ECON402 Study Guide - Quiz Guide: Autoregressive Model, Independent And Identically Distributed Random Variables, Utility

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chocolatekangaroo714

20 May 2018

School

University of Waterloo

Department

Economics

Course

ECON402

Professor

Kate

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University of Waterloo

Department of Economics

ECON 402 S14

Assignment # 4

Due: July 29, 11:59pm

Explain/Derive/Show work for full marks.

1. Consider an economy consisting of a constant population of infinitely lived individuals. The

representative individual maximizes the expected value of:

∑

t=0

∞u(Ct)

(1+ρ)t, ρ>0

The instantaneous utility function,

u(Ct)=Ct−θCt

, where

Assume that C is always in the rage where u'(C) is positive.

Output is linear in capital, plus an additive disturbance: Yt=AKt + et. There is no depreciation; thus Kt+1=

Kt + Yt – Ct. and the interest rate is A; assume A=

. Finally, the disturbance follows a first-order

autoregressive process:

et=ϕet−1+εt

, where

0<ϕ<1

, and where the

εt

's are mean-zero, i.i.d. shocks.

a) Find the first-order condition (Euler) relating Ct and expectations of Ct+1. (4)

b) Suppose that most of output is consumed, i.e. you are given that

Ct=A Kt+A

1+A−ϕ et

, solve for

Kt+1 as a function of Kt and et. (3)

c) Explain & show the effects of a one-time positive shock to

εt

on the paths of Y, K & C?(6)

2. Answer each of the following questions using the search&matching model(10.6):

a) Explain & graph the effect on E* of a fall in the job break up rate. (3)

b) Explain & graph the effect on E* of a rise in the cost of maintaining a job, c. (2)

c) Explain & graph the effect on E* of a rise in effectiveness of matching, K. (3)

d) If φ=0.5, but the arrival rate of jobs offers is lower than the arrival rate of workers, explain who will get

a greater share of the production value of the match, y.(2)

3. Consider 3 types of search model: wage posting with homogeneous workers and no on the job search,

wage posting with heterogeneous workers and no on the job search, and wage posting with homogeneous

workers as well as on the job search.

a) Explain which model would be most appropriate to use, if you wish to investigate wage differentials

across industry. (2)

b) Explain which model would be most appropriate to use if you wished to explain wage variance within

industry. (2)

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Document Summary

Explain/derive/show work for full marks: consider an economy consisting of a constant population of infinitely lived individuals. The representative individual maximizes the expected value of: U (ct ) (1+ )t , >0 t = 0. Assume that c is always in the rage where u"(c) is positive. Output is linear in capital, plus an additive disturbance: yt=akt + et. Kt + yt ct. and the interest rate is a; assume a= . Finally, the disturbance follows a first-order autoregressive process: where >0. , where 0< <1 , and where the t. "s are mean-zero, i. i. d. shocks: find the first-order condition (euler) relating ct and expectations of ct+1. (4, suppose that most of output is consumed, i. e. you are given that ct= a k t+

Related Questions

Let us have an economy that lives only for 2 periods. The representative household has a utility function that values consumption goods for any period as follows

u(c) = ln(c).

The valuation throughout both periods can be represented as a total utility function that discounts future utility with a Î² discount parameter as follows,

U(c1, c2) = u(c1) + Î²u(c2) = ln(c1) + Î² ln(c2),

where ct represent consumption good in period t for every period t = 1, 2. The budget constraint for this person in every period is expressed as

(1+Ïtc)ct + xt =(1âÏtI)(wtnt +rtkt), t=1,2.

where xt represent investment in period t, wt the wages earned by hours worked, rt the return of capital in period t, nt the hours worked supplied by the household in every period t, kt the stock of capital that the household has in period t, Ïtc the tax rate imposed by the government in consumption goods in every period t, and ÏtI the tax rate imposed by the government to total household income in every period t. In every period, the total amount of hours that the representative household is of H Ì = 1. Investment in every period can be defined by the following law of motion of capital

kt+1 = xt +(1âÎ´)kt, t=1,2.

Acknowledge that k3 = 0 because there is no period 3 in this economy. The household starts with a given amount of capital k Ì1 > 0. The representative firm produces good by the following production function in every period t,

Y_t =z_tK_tÎ±N_t1âÎ±, t=1,2.

There is a government that covers government expenditures in period 1 and period 2 by taxing capital rent and labor. Hence, their budget can be expressed as

Ïtcct +ÏtI(wtnt +rtkt)=gt, t=1,2.

a) Before writing any competitive equilibrium definition or the Social Planner Problem, can we modify the budget constraint in â¨order to get rid a variable in every period? Explain.

b) Define the Social Planner Problem.

c) Solve the Social Planner Problem. This is, find the Euler equation in terms of exogenous variables and the only endogenous variable k2.

d) Define the tax distorted competitive equilibrium of this economy.

e) Solve the representative household problem.

f) Solve the representative firm problem in period 1 and 2.

g) Find the optimal Euler equation in terms of exogenous variables and the only endogenous variable k2.

h) Imagine that the government can only implement income taxes. This is Ïtc = 0 for every period t = 1, 2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem?

i) Imagine that the government can only implement consumption taxes. This is ÏtI = 0 for every period t = 1,2. Can the government implement a tax plan that achieves the Pareto allocation reached by the Social Planner Problem? â¨

Refer to the example of the Cobb-Douglas production function discussed in class, where the GDP (Y), Total Employment (X2 ), and Stock of Fixed Capital (X3 ), are measured for the output, labor input and capital input, respectively. The following model is fitted and the R output is below. The data are yearly data from 1955 to 1974.

lnY = B1 +B2lnX2 + B3lnX3 + u

a. Interpret the coefficient estimate of B3 specific to this problem.

b. Find the rate of change of Y with respect to the change of X2 , at Y=218506.3 and X2=11152.6, and X3 = 340072.45.

c. Find the estimate of the return to scale parameter. (3 pts.)

d. Is there strong evidence that the GDP is inelastic relative to the stocks of fixed capital? Justify your answer at a=0.05.

e. Test if there is a positive (first order) autocorrelation in the error term using a=0.05 . Make sure to state the null and alternative hypotheses, to provide the values of the test statistic and the critical value(s), and to draw a conclusion. (6 pts.)

Regression output for Problem 3.

> fit<-lm(log(y) ~ log(x2) + log(x3))

> summary(fit)

Call:

lm(formula = log(y) ~ log(x2) + log(x3))

Residuals:

Min 1Q Median 3Q Max

-0.057142 -0.016250 -0.005793 0.023512 0.038739

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -1.65233 0.60619 -2.726 0.0144 *

log(x2) 0.33972 0.18569 1.830 0.0849 .

log(x3) 0.84600 0.09335 9.063 6.42e-08 ***

Signif. codes: 0 0.001 0.01 0.05 0.1 1

Residual standard error: 0.02829 on 17 degrees of freedom

Multiple R-squared: 0.9951, Adjusted R-squared: 0.9945

F-statistic: 1719 on 2 and 17 DF, p-value: < 2.2e-16

Durbin-Watson Statistic = 0.425618

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ECON402 Study Guide - Quiz Guide: Autoregressive Model, Independent And Identically Distributed Random Variables, Utility

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