# intermediate microeconomics problem set need calculus knowledge

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

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

ð‘ƒ(ð‘„ð·) = ð›¼ âˆ’ ð›½ð‘„ð·

ð‘ƒ(ð‘„ð‘†)= ð›¿ + ðœ“ð‘„ð‘†,

respectively, where ð‘„ð· (conversely, ð‘„ð‘†) denotes quantity demanded (conversely, quantity supplied) and all lower-case Greek letters denote positive parameters such that ð›¼ > ð›½ð‘„ð· > 0 and

ð›¼ > ð›¿

(a)Solve for the market equilibrium price (ð‘ƒâˆ—) and quantity (ð‘„âˆ—) and show this solution on a supply-demand graph.

ð›¿ = 2, ðœ“ = 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 ð›¼ = 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 (ð‘€ð‘…ð‘†ð‘¥,ð‘¦) for each of the following utility functions: (a)ð’°(ð‘¥, ð‘¦) = 3ð‘¥ + ð‘¦

(b)ð’°(ð‘¥, ð‘¦) = ð‘¥2ð‘¦2

(c)ð’°(ð‘¥, ð‘¦) = ð‘¥ð‘¦

ð‘¥+ð‘¦

(d)ð’°(ð‘¥, ð‘¦) = min{ð‘¥, 2ð‘¦}

(e)ð’°(ð‘¥, ð‘¦) =ln ð‘¥ + ln ð‘¦

# 3Utility and consumption choices

Sam Spade is a detective. He consumes only two goods: rot gut whiskey (ð‘¤) and unfiltered cigarettes (ð‘). Suppose that his utility function is given by:

1

ð’°(ð‘¤, ð‘) = ð‘¤2(ð‘ + 1)3

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

ð’°(ð‘¤, ð‘) =

ð‘¤6(ð‘ + 1)4

.

7

1 1

ð’°(ð‘¤, ð‘, ð‘) = ð‘¤2(ð‘ + 1)3ð‘2 .

Calculate Samâ€ s new ð‘€ð‘…ð‘†ð‘¤,ð‘ 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:

• ð’°(ð‘¥ , ð‘¥ ) = ð‘¥ âˆ’ (1 âˆ’ ð‘¥0.5 2
• ð’°(ð‘¥ , ð‘¥

1221 )

) =ð‘¥1âˆ’1

for ð‘¥

> 1, 0 < ð‘¥ < 2

12(ð‘¥2âˆ’2)212

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

• Write down the consumersâ€ utility maximization problem and the associated Lagrangian function;
• Derive the first-order (necessary) conditions (FOCs) with respect to ð‘¥1, ð‘¥2, and ðœ†
• Compute and sign the comparative statics terms ð‘–, ð‘–, and
• Compute the own-, cross-, and income-elasticities of demand corresponding to the demand functions calculated in part (iii) (i.e., ðœ‚ð‘–ð‘– , ðœ‚ð‘–ð‘—, ðœ‚ð‘€).

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

 1

Use the FOCs from part (ii) to solve explicitly for the consumerâ€ s demand functions, ð‘¥
âˆ—

 2

and ð‘¥
âˆ— , as expressed in terms of all relevant model parameters;

ðœ•ð‘¥âˆ— ðœ•ð‘¥âˆ—

ðœ•ð‘¥âˆ—

ð‘–

âˆ€ ð‘–, ð‘— = {1,2} , ð‘– â‰  ð‘—.

ðœ•ð‘ð‘– ðœ•ð‘ð‘—

ðœ•ð‘€

Provide an economic interpretation of your results and note any â€œunusualâ€ characteristics of these demand functions.

## “Looking for a Similar Assignment? Get Expert Help at an Amazing Discount!” The post intermediate microeconomics problem set need calculus knowledge appeared first on Graduate Paper Help.

## "Is this question part of your assignment? We Can Help!"  