• choice under uncertainty

• general equilibrium models

• game theory, signalling, asymmetric info

• externalities and public goods

utility functions to examine expansion paths

• are there infinitely many ces utility functions?

• alpha + beta != 0 necessarily

• CES utility : ANALYSIS

• u = \alpha \ln x + \beta \ln y : why is mux = f(x) here, but the cb version is mux = f(x,y) ?

• if this and cob douglas are equivalent, how come here mux = 1/x (which is a fn only of x), but with cb, its a function of x and y?

• well, its not a big deal, bc utility is ordinal.

• u = x^{\alpha} + \beta \ln y : [corner solution, lagrangian with inequalities]

• u = x + y + \ln y but what about x+z+lny, where z and y are the same item?

• u = x - \frac{1}{y}

• u = \ln x + \ln (\ln y)

• draw x-y expansion path with increasing M, with budget lines thru the path as px/py changes
• CES, elasticity of sub : comparative statics? is that partial y / partial x etc?
• Stone-Geary

• DO WE EVER ASSUME THAT u_xy > 0 ?

  on optimization, or the assumption thereof

  why do firms and consumers have to max profit, they cant do it, why should we model them like they can? why dont we think of profit max and consumer max as better technology? instead of a given? ie. the better our technology, the harder to max, and as you adjust to it, you get better at optimizing

  i guess at the core, its easier to make a mistake than to not make a mistake. accuracy depends on knowledge, which increases with investment in nerds, and ppl who listen to them

  if it is easier to make a mistake than to not, then most ppl will make mistakes, some will do right by hiring nerds and increase their odds, but movement in and out of economy by new firms means that the default will be mistakes

  inferior goods

  ** are there any functions which have a negative sub effect? ** ie. px down, you feel richer, you buy less of x as you get richer, so you actually sub away from it (x is an inferior good) ** consider this in the context of exchange rates too

  GE, and gov intervention

  why would gov be needed if ppl made perfect decisions?

  we shouldnt assume ppl make perfect decisions in a bubble, and that they miss other, obviously good decisions

  ppl make bad decisions more than good, which is why we need the gov

utility functions

• utility : u(\mathbf x) = u(x_1, x_2, \dots , x_n)

• marginal utility : the increase in utility from an increase in a good, all else being equal : \nabla {u(\mathbf x)} = \big [ \frac{\partial u}{\partial x_1} \frac{\partial u}{\partial x_2} ... \frac{\partial u}{\partial x_n} \big ]^T = \big [ u_1(\mathbf x) \dots u_n(\mathbf x)]^T

• marginal utility :
• all mu > 0 means is that mu_x(x,y) will be in the first quadrant. utility can be +ve or -ve, but mu will be in Q1

• non-satiation : marginal utility is always positive; more is always better; each additional unit always increases utility : u_x \gt 0

• decreasing marginal utility : while marginal utility always increases with x, it increases less and less. Each additional unit is good, but not as good as the one before it : u_{xx} \lt 0

• idc : indifference curve : the map between x and y where du = 0. In the case of u = x^{\alpha}y^{1-\alpha}, we get y = \Big ( \dfrac{\bar u}{x^{\alpha}} \Big ) ^{\frac{1}{1-\alpha}}

• mrs : marginal rate of substitution : the negative of the slope of the indifference curve

  • since the idc is given by du = \nabla {u(\mathbf x)} \cdot \mathbf {dx} = 0, if utility is u(x,y), then we have : du = \frac{\partial u}{\partial x} dx + \frac{\partial u}{\partial y} dy = 0 which gives \frac{dy}{dx} = -\frac{u_x}{u_y}

  • The slope of the indifference curve is \frac{dy}{dx} = -\frac{u_x}{u_y} and so the mrs is defined as : MRS_{xy} = \frac{u_x}{u_y}

• decreasing mrs : requires some ANALYSIS

• elasticity of substitution : requires some ANALYSIS

• homogeneous : f(a \cdot \mathbf{x}) = a^k \cdot f(\mathbf{x}) is homogenous degree k

• homothetic : special case of homogeneous where k=1 : f(a \cdot \mathbf{x})=a \cdot f(\mathbf{x}) Example : Cobb-Douglas : u(ax,ay)=(ax)^{\alpha}(ay)^{1-\alpha}=a(x^{\alpha}y^{1-\alpha})=a \cdot u(x,y)

• monotonic transformation : mapping between ordered sets which preserves or reverses the given order (ie. y=2x)
• concave, convex : {log(x), log(-x), -x²} are concave
• quasiconcave, quasiconvex

