ECO 202 (Section: LEC5101) Test 2
1. (15 pts) Consider the consumption model discussed in Ch.16, the consumer has the following utility function:
U = ln(c) + βln(c
′
)
where c is the consumption in the current period, c
′
is the consumption in the future period and β is the discount factor.
The consumer has income of ✩15,000 today and ✩20,000 in the future. Also, the consumer has an initial assets (f) of ✩10,000. The consumer can save or borrow at r = 0.02. The discount factor (β) is equal to 0.95.
In this economy, the government has opted to implement a consump-tion tax. This tax will deduct τ c from current period income and τ
′
c
′ from future period income. Here, τ and τ
′
represent the rates of the consumption tax in current period and in future period.
(a) (2 pts) Setup the budget constraint. Shows the current period budget constraint, future period budget constraint and the life-time budget constraint (final answer should be using the number provided).
(b) (3 pts) Derive the Euler equation.
(c) (3 pts) Compute the optimal c and c
′
(d) (3 pts) The government decide to increase only τ in current period (but not τ
′
in future). How does it affect the optimal consumption level?
(e) (4 pts) The government has chosen to smooth out consumption tax over two periods. This involves increasing both τ and τ
′
in such a way that τdcd + τd
′
c
′
d = τece + τe
′
c
′
e
. Here, the subscript. d denotes the set of τ and c under the tax policy in part d, while the subscript. e indicates the variables under the policy in part e. How would this policy impact consumption in each period, and what are the reasons behind these changes?
2. (15 pts) In the Bathtub model, the labor force L¯ can exist in two states: employed (Et) or unemployed (Ut) during period t. The evolu-tion of unemployment over time is described by the equation:
∆Ut+1 = ¯sEt − fU¯
t
where ∆Ut+1 is the change in employment, ¯s is the job separation rate and f
¯ is the job-finding rate.
The economist has presented labor market data in the table below.
|
|
Separation Rate (%)
|
Finding
Rate (%)
|
labour Force (millions)
|
|
2010
|
0.25
|
4.5
|
153.9
|
|
2013
|
0.35
|
3.8
|
155.4
|
|
2015
|
0.30
|
4.1
|
157.1
|
|
2018
|
0.20
|
3.3
|
162.1
|
(a) (3 pts) Derive the steady state unemployment rate.
(b) (2 pts) What is the natural rate of unemployment in each of these years?
(c) (3 pts) What is the number of employment in each of these years?
(d) (3 pts) What would happen to the natural rate of unemployment if government implement a policy such that it would lower the separation rate. Explain using the natural rate of unemployment formula. State all the assumptions required to make a conclusion.
(e) (4 pts) The job-finding rate is typically represented as a function of both job search effort and the vacancy rate. While the vacancy data is considered reliable, measuring job search effort is more challenging. Aside from using official surveys and survey designed for this purpose, how do economists estimate the search effort?
3. (10 pts) Kappa Bistro is thinking about acquiring a second stove by borrowing $900 from the bank at a 10% interest rate. Given the current setup, adding one more stove increases the meal production per hour by 15, with each meal being sold for $24.
(a) (4 pts) Considering a depreciation rate of 10%, should Kappa Bistro invest in the new stove? Explain your reasoning.
(b) (2 pts) The data analyst estimated the production function to be Y = 24K1/2L
1/2
for the bistro. Calculate the marginal product of capital (MPK) in term of K and L.
(c) (4 pts) Now, Kappa Bistro is considering the purchase of an additional stove. Given the same prices and depreciation rate as discussed in part a, and assuming Kappa Bistro will only hire four unit of labor (L = 4) regardless of the amount of K, calculate the optimal number of stoves Kappa Bistro should acquire to maximize its profit, utilizing the previously calculated MPK from part b.