What quantity will maximize industry revenue pq derive the

Question

Assignment

Imagine an agricultural market (oranges) in which we have year to year data on price and output. We know that demand and supply are linear:

DEMAND       Q^d = α₀ + α₁P,             α₀> 0, α₁ < 0                      (1)
SUPPLY        Q^s = β₀ + β₁P,              β₀, β₁> 0, α₀ - β₀ > 0          (2)
EQUIL          Q^d (P*)= Q^s (P*) = Q*                                             (3)

(A) Derive the equilibrium price and quantity functions

(B) The econometricians tell us that: α0 = 100, α1 = -4 , α0 = 0, α1 = R, with R varying from year to year with weather conditions. Derive the equilibrium price and quantity functions in this special case.

(C) Prove that, as R varies, the equilibrium values of P and Q trace out ('Identify") the demand curve (not the supply curve).

A consumer faces a standard 2-good purchasing environment, say X and Y, with fixed prices (Px and PY) and fixed income I. She has the (twice differentiable) utility function

U = u(X,Y)

with the usual properties that "more is better" and the marginal rate of substitution diminishes in X.

(A) Illustrate graphically the individual's optimal choice of X and Y in this two good world. Explain briefly what you are doing. Be sure to define all concepts used.

(B) Using Lagrangian techniques, derive the demand function to extent possible. Explain the parallels between the first order conditions and the graphical method of deriving the optimal choice of X and Y

(C) In words, what does our analysis "tell us," if anything, about the consumer's decision to allocate income between X and Y? That is, how is predicted behavior restricted?

A consumer faces a standard 2-good purchasing environment, say X and Y, with fixed prices (Px and PY) and fixed income I. She has the utility function

U = XY + X

(A) Formally demonstrate that X is a good, not a bad (pollution).

(B) Using Lagrangian techniques, derive the demand functions for X and Y.

(C) Formally demonstrate that MRS is decreasing in X as is necessary for an interior solution for this maximization problem.

(D) Is Y a normal good or an inferior good? Prove your answer formally

(E) What is the elasticity of X with respect to its own price P_x? Define and then derive

(F) What is the cross-price elasticity of X with respect to P_Y? Define and then derive.

Imagine two rural labor markets (A and B), which in isolation have the following structures:

Market A

Demand:         L_A^d = α₀^A + α₁^AW,       α₀^A  > 0, α₁^A < 0                      (1)
Supply            L_A^S = β₀^A + β₁^AW,       β₁^A > 0                                    (2)

Market B

Demand:         L_B^d = α₀^B + α₁^BW,       α₀^B  > 0, α₁^B < 0                      (3)
Supply            L_B^S = β₀^B + β₁^BW,       β₁^B > 0                                    (4)

Imagine that, courtesy of the government, a high speed train ("the train to nowhere") is built to link the two rural markets, dropping transport costs to zero between the two localities.

A. Derive the formula for equilibrium wages in the single labor market created by this innovation.

B. Econometricians have been active, and, using data before the change, they estimate the parameters of the two markets:

α₀^A = 100, α₁^A = -2, β₀^A = 100, β₁^A = 3
α₀^B = 50, α₁^B = -3, β₀^B = 50, β₁^B = 2

What is the impact of the single market on wages in the two localities?

C. How many residents of B end up working in A?

Assume an industry has a linear demand function of the form:

P = α₀ + α₁Q, with α₀ > 0, α₁ < 0

(A) What quantity will maximize industry revenue (P*Q)? Show your work.

(B) Derive the price elasticity at that revenue maximizing quantity? Show your work.

Consider the widow in Moscow in 1960 who has only a single good (food F) from which to "choose" and an income of I and a price for the good of PF She has a utility function, U = u(F), that has the usual property that more is better.

(A) Derive her demand function for F. Explain briefly what you are doing and why.

(B) What is her income elasticity of demand? Show your work.

Times are a little brighter and two goods become available to the widow, food (F) and other goods (Y). She has a utility function of the form

U = F + FY

which she would like to maximize subject to her budget constraint:

I = P_FF + P_YY

(A) Derive her demand functions for X and Y. USE LAGRANGIAN TECHNIQUES TO OPTIMIZE.

(B) Prove that her marginal rate of substitution is convex to the origin ("bubbles in") so that these calculations do represent a maximum.

(C) What is her income elasticity of demand?

(D) What is her price elasticity of demand?