. Set up and solve the profit maximization problem\(\Pi=15x-9y-.04x^{2}-.01xy-.01y^{2}-4x-2y-500\)

Suppose \(Q=F(K,\ L)\) is a firm's production function. The firm sell its output at pricepand must compensate the labor and capital employed at wagewand interestr. The firm's profit maximization problem is then given by\(\Pi=pF(K,\ L)-rK-wL\)Find the FOCs

A firm supplies \(x_{d}\) in the domestic market and \(x_{w}\) in the world market. The firm has market power in the domestic market but must face competition in the world market, so, \(p_{d}=f(x_{d})\) and \(p_{w}=\bar{p}\). The firm's costs are given by \(c=c(x_{d}+x_{w})\). Set up and solve the firm's profit maximization problem.\(\Pi=f(x_{d})x_{d}+p_{w}x_{w}-c(x_{d}+x_{w})\)FOCs\(\frac{\partial\Pi}{\partial x_{d}}=f^{'}_{x_{d}}(x_{d})x_{d}+f(x_{d})-c^{'}_{x_{d}}=0\)\(\frac{\partial\Pi}{\partial x_{w}}=p_{w}-c^{'}_{x_{w}}=0\)What is the relationship between the two prices when domestic price elasticity is a constant equal to 2?\(\epsilon_{d}=\frac{\partial x_{d}}{\partial p_{d}}\frac{p_{d}}{x_{d}}=-2\)Solving for \(p_{d}\) yields\(-.5p_{d}=\frac{\partial x_{d}}{\partial p_{d}}\frac{1}{x_{d}}\)Because \(p_{d}=f(x_{d})=c^{'}_{x_{d}}-f^{'}_{x_{d}}x_{d}\), where \(1/f^{'}_{x_{d}}x_{d}=\frac{\partial x_{d}}{\partial p_{d}}\frac{1}{x_{d}}\), and \(c^{'}=p_{w}\):\(p_{d}=2p_{w}\)

Suppose \((x_{0}, y_{0})\) is a stationary point for a function in a convex set *S*, andi

- If, for all (x, y) is
*S*,

\(f^{''}_{x}\leq 0,\ f^{''}_{y}\leq 0, \& f^{''}_{x}\cdot f^{''}_{y}\geq (f^{''}_{x, y})^{2}\)

then, \((x_{0}, y_{0})\) is a maximum point

- If, for all (x, y) is
*S*,

\(f^{''}_{x}\geq 0,\ f^{''}_{y}\geq 0, \& f^{''}_{x}\cdot f^{''}_{y}\geq (f^{''}_{x, y})^{2}\)

then, \((x_{0}, y_{0})\) is a minimum point

- If \(f^{''}_{x}\cdot f^{''}_{y}< (f^{''}_{x, y})^{2}\),

then, \((x_{0}, y_{0})\) is a saddle point

- If \(f^{''}_{x}\cdot f^{''}_{y}= (f^{''}_{x, y})^{2}\)

then, \((x_{0}, y_{0})\) could be any of the three

First order (necessary) conditions only tell us whether a point is an extreme point or not, but cannot tell us whether the point is a maximum, a . minimum, or a saddle point.

A firm has three factories, each producing the same item. Letx,y, andzdenote respective output quantities that the three factories produce in order to fulfill an order for 2000 units in total. Hence, \(x+y+z=2000\). The cost functions for the factories are\(c_{ 1}(x)=200+\frac{1}{100}x^{2}\)\(c_{2}(y)=200+y+\frac{1}{300}y^{3}\)\(c_{3}(z)=200+10z\)SolutionMin \(c(x, y, z)=600+\frac{1}{100}x^{2}+y+\frac{1}{300}y^{3}+10z\)Substituting \(z=2000-x-y\):Min \(c(x, y, z)=600+\frac{1}{100}x^{2}+y+\frac{1}{300}y^{3}+10(2000-x-y)\)FOCs\(\frac{1}{50}x-10=0\implies x=500\)\(1+\frac{1}{100}y^{2}-10=0\implies y=30\)What are the second order conditions?

Linear models with quadratic objectives

A firm faces the following two demand functions (What kind of firm is this?):\(P_{1}=a_{1}-b_{1}Q_{1}\)\(P_{2}=a_{2}-b_{2}Q_{2}\)The firm's cost function is further given by\(C=\alpha(Q_{1}+Q_{2})\)We can now derive the profit function and maximize it\(\Pi=P_{1}Q_{1}+P_{2}Q_{2}-\alpha(Q_{1}+Q_{2})\)The profit maximization problem is thus given by\(\Pi=a_{1}Q_{1}-b_{1}Q_{1}^{2}+a_{2}Q_{2}-b_{2}Q_{2}^{2}-\alpha(Q_{1}+Q_{2})\)What are the first order conditions? What are the profit maximizing prices and quantities?

Let a firm be a monopsonist (what does this mean?) in the factor market, while also facing competitive pressures in the output market (what does this mean?). The firm's production function is given by \(Q=L_{1}+L_{2}\). The firm faces two separate labor supplies:\(w_{1}=\alpha_{1}+\beta_{1}L_{1}\)\(w_{2}=\alpha_{2}+\beta_{2}L_{2}\)Set up and solve the firm's profit maximization problem

- Find all stationary points (FOCs)
- Find the largest and smallest values on the boundary of the function
- Compute the values at all points in (I) and (II)
- Derive the SOCs

Suppose \(f(\bold{x})=f(x_{1}, ..., x_{n})\) is defined over a set \(S\in\ R^{n}\) and let *F* be a function of one variable defined over the range of *f*. Define *g* over *S* by

\(g(\bold{x})=F(f(\bold{x}))\)

Then

- If
*F*is increasing and c maximizes (minimizes)*f*over*S*, then c also maximizes (minimizes)*g*over*S*. - If
*F*is strictly increasing, then c maximizes (minimizes)*f*over*S*if and only if c maximizes (minimizes)*g*over*S.*

A firm's profit function is given by

\(\hat{\Pi}(K, L, p, r, w)=pF(K, L)-rK-wL\)

FOCs

\(\frac{\partial \hat{\Pi}}{\partial K}=pF^{'}_{K}-r=0\implies F^{'}_{K}=\frac{r}{p}\)

\(\frac{\partial \hat{\Pi}}{\partial L}=pF^{'}_{L}-w=0\implies F^{'}_{L}=\frac{w}{p} \)

We can now write the *value function* for the firm's maximum profits:

\(\hat{\Pi}^{*}(K^{*}(p,r,w),\ L^{*}(p,r,w),\ p,r,w)=\hat{\Pi}^{*}(p,r,w)\)

