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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.

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›¿ + ðœ“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, ðœ“ = 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)?

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‘¦

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).

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 ðœ†

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., ðœ‚ð‘–ð‘– , ðœ‚ð‘–ð‘—, ðœ‚ð‘€).

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

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

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

ðœ•ð‘¥âˆ— ðœ•ð‘¥âˆ—

ðœ•ð‘¥âˆ—

ð‘–

âˆ€ ð‘–, ð‘— = {1,2} , ð‘– â‰ ð‘—.

ðœ•ð‘ð‘– ðœ•ð‘ð‘—

ðœ•ð‘€

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

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October 22, 2020