Income & sub effects & other topics

• Marshallian : max Utility given Budget : x=f(p,I)
• Hicksian : min Cost given Utility : h(p,u̅) = arg min Σ p_ix_i subject to u(x)= u̅
• Hicksian is also called the compensated demand function
• L = p_xx+p_y - \lambda(U_0 - \alpha \ln x - \beta \ln y) becomes x_c^* = U_0 \cdot \Big ( \dfrac{p_y}{p_x} \cdot \dfrac{\alpha}{1-\alpha} \Big ) ^{1-\alpha}
• uncompensated substitution effect : \frac{\partial x_c^*}{\partial p_x} \Big \vert _{U=U_0} < 0
• slutsky decomp : total derivative of h(p,u)=x(p,m)
• PROVE SUB EFFECT IS ALWAYS NEGATIVE for convex preferences - needs diminishing mrs
• youngs theorem
• shepherds lemma
• COMPARE : marshallian demand curve to hicksian (compensated) demand curve
• COMPARE : M v H, slutsky equation. check this out. INFERIOR V NORMAL
• gross complements, substitutes : for log pref, that d px causes no change in y, they are not subs at all.
• envelope theorem
• hotelling lemme
• roys identity
• expenditure fn e(p,u) = e(p,v(p,I)) : basically, plug in hicksian allocation into cost function
• indirect utility v(p,I) = v(p,e(p,u)) : basically, plug in marshallian allocation into utility function
• compensating variation: amt you have to pay someone to restore their initial utility

Firms

production functions

• mp > 0
• diminiishing returns
• diminishing MRTS (snyder p309)
• positive cross productivity d/dk(df/dL) > 0 [reverse, by youngs theorem too], comment on p309
• CRS, constant returns to scale (homothetic)

• CES production: q=f(k,l) = (k^{\rho}+L^{\rho})^{\gamma/\rho} , where rho < 1, gamma > 0

• elasticity of sub for ces, check out cases, same almost as utility

oligopoly

• Cournot duopoly

• what is good in the short term might be bad in the long term

pcomp

• profit max under pcomp : \pi = pq-c(q) \rightarrow p=MC

• \pi = p \ln z - wz , where q = \ln z, so z=e^q

• \pi = p \ln e^q - we^q

• \pi = p q - we^q

limiting factor

• CRS : ie. f(L,K) where f(aL,aK)=af(L,K). IF profit = f(L,K) - wL -rK, then the firm can keep increasing profit as long as it increasing L and K in proportion, such that MPL/w=MPK/r : its expansion path

• but if f(L\bar{K}), then if you tried following the expansion path, youd get a smaller amt for L, because the idea is moving L-K together, in which case u can increase profit forever, but if not, you max L as tho K is fixed, which it is, and that goes until MPL = w/p. the ratios MP_i/p_i will not be equal.

  We cannot observe MPL/w = MPK/r, because I will calculate it.

  the expansion path it gives is a product of that equality.

  which does not hold. the equality will give z2

  but then theres the added bit from maxing at that k_supplied.

  if u cud spend $ how u wanted, youd want this ratio

  (1) : z_2^{expansion} = \dfrac{\beta}{1- \beta} \dfrac{p_2}{w} \cdot k

  you would choose to expand along that path

  but if k is limited, then you can no longer follow this path.

  so u put more money into z than this path suggests

  so while your profits go up by adding to z

  they dont go up as much as they wud if you could spend more on k

  since k is limited, z_2^* = z_2^{expansion} + squeezing z

is z**alpha*(k+1)**(1-alpha) CRS? not exactly, but it will keep buying z and k. show it

Choice under undertainty

• a fair bet : E[x] = 0 , for example E[x] = 0.5 \cdot (x) + 0.5 \cdot (-x) = 0

• St Petersburg paradox

• expected utility : E[u(x)] = \displaystyle\sum p(x) \cdot u(x)

• von morgenson nonsense, and pi
• expected utility : of E[u(w)], w=w₀ + x, where x = {k,-k} with p=0.5, remembering that E[f(x)] = Σp_if(x_i)

• risk aversion :

  • u(w) is concave, meaning u'(w) \gt 0 and u''(w) \lt 0

  • this agent avoids a fair bet : u(w) \gt E[u(w+x)], where E[x] = 0

• insurance : pay to avoid risk, where risk is x such that where E[x] < 0

• risk aversion : Pratt's measure r(w) = -u''/u'

• risk aversion : analysis via Taylor series expansion

• risk aversion and wealth : the problem with u=lnw

• risk aversion and wealth : exponential wealth utility u = -e^-Aw and CARA

  • A is what makes this person risk averse or not
  • a person would pay h to avoid taking a fair bet of x, regardless of how wealthy they are
  • hence the name, CARA - thats a bit weird
  • A = 0.000001, u = 1-e^{-Aw}, compare the same bet for different levels of initial wealth