\(\partial\hat{\Pi}^{*}/\partial p=\frac{\partial \hat{\Pi^{*}}}{\partial K^{*}}\frac{\partial K^{*}}{\partial p}+\frac{\partial \hat{\Pi^{*}}}{\partial L^{*}}\frac{\partial L^{*}}{\partial p}+\frac{\partial \hat{\Pi^{*}}}{\partial p}=0+0+\frac{\partial\hat{\Pi^{*}}}{\partial p}\)

Because \(\)\(\hat{\Pi^{*}}=pF(K,\ L)-rK-wL\):

\(\frac{\partial\hat{\Pi^{*}}}{\partial p}=F(K^{*},\ L^{*})=Q^{*}\)

The last equation is known as Hotelling's Lemma, and describes the effect of a change in price on a firm's profits (the same can be done wrt a change in the hiring price of capital and labor)

Econ 365: List of important notions

(In no particular order)EntrepreneurshipTransaction costsProperty rightsInstitutionsExtractive institutionsRent-seekingAsymmetric informationPrincipal-agent problemPrisoners' dilemmaCoordination gameMonopolyMonopolistic competitionPublic goodSelf-enforcing equilibriumParadox of the stateCredible-commitment problemState capacityLegal capacityFiscal capacityMarket-preserving federalismLaw merchantCommon lawPublic choiceSuperstitious beliefsPath dependenceMises' impossibility theorem

Econ 340: Week 11

Tools of comparative staticsWhat is the point of comparative statics? Identifying the effects of a change in one or more parameters on the solution to an optimization problemSimple chain ruleImagine we are dealing with a function of two variables, z=f(x, y)Both x and y are themselves functions of a third variable, t: \(x=f\left(t\right),\ y=g\left(t\right)\)Thus, the original function is really a composite function: \(z=F\left(f\left(t\right),\ g\left(t\right)\right)\)In order to identify the effect of a change in t on z, we must take the total derivative of this composite function wrt t \(\frac{dz}{dt}=\frac{dz}{dx}\ \frac{dx}{dt}+\frac{dz}{dy}\ \frac{dy}{dt}\)Example 1\(z=F\left(x,\ y\right)=x^2+y^3\), where \(x=t^2\) and \(y=2t\)\(z=\left(t^2\right)^2+\left(2t\right)^3=t^4+8t^3\) \(\frac{dz}{dt}=4t^3+24t^2\)Example 3Take the following demand function: \(D=D\left(p,\ m\right)\)What are p & m?We now assume that both are functions of t: \(p=p\left(t\right)\), \(m=m\left(t\right)\)Hence, we obtain the composite demand function \(D=D\left(p\left(t\right),\ m\left(t\right)\right)\)What are the dynamics of consumption over time?\(\dot{D}=\frac{dD}{dt}=\frac{\partial D}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial D}{\partial m}\frac{\partial m}{\partial t}\)What is the rate of change of demand for this good?\(\frac{\dot{D}}{D}=\frac{1}{D}[\frac{\partial D}{\partial p}\frac{\partial p}{\partial t}+\frac{\partial D}{\partial m}\frac{\partial m}{\partial t}]=\frac{p}{D}\frac{\partial D}{\partial p}\frac{\dot{p}}{p}+\frac{m}{D}\frac{\partial D}{\partial m}\frac{\dot{m}}{m}\)Thus, the rate of change of demand over time is a function of both the price elasticity and income elasticity of demandChain rules for many variablesTake the following function \(z=F\left(x,\ y\right)\)x and y are both functions of two other variables\(x=f\left(t,\ s\right),\ y=g\left(t,\ s\right)\)Thus, \(z=F\left(f\left(t,\ s\right),\ g\left(t,\ s\right)\right)\)Example 1 \(z=x^2+2y^2\) Where \(x=t-s^2,\ y=t\cdot s\) Thus, \(z=\left(t-s^2\right)^2+2\left(t\cdot s\right)^2\) \(\frac{\partial z}{\partial t}=F_x^{'}\ \frac{\partial x}{\partial t}+F_y^{'}\ \frac{\partial y}{\partial t}=?\) \(\frac{\partial z}{\partial s}=F_{x}^{'}\ \frac{\partial x}{\partial s}+F_{y}^{'}\ \frac{\partial y}{\partial s}=?\)Example 3Let the following be the output function of an economy:\(Y=F(K, L, T)=AK_{t}^{a}L_{t}^{b}T_{t}^{c}\)How does output vary over time?\(\dot{Y}=aAK_{t}^{a-1}L_{t}^{b}T_{t}^{c}\dot{K}+bAK_{t}^{a}L_{t}^{b-1}T_{t}^{c}\dot{L}+cAK_{t}^{a}L_{t}^{b}T_{t}^{c-1}\dot{T}\) Dividing both sides by \(Y=AK_{t}^{a}L_{t}^{b}T_{t}^{c}\) we obtain the rate of change of output over time:\(\frac{\dot{Y}}{Y}=a\frac{\dot{K}}{K}+b\frac{\dot{L}}{L}+c\frac{\dot{T}}{T}\)Implicit differentiation along a level curveLet F be a function of two variables, x and yWe might want to know my how much we have to vary the quantities of both in order to leave the absolute value of the function constant: \(F(x, y)=c\)In order to find this, we must implicitly define one of the two variables as a function of the other \(F(x, f(x))=c\)We now define \(u(x)=F(\cdot)\) and take the partial derivative wrt x: \(u^{'}(x)=F^{'}_{x}\cdot 1+ F^{'}_{f(x)}\cdot f^{'}_{x}=0\)Replacing \(f^{'}(x)=y^{'}\) and solving for the latter: \(y^{'}=-\frac{F^{'}_{x}}{F^{'}_{y}}\)Example 1 \(F(x, y)=x\cdot y=5\) \(y^{'}_{x}=?\)Example 4\(D=D(t, p)\)Where t is the per unit tax and p is own price\(S=S(p)\)Equilibrium requires\(S(p)=D(t, p)\)Find \(\frac{dp}{dt}\) \(F(t, p)=D(p, t)-S(p)=0\)\(D^{'}_{t}\cdot 1+D^{'}_{p}\cdot \frac{p}{t}=S^{'}_p\cdot \frac{dp}{dt}\) Solving for \(\frac{dp}{dt}\) \(\frac{dp}{dt}=\frac{D^{'}_{t}}{S^{'}_{p}-D^{'}_{p}}\)More general casesExample 3What are the expected gains from searching?\(\Pi(t)=[p^{o}-p(t)]x^{o}-wt\)How do we interpret this function?\(\dot{\Pi(t^{*})}=-\dot{p}(t^{*})x^{o}-w=0\)Thus, optimum search yields:\(\dot{p}(t^{*})x^{o}=w\) Derive this graphically

Econ 340: Homework #2

Use the following notation for the demand function

\(D: Q(P)\)

Econ 340: Week 10

Handout from week 9

- The function
*f*has a**local extreme maximum (minimum)**at*c*, if there exists an interval (*a*,*b*) about*c*, such that \(f\left(x\right)\le\left(\ge\right)f\left(c\right)\ \forall\ x\ \in\left(a,\ b\right)\subset D\)

- Interior points in \(I\) where \(f^{'}\left(x\right)=0\)
- End points of \(I\) (if included in \(I\))
- Interior points in \(I\) where \(f^{'}(x)\) does not exist

Find the local maxima and minima for the function\(f(x)=\frac{1}{9}x^{3}-\frac{1}{6}x^{2}-\frac{2}{3}x+1\)FOC\(f^{'}(x)=0 \iff \frac{1}{3}x^{2}-\frac{1}{3}x-\frac{2}{3}=0\)\(x^{2}-x-2=0\)\(x_{1,\ 2}=\frac{1\pm \sqrt{9}}{2}\implies x_{1,\ 2}=\frac{1\pm 2}{2}=1,\ 2\)What is \(f^{'}(x)\) to the left of \(x=1\), what is it to the right?What is \(f^{'}(x)\) to the left of \(x=2\), what is it to the left?

