Economics

Function of price and income. X* = X(Px, Py, I) and Y* = Y(Px, Py, I). Assumption that nominal income and other prices held constant.

homogenous of degree k if f(tx1, tx2...txn) = (t^k)f(x1, x2...xn).. To check, f(tx, ty) = tf(x,y). 1. replace x with tx and y with tx in function. 2. factor out t.

doubling all arguments will double the value of f(x) aka constant returns to scale

doubling all arguments will leave f(x) unchanged

zero --> meaning changing all prices and income in same propotions will not effect quantity of goods demanded

-Px/Py ratio remains unchanged, utility max conditions also require MRS stays constant -Budget line will remain parallel and shift either in or out with new utility demand functions -Increases to Income increases quantity demanded, decreases to I decrease Q demanded

Consumer will adjust consumption of x,y to make MRS = to new price ratio. New point will be on original indifference curve but at point of tangency = to slope of new budget line. (page 146)

Consumer must move to new indifference curve because price changes also change "real" income -- price increase, lower income. price decrease, higher income.

Increases in income decreases amount of good consumed

Increases in income increase amount of good consumed

Outward shift in the demand curve caused by shift in income, another good price, or preferences

Movement along a given demand curve by change in price of x (px)

Also called compensated demand. Function of price and utility. Xc = Xc(Px, Py, U). Holds real income - Utility - constant. No income effect, only substitution.

Hicksian demand function can be found by differentiating expenditure function E(Px, Py, U)

1. Derive indirect utility function, V 2. V = U 3. Isolate I to get expenditure (I = ...)

1. Optimize x* and y* 2. Insert optimitzed x* and y* into original utility function

Marshallian, income is held constant. Hicksian, utility is held constant.

Derive substitution and income effects using concept of duality. dX/dPx with U constant (substitution effect) - XdX/dI (income effect). For substitution, differentiate compensated demand function and use V in place of U.

Measures change in quantity x demanded in response to change in good x own price. Ex,px = (dX(Px, Py, I)/dPx) * (Px/X)

Measures change in quantity x demanded in response to change in income. Ex, I = (dX(Px, Py, I)/dI) *(I/X)

Responsiveness of one variable to a change in another. Units cancel out. Ey,x = dy(x)/dx * x/y

Measures change in quantity x demanded in response to change in price of some other good (y). Ex, Py = (dX(Px, Py, I)/dPy) * (Py/X)

Elastic means variable affects other variable significantly. Inelastic means effect is small.

How much a consumer needs to be compensated for a price increase. Integral of hicksian demand evaluated at original utility and original price minus hicksian demand at original utility and new price.

How much a consumer is willing to pay for right to consume as much as they want of x at price px0. Area below compensated demand curve and above market price.

How much a consumer is willing to pay to see price return to old levels.

Applies envelope theorem to indirect utility function and allows us to get expenditure function from marshallian demand. x(Px, Py, I) = -dV/dPx / dV/dI

In mathematical terms, if you have a function that depends on a choice variable (like your path) and parameters (like the characteristics of the landscape), the envelope theorem states that the derivative of the maximum value of that function with respect to the parameters is equal to the derivative of the function with respect to those parameters, evaluated at the optimal choice of the variable.

any constrained maximization problem has an associated dual problem in constrained minimization

marginal benefit of x / marginal cost of x

any 2 points in the set can be joined by a line contained completely within set.

Slope of utility curve. -dy/dx. Utility must have diminishing MRS, with assumption that people become less willing to trade away y to get more x

Utility of X^alphaY^beta, typically alpha + beta = 1. Will be homogenous of degree one.

U(x,y) = min(alpha*x, beta*y) where y/x = alphaeta and is ratio of y consumed to x.

Homogeneity of degree 1: doubling all inputs doubles output. Perpendicular lines.

Linear, same MRS at all points. U(x,y) = alpha*x + beta*y. MRS is not diminishing. Parallel lines. Homogenous degree one.

constant elasticity of substitution. U(x,y) = [X^delta + y^delta]^1/delta where delta is less than or equal to 1 and not zero.

Max U (x,y) s.t. I = PxX + PyY ---> Dual: Min E(x,y) s.t. Ubar = U(x,y)
Indirect utility function U* = V(Px, Py, I) <-(inverses-> Expenditure function E*=E(Px,Py,Ubar)
--> V=V(Px,Py,I) when I = e, e=e(Px,Py,Ubar) when V=Ubar
Use Roy's Identity to get Marshallian demand from Indirect --> x(Px,py,I) = -dV/dPx / dV/dI
Use Shephard's lemma to get Hicksian demand from Expenditure --> Xc(Px,Py,U) =dE/dPx

1. Get expenditure function 2. partial derivative of E wrt px = hicksian demand for x 3. partial derivative of E wrt py = hicksian demand for y

Measure of additional output from using one more unit of input. dq/dk marginal physical product of capital, dq/dl marginal physical product of labor

RTS(l for k) = -dk/dl | q=qo = MPk/MPl

Constant returns to scale -- marginal product of any input depends only on ratio of capital to labor input

= percent change of k/l over percent change in RTS or d(k,l)/dRS * RTS / k/l

MR(q*)=MC(q*). Find derivatives of revenue and cost functions and set equal to solve for q.

Profit = Revenue (Price * Quantity) - Cost

1. Find SMC by taking derivative of C(q) 2. Set P = SMC and solve for q* --> this is quantity supplied

MC = MR = P

dC/dw and dC/dr of prices for L and K

Simple profit maximization problem. Derivative of PQ*q-CQ= profit, set equal to zero and solve for q*. Plug q* back into P.

Sub q* back into conditional input demand functions

Multiply firm supply curve by number of firms