Problem Set 4, Vergote PDF

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This one is pretty correct yo...

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Linh Nguyen, Andrew Sirenko Intermediate Microeconomics Professor Wouter Vergote

Problem Set 4 Problem 1, #1 As price increases from $5 to $15, our budget curve goes from an x-intercept of 18 to 6 units of cheese (Because we can only buy 6 cheese max instead of 18), so we shift the optimal bundle from point A to B for a total effect of 4 less Gouda Change. We can split this into Substitution and income effects. The pure substitution effect results from the change in the relative price of good x (Gouda cheese) to good y => on the graph results in a change in the slope of the budget constraint (A -> C) SE = xA

−¿ xC = 6 −¿ 4 = 2 (lbs of cheese)

The pure income effect results from the change in the amount of John’s perceived wealth => on the graph results in a parallel shift of the budget constraint. (C -> B) IE = xC −¿ xB = 4 −¿ 2 = 2 (lbs of cheese)

Problem 1, #2 Compensating variation is the amount of compensation needed for John to remain at the same utility level after the price increase. It is signified by the monetary value of the parallel shift of the budget constraint from the line that crosses point B to the line that crosses point C.

John’s current budget: px(xinitial) = 18(5) = $90 => pyy* = 90 John’s desired budget after the price change to maintain the same utility level: px’(9) = 15(9) = 135 => CV = 135 - 90 = $45

Problem 1, #3 The consumer surplus will not be an exact measure of the monetary value he associates with the price change because there is an income effect, which yields the area under the demand curve above the initial price (consumer surplus) to be different than the compensation variation. However, we know the range of the Change in consumer surplus lies between CV and EV, Specifically (since Price increases CV>=∆CS >= EV), but we would need the demand curve to find the EXACT value.

Problem 2, #1 U (Z , C)=√ ❑ => MUZ =

1 1 , MUC = 2 √❑ 2 √❑

A quasilinear function can be represented by the form u(x1, x2 ) = v(x1) + x2, or that the marginal utility of one of the two goods is constant. In this case, both goods have non-constant marginal utility functions. Thus, the utility function is not quasi-linear.

Problem 2, #2

Problem 3, #1 U(Z, C) = C+2 √ ❑ A quasilinear function can be represented by the form u(x1, x2 ) = v(x1) + x2, or that the marginal utility of one of the two goods is constant. In this case, good C has a constant marginal utility. Thus, the utility function is quasilinear.

Problem 3, #2 max U(Z, C) = C+2 √ ❑ Z, C subjected to: C + wZ = wT max L(Z, C, λ) = C+2 √❑ Z, C, λ

−¿ λ(C + wZ

−¿ wT)

First-order Conditions: ∂L = ∂Z ∂L =1 ∂C

1 √❑

−¿

λw = 0 => λw =

1 √❑

(1)

−¿ λ = 0 => λ = 1(2) (the constraint is binding)

∂L = C + wZ −¿ wT= 0 => C + wZ = wT (3) ∂λ From (2), (3) =>

1 = w => Z = √❑

1 w2

(5)

Problem 3, #3 Labor income = Consumption of other goods = C From (1) : C = wT C = w(T C = wT

−¿ wZ −¿ −¿

1 2 ) w 1 w

=> Raymond’s labor income does depend on wage rate.

Problem 3, #4 1 Z= w2 Labor = L = 24 - Z = 24 ∂L = ∂w

2 w3

1 w2

=> decreasing as w increases for all w => L increases as w increases

=> Raymond works more as wage increases

Question 3, #5 Raymond has non-labor income equal to A > 0 => new budget constraint: C + wZ = wT + A => C = A + w(24 - Z) => new utility function: U(C, Z) =

C+2 √❑

= A + w(24 - Z) + 2 √ ❑

First-order conditions: ∂U = -w + ∂Z

1 =0 √❑

=> w =

1 √❑

=> Z =

1 => this yields the same function as question 3, #2/ w2

Since A is a constant, it does not have an effect on the utility maximization function. Thus, with a non-labor income, the answer to (2), (3) and (4) would remain the same. One could also note that because the function is quasi-linear, we would know that any constant on-labor income has no effect on these answers....