Find the local maxima and minima for the function\(f(x)=x^{2}e^{x}\)FOC\(f^{'}(x)=0 \iff 2xe^{x}+x^{2}e^{x}=0\)\(e^{x}(2x+x^{2})=0\)\(e^{x}\neq0\)\(x(2+x)=0\)\(x=0,\ x=-2\)Do the same as for example 2 to establish which one is a minimum and which a maximum

If \(f(x)\) is twice differentiable in an interval \(I\), and *c* is an interior point of \(I\), then:

- \(f^{'}(c)=0\ \&\ f^{''}(c)<0 \implies x=c\) is a strict local maximum
- \(f^{'}(c)=0\ \&\ f^{''}(c)>0 \implies x=c\) is a strict local minimum
- \(f^{'}(c)=0\ \&\ f^{''}=0 ?\)

Re-do example 1 using the second derivative test

\(\Pi(Q)=R(Q)-C(Q)-tQ\)\(\Pi^{'}(Q)=0\ \iff R^{'}(Q^{*})-C^{'}(Q^{*})-t=0\)What are the comparative statics of the FOC wrt \(t\)?\(f^{''}(x)\)\(\frac{\partial \Pi^{'}(Q)}{\partial t}=\frac{\partial R^{'}}{\partial Q^{*}}\frac{\partial Q^{*}}{\partial t}-\frac{\partial C^{'}}{\partial Q^{*}}\frac{\partial Q^{*}}{\partial t}-1=0\)\(R^{"}\frac{\partial Q^{*}}{\partial t}-C^{"}\frac{\partial Q^{*}}{\partial t}=1\)Solving for \(\frac{\partial Q^{*}}{\partial t}\):\(\frac{\partial Q^{*}}{\partial t}=\frac{1}{R^{''}-C^{''}}\)Since \(R^{''} <0,\ C^{''}>0\)\(\frac{\partial Q^{*}}{\partial t}<0\)

- The point
*c*is called an**inflection point**for the function*f*if there exists an interval (*a, b*) such that

1. \(f^{''}(x)\geq 0\ \in (a,\ c)\) and \(f^{''}(x)\leq 0\ \in (c,\ b)\)

or

2. \(f^{''}(x)\leq 0\ \in (a,\ c)\) and \(f^{''}(x)\geq 0\ \in (c,\ b)\)

\(f(x)=x^{4}\)\(A, a, b\)\(f^{'}(x)=0\iff\ 4x^{3}=0\)\(f^{''}(x)=12x^{2}\)Thus, \(f^{''}(x)=0 \iff x=0\)Since \(f^{''}(x)\) does not change sign at \(x=0\), this is not an inflection point

Redo example 1 from above:\(f(x)=\frac{1}{9}x^{3}-\frac{1}{6}x^{2}-\frac{2}{3}x+1\)

A function is said to be aconcave functionif the line segment joiningtwo points on the graph isanythe graph orbelowthe graphonA function is said to be aconvex functionif the line segment joiningtwo points on the graph isanythe graph orabovethe graphonA function is said to be astrictly concave functionif the segment joiningtwo points on the graph isanythe graphbelowA function is said to be astrictly convex functionif the segment joiningtwo points on the graph isanythe graphabove

- A function of two variables
*x*and*y*with domain*D*is a rule that assigns a specified number*f(x, y)*to each point (x, y) in*D*

\(F(x, y)=Ax^{a}y^{b}\), where \(A, a, b\) are constantsWithxandybeing two factors of production, F is called a Cobb-Douglas or Wicksell production function

The Cobb-Douglas function is a function characterized by homogeneity of degreea+b

- \(f(x, y)=\sqrt{x-1}+\sqrt{y}\)
- \(g(x, y)=\frac{2}{(x^{2}+y^{2}-4)^{.5}}+\sqrt{9-(x^{2}+y^{2})}\)

- When taking the derivative of a function of two variables with respect of
variable, we treat the other variable*one*it were a constant*as if*

\(z=x^{3}+2y^{2}\)

\(\frac{\partial z}{\partial x}=3x^{2}\)

\(\frac{\partial z}{\partial y}=4y\)

- If \(z=f(x, y)\), then \(\partial z/ \partial x\) denotes the partial derivative of the function wrt
*x*when*y*is held constant, and \(\partial z/ \partial y\) is the partial derivative of the function wrt*y*when*x*is held constant

- \(f(x, y)=x^{3}+x^{2}y^{2}+x+y^{2}\), find the partial derivatives for this function
- \(f(x, y)=\frac{y(x^{2}+y^{2})-2x^{2}y}{(x^{2}+y^{2})^{2}}\), find the partial derivatives for this function

\(x=Ap^{-1.5}m^{2.08}\)\(\frac{\partial x}{\partial p}=-1.5Ap^{-2.5}m^{2.08}\)\(\frac{\partial x}{\partial m}=2.08Ap^{-1.5}m^{1.08}\)

\(f^{'}_{x}(x, y)=\lim_{h\to 0}\frac{f(x+h,\ y)-f(x,\ y)}{h}\)

\(f^{'}_{y}(x, y)=\lim_{k\to 0}\frac{f(x,\ y+k)-f(x,\ y)}{k}\)

Let \(Y=F(K,\ L)\) be the number of units produced whenKunits of capital andLunits of labor are employed in the production process. What is the interpretation of \(F^{'}_{K}(100, 50)=5\)?

