intermediate microeconomics problem set need calculus knowledge

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October 12, 2020
Discussion: Causes of Delinquent Behavior
October 12, 2020

intermediate microeconomics problem set need calculus knowledge

Problem Set 1

Due electronically by the due date to be announced in class and/or through Blackboard

Instructions: Answer all parts of the following four problems. To receive full credit all work must be shown. Illegible or sloppy work will receive a grade of zero. Late problem sets will not be accepted. Be sure to carefully follow any additional instructions given below.

1Supply-Demand analysis

Let the inverse market demand and supply curves for an arbitrary good be given by

and

𝑃(𝑄𝐷) = 𝛼 − 𝛽𝑄𝐷

𝑃(𝑄𝑆)= 𝛿 + 𝜓𝑄𝑆,

respectively, where 𝑄𝐷 (conversely, 𝑄𝑆) denotes quantity demanded (conversely, quantity supplied) and all lower-case Greek letters denote positive parameters such that 𝛼 > 𝛽𝑄𝐷 > 0 and

𝛼 > 𝛿

(a)Solve for the market equilibrium price (𝑃∗) and quantity (𝑄∗) and show this solution on a supply-demand graph.

𝛿 = 2, 𝜓 = 3. Determine the corresponding equilibrium values for the market price, quantity transacted, and own-price elasticity of market demand. Repeat this exercise now assuming that 𝛼 = 20 and all other estimated model parameters remain the same.How do your results comport with the comparative statics predictions derived in parts (b) and (d)?

2Marginal rates of substitution

Find the marginal rate of substitution (𝑀𝑅𝑆𝑥,𝑦) for each of the following utility functions: (a)𝒰(𝑥, 𝑦) = 3𝑥 + 𝑦

(b)𝒰(𝑥, 𝑦) = 𝑥2𝑦2

(c)𝒰(𝑥, 𝑦) = 𝑥𝑦

𝑥+𝑦

(d)𝒰(𝑥, 𝑦) = min{𝑥, 2𝑦}

(e)𝒰(𝑥, 𝑦) =ln 𝑥 + ln 𝑦

3Utility and consumption choices

Sam Spade is a detective. He consumes only two goods: rot gut whiskey (𝑤) and unfiltered cigarettes (𝑐). Suppose that his utility function is given by:

1

𝒰(𝑤, 𝑐) = 𝑤2(𝑐 + 1)3

vs. 2 bottles of whiskey and 6 packs of cigarettes.

𝒰(𝑤, 𝑐) =

𝑤6(𝑐 + 1)4

.

7

1 1

𝒰(𝑤, 𝑐, 𝑝) = 𝑤2(𝑐 + 1)3𝑝2 .

Calculate Sam’s new 𝑀𝑅𝑆𝑤,𝑐 and discuss the impact of poetry on this expression relative to the result found in part (a).

4Comparative statics and demand [harder]

For each of the following two-good utility functions:

  • 𝒰(𝑥 , 𝑥 ) = 𝑥 − (1 − 𝑥0.5 2
  • 𝒰(𝑥 , 𝑥

1221 )

) =𝑥1−1

for 𝑥

> 1, 0 < 𝑥 < 2

12(𝑥2−2)212

(do not concern yourself with second-order conditions when solving this problem)

  • Write down the consumers’ utility maximization problem and the associated Lagrangian function;
  • Derive the first-order (necessary) conditions (FOCs) with respect to 𝑥1, 𝑥2, and 𝜆
  • Compute and sign the comparative statics terms 𝑖, 𝑖, and
  • Compute the own-, cross-, and income-elasticities of demand corresponding to the demand functions calculated in part (iii) (i.e., 𝜂𝑖𝑖 , 𝜂𝑖𝑗, 𝜂𝑀).

(assume throughout that all endogenous variables are strictly positive);

1

Use the FOCs from part (ii) to solve explicitly for the consumer’s demand functions, 𝑥
∗

2

and 𝑥
∗ , as expressed in terms of all relevant model parameters;

𝜕𝑥∗ 𝜕𝑥∗

𝜕𝑥∗

𝑖

∀ 𝑖, 𝑗 = {1,2} , 𝑖 ≠ 𝑗.

𝜕𝑝𝑖 𝜕𝑝𝑗

𝜕𝑀

Provide an economic interpretation of your results and note any “unusual” characteristics of these demand functions.

 

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