• normally distributed risk E[u]=\int u(w) \cdot f(w,\mu,\sigma) dw, where f \backsim N(\mu,\sigma^2)

• relative risk aversion : an upgrade in thinking rr(w) = wr(w) = -W u''/u'

• CONSTANT RELATIVE RISK AVERSION
  • its not really sensible to think a person will look at the same gamble x the same regardless of their wealth
  • makes more sense to think a person will regard a bet of y% of their wealth the same tho, regardless of their wealth
  • u(w,R) = \begin{cases} \dfrac{w^R}{R} &\text{if } R \lt 0 \\ \ln W &\text{if } R = 0 \end{cases}
  • constant relative risk aversion, CRRA
  • empirical evidence** is R = (-3, -1) *** from where? CRRA

  • crra, (p221) the person is concerned with the percentage of loss. ln(w-fw) = 0.5*ln(w-xw) + 0.5*ln(w+xw) -> f = 1-(+sqrt(1-x**2))

  • so that result means that such a person will pay f% to avoid a x% fair bet, no matter how much wealth they have. that's kind of an odd result. a gazillionaire would have f% to avoid an x% fair bet, and some poor fella would do the same (if they had the same level of R in the crra)

• risk aversion and wealth : a summary

• How to deal with risk :

  • insurance

  • diversification

  • options

  • pricing an insurance product to max profit based on known risk

  • portfolio problem

f(x) = \dfrac{1}{\sigma \sqrt{2 \pi}} \cdot e^{-\frac{1}{2} \cdot ( \frac{x - \mu}{\sigma} ) ^2}

why do i remember a bit where i integrated a big scary looking integral, it might have been the expected value of the normal distribution

General equilibrium

• edgeworth box : 2 person endowment and exchange

• trade can be good, but it can make you vulnerable

• robinson crusoe : 1 person - 1 firm (y=lnz) - 1 good economy

Agents are consumers and owners of the firms. Amy has a net, and bob has a hoe. amy can catch x fish, and bob can grow y apples but any can catch more fish is bob helps, and bob can grow more apples in amy helps so bob helps amy catch fish, and amy gives bob fish as payment and amy helps bob grow apples, and bob gives amy apples thats a GE model 2 consumers, 2 firms, 2 goods the agents are both consumers and owners of the firms

• 2x1x1 economy : 2 consumers + 1 firm + 1 good
• 2x2x2 economy : 2 consumers + 2 firms + 2 goods : single product firms w log production

• government : all consumers want a good [water, electricity, roads] and firms wont provide it. it's free for consumers, but they pay for it in income tax. and firms pay tax also. so there is a personal income tax and corporate income tax rate.

• 3x3x3 single product firms

• 1x2x2 1 consumer, 2 firms, 1 intermediate good, 1 final good

• 1 consumer, 2 firms. 1 makes capital. 1 makes goods. consumer owns both.

  • consumer want c and b
  • 1 firm, makes capital and output
  • it hires z = z1 + z2, the consumer only supplies n, doesnt see the rest
  • the firm makes f(L=z1, K=0) output, but K=g(z2) capital
  • next year, the firm makes f(z1, K) output, but K=g(z2) capital again
  • or, 1 consumer, 2 firms, 1 firm makes capital, the other uses that output to make goods
  • the consumer works and receives goods.
  • consumer saves.

• before anything else, add multiple periods

  • year 1: young max u=f(c1,c2,b1,b2) and there is money and they can save
  • young have endowment of money and leisure, old have money and leisure, but more money, and want leisure more
  • firms max profit like always, producing one good, but using 2 kinds of labor, young and old

  • consumer max: \alpha_i \ln c_i + \beta_i \ln b_i

  • budget : leisure_endowment*wage + savings + money_endowment + dividends

• high and low skill workers. with firms hire both

• GE: 2 consumers, 2 firms, 2 goods. one firm's output is demanded by both consumers and the other firm.

• there is a bank, and every agent can borrow money. the bank can create money. and charge interest per period. there need to be 2 periods, in order to compare today and tmrw.

• multiple periods

• money exists : consumers have endowment of TIME + MONEY, firms have MONEY. consumers want money since it gives utility, but also to spend it. the amt of money they want depends on their age.

• ** GE: 2x2x2_economy where both firms make both goods
• ** GE-07: 2x2x2: 2 consumers have oil, each owns a firm and makes goods that both want

• signalling games education

• simultaneous oligopoly. compete on price. show nash eq.