- \(\frac{\partial^{2} z}{\partial x^{2}}\) is the second order partial derivative of
*z*with respect to*x* *\(\frac{\partial^{2} z}{\partial y^{2}}\)*is the second order partial derivative of*z*with respect to*y*

Find the second partial derivatives of the two functions from example 1.1 above

\(f(x,\ y)=x^{3}e^{y^{2}}\)Find the first and second order derivatives wrt x and y

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**A function***f***of***n***variables****domain***D*is a rule that assigns a specified number \(f(\bold{x})=f(x_{1},..., x_{n})\) to each*n*-vector \(\bold{x}=(x_{1},..., x_{n})\) in*D*

Thedemand for sugarin the US in the years 1929-1935 was estimated to be approximated by the following linear function:\(x=108.83-6.0294p+0.164w-0.4217t\)Wherepis the price of sugar,wthe individual's wage, andtthe yearThedemand for beerin the UK has been estimated to be approximated by the following Cobb-Douglas function:\(f(\bold{x})=1.058x_{1}^{.136}x_{2}^{-.727}x_{3}^{.914}x_{4}^{.816}\)

The general for of a Cobb-Douglas function of many variables is given by

\(f(\bold{x})=Ax_{1}^{\alpha_{1}}...x_{n}^{\alpha_{n}}\)

Taking the log of both sides yields

\(\ln{f(\bold{x})}=\ln{A}+a_{1}\ln{x_1}+...+a_{n}\ln{x_{n}}\)

Thus, we say that the Cobb-Douglas function is **log-linear**, because the log of the function is a linear function of the variables

Any function of *n* variables that can be constructed from continuous functions by combining the operations of addition, subtraction, multiplication, division, and functional composition is continuous wherever is defined

- \(f(x,\ y,\ z)=x^{2}y+8x^{2}y^{5}z-xy+8z\), this function is merely the sum of products of positive powers, thus,
*f*is continuous \(\forall\ x,\ y,\ z\) - \(g(x,\ y)=\frac{xy-3}{x^{2}+y^{2}-4}\), the function is continuous \(\forall\ x,\ y\) such that \(x^{2}+y^{2}\neq 4\)

If \(z=f(\bold{x})=f(x_{1},..., x_{n})\), then \(\partial f/\partial x_{i}, \forall\ i=1,..., n\) means that the partial derivatives of \(f(x_{1},..., x_{n})\) wrt \(x_{i}\) when all the other variables \(x_{j}\ (j\neq i)\) are held constantSo, provide they all exist, there arenpartial derivatives of first order

\(z=5x_{1}^{2}+x_{1}x_{2}^{3}+x_{3}^{3}-x_{2}^{2}x_{3}^{2}\)Find the partial derivatives wrt to all three independent variables

If \(z=f(\bold{x})\), then the two second-order cross-partial derivatives, \(z^{''}_{ij}\) and \(z^{''}_{ji}\) areusuallyequal

\(\partial z/ \partial x_{i}=\lim_{h\to 0}\frac{f(x_{1},..., x_{i}+h,...,n_{n})-f(x_{1},...,x_{i},...,x_{n})}{h}\)

Take the following agricultural production function\(F(K, L, T)=AK^{a}L^{b}T^{c},\ (A,\ a,\ b,\ c>0)\)Find the first and second order partial derivatives and evaluate the signs

Take the following utility function\(u(x, z)\)Where x is consumption and z is pollution\(u^{'}_{x}>0\ \&\ u^{'}_{z}<0\)What does the cross partial look like?

Take the demand function of the form\(D=Am^{\alpha}p_{i}^{-\beta}p_{j}^{\gamma}\)Where \(m\) is income, \(p_{i}\) is the own price, and \(p_{j}\) is the price of a substitute goodThe formula for the proportion of income spent on own good is given by\(\frac{p_{i}D_{i}}{m}=Am^{\alpha-1}p_i^{1-\beta}p_{j}^{\gamma}\)When \(\alpha>1\), this fraction is increasing in income. Thus, we refer to the good as luxury good. When \(\alpha<1\), the fraction is decreasing income. We refer to this good as an inferior good.

\(\)

Econ 340: Week 9

\(D:\ Q_d=a-bP \)\(S:\ Q_s=\alpha+\beta(P-t)\)Equilibrium\(a-bP=\alpha+\beta(P-t)\)\((\beta+b)P=a+\alpha+\beta t\)\(P^{*}=\frac{a+\alpha+\beta t}{\beta+b}\)\(Q^{*}=a-b\frac{a+\alpha+\beta t}{\beta+b}\)Taking the first derivative, we can show that an increase in taxation increases the equilibrium price level, while the equilibrium quantity falls

\(D: Q_d=a-b(P+t)\)\(S:\ Q_s=\alpha+\beta P\)Equilibrium\(a-b(P+t)=\alpha+\beta P\)\(P(\beta+b)=a-\alpha-bt\)\(P^{*}=\frac{a-\alpha-bt}{\beta+b}\)\(Q^{*}=\alpha+\beta\frac{a-\alpha-bt}{\beta+b}\)Taking the first derivative, we can show that the equilibrium price is decreasing in the tax, while the equilibrium quantity is decreasing in t

- Maxima and minima are both
*extreme points* - Definition of a maximum: \(c\in D\) is a maximum point for \(f\), \(\iff f(x)\leq f(c)\ \forall\ x\in D\)
- Definition of a minimum: \(d\in D\) is a minimum point for \(f,\ \iff f(x)\geq g(d)\ \forall\ x\in D\)
- Thus, f(c) is the
*maximum value*of the function, while f(d) is its*minimum value* __Strict__maxima and minima: \(\leq \& \geq\) now become \(< \& >\)

Find the maximum of the function \(f(x)=3-(x-2)^2\)Remember that \((x-2)^2\) is positive for any value of x. Thus, the function must be maximizes at \(f(x)=3\)Does the function have a minimum?What happens to the function as x goes to \(\infty\)?

\(g(x)=\sqrt{x-5}-100\)Remember that x must be larger than 5, which means that the lowest value the function can take is \(f(x)=100\)Does the function have a maximum?

- Stationary points: If f(x) is differentiable with both a minimum and a maximum at some interior point c of its domain, the tangent line must be horizontal. Unfortunately, this is a necessary but not a sufficient feature for a maximum or a minimum. This is because the same is true for
*all*stationary points. - Formally, if a function is differentiable over the interval \(I\) and \(c\) is an interior point of \(I\) and \(f^{'}(x)=0\) for \(x=c\), then \(c\) is a stationary point

Assume \(c\) is a maximum point of \(f\) over the interval \(I\). Since \(c\) is an interior point of \(I\), then there exist an \(h\) such that \(c+h\in I\)By definition of a maximum, \(f(c+h)\leq f(c)\), which implies \(f(c+h)-f(c)\leq 0\)For a small \(h\) enough, we can write Newton's quotient\(\frac{f(c+h)-f(c)}{h}\leq 0\)The limit for \(h\to 0\)from the rightyields:\(\lim_{h\to 0}^{+}\frac{f(c+h)-f(c)}{h}= 0\)The limit for \(h\to 0\)from the leftyields:\(\lim_{h\to 0}^{-}\frac{f(c+h)-f(c)}{h}= 0\)Thus, \(f^{'}(c)=0\)QED

