Intermediate Microeconomics

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Intermediate Microeconomics

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Intermediate Microeconomics NYUAD, Spring 2018

Problem 1: Non-linear pricing

Suppose that there two goods X and Y , available in arbitrary nonnegative quantities (so the the consumption set is R2 +). The consumer

has preferences over consumption bundles that are strongly monotone,

strictly convex, and represented by the following (differentiable) utility

function:

u(x, y) = y + 2α√x,

where x is the quantity of good X, and y is the quantity of good Y ,

and α ≥ 0 is a utility parameter.

The consumer has strictly positive wealth w > 0. The price of good

Y is pY = 1. However, the price of good X depends on the quantity

of good X that the consumer purchases. In particular, pX(x) = √1x,

where x is the quantity of good X the consumer purchases. (Note that

pX(x) is the price per unit when the consumer purchases x units).

(1) In an appropriate diagram, illustrate (i) the indifference map for

the consumer, and (ii) the consumer’s budget set. Make sure you

label diagrams clearly, and include as part of your answer any calculations about the slopes and intercepts of the indifference curves

and the budget line.

(2) Formulate and solve the consumer’s utility maximization problem,

and find the demand and value functions. Your demand and value

functions should be functions of the parameters w and α.

Intermediate Microeconomics

(3) Suppose w = 10. In an appropriate diagram, illustrate the demand

function for good X, x(α|w = 10), and for good Y , y(α|w = 10).

Intermediate Microeconomics

Problem 2: Endowment effect

Suppose that there two goods X and Y , available in arbitrary nonnegative quantities (so the the consumption set is R2 +).

Instead of being endowed with a fixed amount of wealth w, the consumer has an initial endowment of the two goods, ¯ x > 0 and ¯ y > 0,

where ¯ x is the quantity of good X the consumer owns and ¯ y is the

quantity of good Y the consumer owns. The consumer has no additional wealth, but the consumer can buy and sell goods X and Y at

the fixed prices pX > 0 (for good X) and pY > 0 (for good Y ).

(1) In an appropriate diagram, illustrate the consumer’s budget set.

Make sure you label the diagrams clearly, and include as part of

your answer, the slope and intercepts of the budget line.

The consumer has preferences over consumption bundles that are strongly

monotone and strictly convex. However, the consumer’s preferences

depend on their initial endowment of the good. (Preferences with this

property are called endowment-dependent preferences and have been

studied widely by economists in recent years). In particular, the consumer’s preferences can be represented by the following “endowmentdependent” utility function:

u(x, y) = min 1 3 log x x¯ + 23 log y y¯ , 23 log x x¯ + 1 3 log yy¯ ,

where x is the quantity of good X, y is the quantity of good Y , and

(¯ x, y¯) >> 0 is the consumer’s initial endowment of the two goods.

Intermediate Microeconomics

(Note that the utility function u : R2 + → R is not differentiable, but

the function f(x, y|p, x, ¯ y¯) = p log x x¯ + (1–p) log yy¯ is differentiable

for any parameters (p, x, ¯ y¯) >> 0).

(2) In an appropriate diagram, illustrate the indifference map for the

consumer. Make sure you label the diagram clearly.

(3) Formulate and solve the consumer’s utility maximization problem,

and find the demand functions. Note that the demand functions

will depend on the parameters (pX, pY , x, ¯ y¯) >> 0.

(4) For what prices (pX, pY ) does the consumer optimally decide to (i)

consume their initial endowment of the goods, (ii) sell some units

of good Y to buy more units of good X, or (iii) sell some units of

good X to buy more units of good Y ?

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Intermediate Microeconomics

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