- In some cases, we can find the maximum or minimum values of a function just by studying the sign of its first derivative
- Suppose \(f(x)\) is differentiable in an interval \(I\) and that is has only one stationary point, \(x=c\). Suppose \(f^{'}(x)\ge 0\ \forall x \in I\) such that \(x\leq c\), whereas \(f^{'}(x)\leq 0\ \forall x\in I\) such that \(x\ge c\). Then, \(f(x)\) is increasing to the left of \(c\) and decreasing to the right of \(c\). It follows that \(x=c\) is a maximum point for f in the interval

\(c(t)=\frac{t}{t^{2}+4}\), \(t\ge 0\)Find the maximum of c(t)\(\frac{dc(t)}{dt}=\frac{(t^{2}+4)-(t)(2t)}{(t^{2}+4)^{2}}=\frac{4-t^{2}}{(t^{2}+4)^{2}}=\frac{(2-t)(2+t)}{(t^{2}+4)^{2}}\)This derivative would be positive for \(t<2\) and negative for \(t>2\). Thus, \(t=2\) is the maximum of the function

- If \(f(x)\) is concave, that is, \(f^{''}(x)\leq 0\), over the interval \(I\), and the function contains a stationary point \(c\) in \(I\), then \(c\) is a maximum
- If \(f(x)\) is convex, that is, \(f^{''}(x)\geq 0\), over the interval \(I\), and the function contains a stationary point \(c\) in \(I\), then \(c\) is a minimum

\(f(x)=e^{x-1}-x\)\(f^{'}(x)=e^{x-1}-1=0\)\(e^{x-1}=1\)\((x-1)\ln{e}=\ln{1}\)\(x-1=0\Longrightarrow x=1\)\(f^{''}(x)=e^{x-1}>0\)Thus, the function is convex, which makes \(x=1\) the minimum

Suppose \(Y(N)\) bushels of wheat are harvested per acre of land when N pounds of fertilizers per acre are used. IfPis the dollar price per bushels of wheat andqis the dollar price per pound of fertilizer, then profits in dollars per acre are\(\pi(N)=PY(N)-qN,\ N\geq0\)Suppose there exists N* such that \(\pi^{'}(N)\geq 0\), for \(N\leq N^{*}\), whereas \(\pi^{'}(N)\leq0\) for \(N\geq N^{*}\). Then, \(N^{*}\) maximizes profits and \(\pi^{'}(N)=0\). That is, \(PY^{'}(N^{*})=q\)Suppose the function \(Y(N)=\sqrt{N}\), \(P=10\), and \(q=.5\).\(10\cdot .5 N^{-.5}=.5\)\(N^{-.5}=.1\)\(N^{*}=100\)

\(\)a) \(c(q)=2q^{2}+10q+32\)b) \(c(q)=aq^2+bq+c\)

A monopolist is faced with the demand function denoting the price \(P(Q)\) when output is \(Q\). The monopolist has a constant average cost \(k\) per unit produced.a) Find the profit function \(\pi(Q)\), and prove that the FOC for maximizing profits is\(P(Q^{*})+Q^{*}P^{'}(Q^{*})=k\)b) By implicit differentiation, find how the monopolist's choice of optimal is affected by changes in \(k\)c) How does the optimal profit react to a change in \(k\)?

- Important question: How can we locate extreme points when the function is not steadily increasing and/or decreasing?
- The extreme value theorem:

Iffis continuous function over a closed, bounded interval [a, b], then there exists a pointdin [a, b] wherefhas a minimum, and a pointcin [a, b] wherefhas a maximum, so that \(f(d)\leq f(x)\leq f(c)\ \forall\ x \in [a, b]\)

- The conditions of the extreme value theorem are
*sufficient*but not*necessary*

- Find all stationary points of
*f*in (*a, b*)– that is, find all points*x*in (*a, b*) that satisfy the equation*f'(x)=0* - Evaluate
*f*at the end points*a*and*b*of the interval and also at all stationary points - The largest function value found in 2. is the maximum value and the smallest function value is the minimum value of the function

Find minimum and maximum of \(f(x)=3x^{2}-6x+5\), \(x\in [0, 3]\)

Find the minimum and maximum value of \(f(x)=.25x^{4}-\frac{5}{6}x^{3}+.5x^{2}-1\), \(x\in [-1, 3]\)

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\(\pi(Q)=R(Q)-C(Q)\)FOC\(R^{'}(Q^{*})=C^{'}(Q^{*})\)Thus,production should be adjusted to a point where the marginal revenue is equal to the marginal cost.

Do example 2, p. 288

Neither a borrower nor a lender be,For loan oft loses both itself and friend,And borrowing dulls the edge of husbandry.

The intertemporal choice problem: how should you allocate consumption over time (say, two periods)?Assume a log utility function:\(U(c_{1},c_{2})=\ln{c_1}+\frac{1}{1+\rho}\ln{c_{2}} \)Where you can only spend on \(c_2\) what you have not spent on \(c_1\), or\(c_{2}=y_{2}-(1+r)(c_{1}-y_{1})\)Substituting into the utility function\(U(c_{1},c_{2})=\ln{c_1}+\frac{1}{1+\rho}\ln{[y_{2}-(1+r)(c_{1}-y_{1})]} \)FOC\(\frac{1}{c_1}+\frac{1}{1+\rho}\frac{1}{y_{2}-(1+r)(c_{1}-y_{1})}\cdot (-(1+r))=0\)Solving for \(c^{*}_{1}\)\(c^{*}_{1}=\frac{y_{2}+ry_{1}}{2+\rho}\)

Econ 306: Intermediate Microeconomics

General informationEnnio E. Piano, Department of Economics and F.A. Hayek Program, George Mason University,Buchanan Hall (formerly Mason Hall), D137-7.epiano@gmu.eduOffice hours: Thursday, 9:00-10:30 AM or by appointment. Buchanan Hall, F.A. Hayek program office, D137-7.Course objectivesThe course is aimed at familiarizing the students with the basic tools of economic analysis, including the theory of consumer behavior, the theory of seller behavior, partial and general equilibrium theory, and game theory. The focus will be on the application of these tools to the analysis of human behavior in all its forms.PrerequisitesPrinciples of Microeconomics (ECON 103), Principles of Macroeconomics (ECON 104), and one semester of calculus (MATH 108 or 133). I expect you to be familiar with the use of algebra to solve simple sets of simultaneous equations, graphing, and to understand the concept of a derivative. ReadingsRequired textbooksHirshleifer, J., Glaezer, D., and Hirshleifer, D. 2005. Price theory and applications: Decisions, markets, and information (7th edition). Cambridge University Press.ArticlesAllen, D. W. "Property rights, transaction costs, and Coase: One more time." (https://drive.google.com/file/d/0B3CY-zqfWBQCRTlLSDltcVJ2Vzg/view?usp=sharing)Buchanan, J. M. "Cost and choice." (https://drive.google.com/file/d/0B3CY-zqfWBQCdnNXU3hOeEZzTTQ/view?usp=sharing)Cowen, T. "Public goods." (https://drive.google.com/file/d/0B3CY-zqfWBQCZjJJdWpXREc0VVk/view?usp=sharing)Shughart III, W. "Public Choice." (https://drive.google.com/file/d/0B3CY-zqfWBQCRlZpTG9DSjN1Q1U/view?usp=sharing)Staten M. and Umbeck J. "Economic inefficiency." (https://drive.google.com/file/d/0B3CY-zqfWBQCQjlzUERzQnpWRlE/view?usp=sharing)Varian, H. "How to build an economic model in your spare time." (https://drive.google.com/file/d/0B3CY-zqfWBQCMFY2dVlkYTBnMW8/view?usp=sharing)Recommended readingsAlchian, A. A., and Allen, W. R. 1969. Exchange and production: Theory in use. Wadsworth.David Friedman. Price Theory: An Intermediate Text. 1985. (Available free online: http://www.daviddfriedman.com/Academic/Price_Theory/PThy_ToC.html)Scheduled outlineAugust 28th-30thClass overview and introduction. Hirshleifer, chapter 1.September 4th-6thPreferences and choice. Hirshleifer, chapter 3; Buchanan, "Cost and choice."September 11th-13thConsumption and the derivation of the laws of demand. Hirshleifer, chapter 4. Quiz 1.September 18th-20thDemand theory, extensions. Hirshleifer, chapter 5. Quiz 2.September 25ht-27thPure exchange. Hirshleifer, chapters 2 and 14. Quiz 3.October 2nd-4thProperty rights, transaction costs, and the Coase theorem. Hirshleifer, chapter 14; Allen, D. W. "Property rights, transaction costs, and Coase: One more time." Quiz 4.October 10th-11thThe economics of time. Hirshleifer, chapter 15. Quiz 5.October 16thMidterm review.October 18thMidterm.October 23rd-25thThe theory of the firm and introduction to production theory. Hirshleifer, chapter 6.October 30th-November 1stThe theory of the competitive firm. Hirshleifer, chapter 7. Quiz 6.November 6th-8thThe theory of the monopolistic firm. Hirshleifer, chapter 8. Quiz 7.November 13th-15thIntroduction to game theory and the theory of oligopoly. Hirshleifer, chapter 10. Quiz 8.November 20th-27thWelfare economics. Hirshleifer, chapter 16; Cowen, "Public goods;" Staten and Umbeck, "Economic inefficiency." Quiz 9.November 29th-December 4thPublic choice theory and "How to build an economic model in your spare time!" Hirshleifer, chapter 17; Shughart, "Public choice." Quiz 10.December 6thFinal review.Other important datesSeptember 5th, last day to add and drop classes without penalty.September 19th, last day to drop with a 33% tuition penalty.September 29th, last day to drop with a 66% tuition penalty.November 22nd-26th, Thanksgiving recess.December 9th, last day of class.December 13th-20th, exam period.GradingYour grade in this course consists of ten weekly quizzes (30%), a midterm exam (35%), and a final exam (35%).Quizzes: Quiz questions are based on the assigned readings and lectures. This means you cannot and will not do well in this class unless you do the assigned readings and attend the lectures. The quizzes will always be taken on Wednesday. I will drop the lowest quiz grade to allow for unexpected events. No make-up quizzes are available. No exceptions. Midterm: The midterm date is October 16th. You cannot make-up the midterm under any circumstance. If you know ahead of time of some extenuating circumstance that will prevent you from taking the midterm, you must contact me ahead of time (i.e. at least 48 hours prior to the exam). In the event that you are excused from taking the midterm upon contacting me, I will shift your midterm grade towards your final (so your midterm will be worth 0% and your final 60%). If you do not contact me ahead of time and do not take the midterm, you will receive a zero. No exceptions. Final: The final exam date is to be announced. The final exam is comprehensive and you must take it on this date. If you know ahead of time of some extenuating circumstance that will prevent you from taking the final on the scheduled date, you must contact me ahead of time (i.e. at least one week prior to the exam). In the event that you are excused from taking the final on the scheduled date upon contacting me, we will arrange an alternate date. If you do not contact me ahead of time and do not take the final on the scheduled date, you will receive a zero. No exceptions.The grading scale is as follows:A+: 97-100%; A: 92-96%; A-: 88-91%; B+: 84-87%; B: 80-83%; B-: 76-79%; C: 70-75%; F:<70%.Academic integrityGeorge Mason University’s Honor Code requires all community members to maintain the highest standards of academic honesty and integrity. Cheating, plagiarism, lying, and stealing are all prohibited. Honor Code violations will be reported to the Honor Committee. Plagiarism is not accepted (statements from Macon web site: http://mason.gmu.edu/montecin/plagiarism/htm#plagiarism). The use of electronic devices is prohibited during an exam or a quiz; failure to comply with this will result in your failure of the assignment and potentially the failure of the class. Make sure to familiarize yourself with the GMU Honor Code, which is stated in the George Mason University Undergraduate Catalog.

Econ 340: Week 6-b

- Suppose \(y\) is a function of \(u\) and \(u\) is a function of \(x\)
- Formally, \(y=f\left(u\right)\) and \(u=g\left(x\right)\)
- What happens when \(x\) changes?
- A change in \(x\) causes a "chain reaction" on the values of \(u\) and, therefore, \(y\)
- In order to find this effect, we use the
**chain rule**:

\(\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}\)

- The chain rule: "if y is a differentiable function of u, and u is a differentiable function of x, then y is a differentiable function of x"

Find the derivative of the following:

a) \(y=u^5\), \(u=1-x^3\)

\(y=(1-x^3)^5 \Longrightarrow \frac{\partial y}{\partial x}=5(1-x^3)^4\cdot -3x^2\Longrightarrow \frac{\partial y}{\partial x}=-15x^2(1-x^3)^4\)

b) \(y=\frac{10}{(x^2+4x+5)^7}\)

\(\frac{\partial y}{\partial x}=-70(x^2+4x+5)^{-8}\cdot (2x+4)\Longrightarrow \frac{\partial y}{\partial x}=-70(2x+4)(x^2+4x+5)^{-8}\)

c) \(y=\left(\frac{x-1}{x+3}\right)^{1/3}\)

\(\frac{\partial y}{\partial x}=\frac{1}{3}(\frac{x-1}{x+3})^{-2/3}\cdot \frac{(x+3)-(x-1)}{(x+3)^2}\)

- We can also write \(y=F(x)=f(g(x))\)
- Thus, \(F^{'}(x)=f^{'}(g(x))\cdot g^{'}(x)\)

- Write \(F(x)=f(g(x))\)
- Use Newton's quotient to write \(F^{'}(x)=\lim_{h\to 0} \frac{f(g(x+h))-f(g(x))}{h}\)
- Define \(k=g(x+h)-g(x)\) and substitute \(g(x+h)=g(x)+k\) into \(f(g(x+h))\)
- Multiply and divide \(F^{'}(\cdot)\) by \(k\): \(F^{'}(x)=\lim_{h\to 0} \frac{f(g(x)+k)-f(g(x))}{k}\cdot \frac{k}{h}=F^{'}(x)=\lim_{h\to 0} \frac{f(g(x+h))-f(g(x))}{k}\cdot \frac{g(x+h)-g(x)}{h}\)
- Taking the limit yields: \(F^{'}(\cdot)=f^{'}(g(x))\cdot g^{'}(x)\)

- We call the derivative of a function its
**first derivative**. If the resulting function is itself differentiable, we call it the**second derivative**of the original function - Notation: \(f^{''}(x)\) is the second derivative of \(f(x)\). \(f^{''}=\frac{d^2f(x)}{dx^2}\)

Find the second derivative of the following equations

a) \(Y=AK^{\alpha}\)

\(Y^{'}=\alpha AK^{\alpha-1}\)\(Y^{''}=\alpha(\alpha-1)AK^{\alpha-2}\)

b) \(L=\frac{t}{t+1}, t\ge 0\)

\(L^{'}=\frac{1}{(t+1)^{2}}\)\(L^{''}=-2(t+1)^{-3}\)

Econ 340: Week 6

- We will first focus on the geometrical relevance off the notion of derivative: the slope of a curve
- Important: Under most circumstances, we always refer to the slope of a curve
*at a given point*of the curve

- Take a function \(y=f\left(x\right)\). Draw it. Draw a straight line tangent to f(x) at point \(\left(a,\ f\left(a\right)\right)\).
- We refer to the
*slope of the tangent*as the**derivative**of f(x) at point a. - The notation for this derivative is: \(f^{'}(a)\) (f prime of a)

- Draw the graph from p. 165
- The formula for the slope of this curve is given by: \(m_{PQ}=\frac{f(a+h)-f(a)}{h}\), where the numerator is the vertical distance between the two points and the denominator is the horizontal distance between the two
- This is also referred to as
**Newton quotient** - Important: what happens when h=0?
- Nevertheless, we want to find a formula for the tangent when h tends towards zero:
- We thus have the following definition of a derivative: \(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)
- We can now use the point-slope formula to derive the equation for our tangent: \(y-f(a)=f'(a)(x-a)\)

Derive the derivative of \(f(x)=x^2\) using the method above

\(f(x)=x^2\) implies \(f(a+h)=(a+h)^2\) and \(f(a)=a^2\).Substituting in the formula for the derivative yields\(f'(a)=\lim_{h\to 0}\frac{(a+h)^2-a^2}{h}\)Basic arithmetic yields\(\)\(f'(a)=\lim_{h\to 0}\frac{a^2+2ah+h^2-a^2}{h}=\lim_{h\to 0}\frac{2ah+h^2}{h}=\lim_{h\to 0}\frac{2a+h}{1}=2a\)

- If
*f*is a relatively simple function, we can find the derivative by following the following rules:

- Add \(h\neq 0\) to \(a\) and compute \(f(a+h)\)
- Compute \(f(a+h)-f(a)\)
- Form the Newton quotient
- Simplify the resulting formula as much as possible, making sure not to have \(h\) in the denominator
- Take the limit as
htends to 0

Example

Use these rules to find the derivative of \(f(x)=x^3\)

Some notational issues

- There are a wide array of notations when it comes to derivatives (this is because Newton and Leibniz "discovered" calculus): \(\frac{dy}{dx};\ \frac{df\left(x\right)}{dx};\ \frac{d}{dx}f\left(x\right)\)

Increasing and decreasing functions

- If \(f\left(x_2\right)\ge f\left(x_1\right)\) whenever \(x_2>x_1\), the function is
**increasing** - If \(f\left(x_2\right)\le f\left(x_1\right)\) whenever \(x_2>x_1\), the function is
**decreasing** - If \(f(x_2)>f(x_1)\) whenever \(x_2>x_1\), the function is
**strictly increasing** - If \(f(x_2)<f(x_1)\) whenever \(x_2>x_1\), the function is
**strictly decreasing**

We can reformulate this conditions in terms of the function's derivative:

- If \(f'(x) \ge 0\) for all \(x\) in the interval
*I*, the function is**increasing**over*I*(and vice-versa) - If \(f'(x)\le 0\) for all \(x\) in the interval
*I*, the function is**decreasing**over*I*(and vice-versa) - \(\)If \(f'(x)=0\) for all \(x\) in the interval
*I*, the function is**constant**over*I*(and vice-versa)

Examples

- Find whether the function \(f(x)=\frac{1}{2}x^2-2\) is increasing or decreasing
- Find whether the function \(f(x)=-\frac{1}{3}x^3+2x^2-3x+1\) is increasing or decreasing

Rates of change

- In economics, we focus on a precise interpretation of the notion of derivative: the derivative is seen as the rate of change of a variable

Rate of change interpretation

- Take a function \(f(x)\). Newton's quotient can then be interpreted as the
*average rate of change*in \(f(x)\). \(f(x)\) changes at every point over the interval. This change is expressed by the difference in the numerator. Dividing this change by the change in \(x\) (e.g., \(x\)) is the average rate of change - Thus, using the interpretation of the Newton's formula from above, the derivative can be interpreted as \(\)the
*instantaneous rate of change*of the function - We can now derive the
*relative rate of change*or*proportional rate of change*: \(\frac{f'(a)}{f(a)}\)

Example

Consider a firm producing some commodity in a given period, with

- \(C(x)\) being the cost function
- \(R(x)\) being the revenue function
- \(\pi(x)=R(x)-C(x)\) is the profit function
We can use Newton's quotient to find the marginal cost of the firm:\(C'(x)=\lim_{h\to 0}\frac{C(x+h)-C(h)}{x}\)For an \(h\) small enough, we can write\(C'(x)\simeq \frac{C(x+h)-C(x)}{h}\)For \(h=1\) we obtain\(C'(x)\simeq C(x+1)-C(x)\)Thus, the interpretation of marginal cost is (approximately) the increase in cost due to an increase in output by one unit

Differentiability and empirical functions

Economists use derivatives to study the rate of change of a wide variety of variables. In some case, this is mathematically improper, as the variable of interest is discrete and not continuous and we cannot define a derivative for discrete functions

- We say that \(f(x)\) has the number
*A*as its limit as \(x\) tends to \(a\), if \(x\) tends to*A*as \(x\) tends to (but is not equal to) \(a\):

\(\lim_{x\to a} f(x)=A\) or \(f(x) \to A\) as \(x\to a\)

- We say that the limit does not exist if \(f(x)\) does not tend to any specific value as \(x\) tends to \(a\)
- Important: writing \(\lim_{x\to a}f(x)=A\) means that we can make \(f(x)\) as close to \(A\) as we want for all \(x\) sufficiently close to (but not equal to) \(a\)

If \(\lim_{x\to a}f(x)=A\) and \(\lim_{x\to a}g(x)=B\), then

- \(\lim_{x\to a}(f(x)\pm g(x))=A\pm B\)
- \(\lim_{x\to a}(f(x)\cdot g(x))=A\cdot B\)
- \(\lim_{x\to a}\frac{f(x)}{g(x)}=A/B, B\neq 0\)
- \(\lim_{x\to a}(f(x))^r=A^r\)

- \(\lim_{x\to 2}(x^2+5x)\)
- \(\lim_{x\to 4}\frac{2x^{3/2}-\sqrt{x}}{x^2-15}\)
- \(\lim_{x\to a}Ax^n\)

- If the limit \(f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}\) exists, then we say that the function
*f*is differentiable at

- If the function is a constant, its derivative is always equal to zero, or \(f(x)=A \Longrightarrow f'(x)=0\)
- What is the intuition behind this?
- We can extend this to derive the following rule: the derivative of a function containing an additive constant is given by

\(y=A+f(x) \Longrightarrow y'=f'(x)\)

- If the constant
*multiplies*the function, then the formula for the derivative is

\(y=Af(x) \Longrightarrow y'=Af'(x)\)

\(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}\)Now write\(g(a)=Af(a)\)Which implies\(g'(a)=\lim_{h\to 0}\frac{g(a+h)-g(a)}{h}\)Substituting from above yields\(g'(a)=\lim_{h\to 0}\frac{Af(a+h)-Af(a)}{h} \Longrightarrow g'(a)=\lim_{h\to 0}A\frac{f(a+h)-f(a)}{h}\)Or\(g'(a)=A\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}=Af'(a)\)QED

Find the derivative of the following functions

- \(y=5+f(x)\)
- \(y=f(x)-\frac{1}{2}\)
- \(y=4f(x)\)
- \(y=\frac{Af(x)+B}{C}\)

\(f(x)=x^a\Longrightarrow f'(x)=ax^{a-1}\), wehere \(a\) is constant

Find the derivative of the following functions

- \(y=x^5\)
- \(y=3x^8\)
- \(y=\frac{x^{100}}{100}\)

\(F(x)=f(x)\pm g(x) \Longrightarrow f'(x)\pm g'(x)\)

\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)+g(x+h)-(f(x)+g(x))}{h}=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}+\lim_{h\to 0}\frac{g(x+h)-g(x)}{h}\)Which implies\(F'(x)=f'(x)+g'(x)\)

\(F(x)=f(x)\cdot g(x)\Longrightarrow F'(x)=f'(x)g(x)+f(x)g'(x)\)

\(F'(x)=\lim_{h\to 0}\frac{F(x+h)-F(x)}{h}\Longrightarrow \lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x)}{h}\)

We now add and subtract \(f(x)g(x+h)\) to the numerator

\(F'(x)=\lim_{h\to 0}\frac{f(x+h)\cdot g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}\Longrightarrow\lim_{h\to 0}\frac{[f(x+h)-f(x)]g(x+h)+f(x)[g(x+h)-g(x)]}{h}\)

Rearranging yields

\(F'(x)=\lim_{h\to 0}g(x+h)\frac{f(x+h)-f(x)}{h}+f(x)\frac{g(x+h)-g(x)}{h}\)

This implies

\(F'(x)=g(x)f'(x)+f(x)g'(x)\)

Find the derivative of

\(h(x)=(x^3-x)(5x^4+x^2)\)

\(F(x)=\frac{f(x)}{g(x)}\Longrightarrow F'(x)=\frac{f'(x)g(x)-f(x)g'(x)}{g(x)^2}\)

- \(F(x)=\frac{3x-5}{x-2}\)
- Find the derivative for the average cost function \(AC=\frac{C(x)}{x}\)

Econ 365: Week 6

- Field that emerged in the 1960s with the publication of
*The Calculus of Consent*(Buchanan and Tullock 1962) - Three driving principles of public choice:

- Rationality postulate
- Methodological individualism
- Politics as exchange

- Contributions of public choice to our understanding:

- Voting
- Bureaucratic behavior
- Theory of regulation
- Dictatorial behavior
- Legislative process (logrolling, agenda setting, industrial organization of congress)

- How economics sees conflict
- The dark side of the force: Exchange vs. Conflict
- Why conflict?

- Informational asymmetries
- Transaction costs

- A simple model of conflict:

\(p_i,\ i\in\left\{1,\ 2\right\}\)\(p_1=1-p_2\)\(p_1=\frac{n_1}{n_1+n_2},\ p_2=\frac{n_2}{n_1+n_2}\)Max \(p_i\Theta-cn_i\)

- Tullock (1971),
*Public Choice*. One of the first contributions in the literature on the economics of conflict

- What is a collective action problem?
- The logic of collective (in)action?
- The importance of size, rewards, and side benefits

- The payoff of the potential participant (siding with revolution):

\(P_r=P_g\cdot\left(L_v+L_i\right)+R_i\left(L_v+L_i\right)-P_i\left[1-\left(L_v+L_i\right)\right]-L_w\cdot I_r+E\)

- See paper for the meaning of symbols
- Note that for large enough groups, \(L_i\simeq0\):

\(P_r\simeq P_gL_v+R_iL_v-P_i\left(1-L_v\right)-L_wI_r+E\)

- Define \(G_r=P_r-P_gL_v\):

\(G_r\simeq R_iL_v-P_i\left(1-L_v\right)-L_wI_r+E\)

- Public good considerations have mostly a marginal effect on the decision of individuals to participate
- Revolutions are more likely to succeed when individuals expect a larger
*private*benefit from participation. What does this mean historically? - Why do regimes target minorities?

Econ 340: Week 4-5

Univariate functionsIntroductionDefinition of univariate function: A function where one variable (usually identified with the letter \(y\)) depends on one other variable (usually identified with the letter \(x\)). A function needs not be formulated mathematically. For example, one can show the same relationship with a table (GDP over time example)A univariate function can also be represented by a graph on a two-dimensional space (example)Basic definitionsFunction: Rule assigning a unique value to our dependent variable given the value of the independent variable. Why is it important that the value of the dependent function be unique?Domain: All the values taken by the independent variable (usually identified with the letter \(D\))Range: All the values taken by the dependent variable (usually identified with the notation , \(f\left(x\right)\) "eff of ex")Functional notationStandard notation for a function is: \(y=f\left(x\right)\), where \(y\) is the dependent variable, \(x\) is the independent variable, and \(f\left(\cdot\right)\) is the rule defining the functionExamples\(f\left(x\right)=x^3\), numerical examples