Time Allowance: Two hours

Answer all questionsfrom Part A within 500 words altogether, and all questionsfrom Part B within 500 words altogether.

Questions in Part A carry 40 per cent of the total mark and questions in Part B carry 60 per cent of the total mark.

In cases where a student exceeds the word limit, the policy of the Economics Department is that the student’s answers up to the word limit will be the ones that count. All remaining answers will be ignored.

PART A

Answer all four questions from this section.

1) Roy Model

Let GU(2)S denote the variance of income distribution in the US, GM(2)X denote the variance of the income distribution in Mexico, and GMX ,US denote the covariance.

(a) Suppose GM(2)X  > GMX ,US  and GU(2)S > GMX ,US , what does the Roy model predict about the selection of immigrants from Mexico to the US and why?

(b) Now suppose GM(2)X > GMX ,US > GU(2)S and the cost of moving becomes lower. Would the average quality of Mexican immigrants increase or decrease and why? Use a diagram.

2) Local Labour Markets

Assume that workers are perfectly mobile as in the Roback model. Each city has an amenity value s where s can affect both workers’ utilities and firms’ costs. Each worker supplies a single unit of labour. Taking s in his location as given, the representative worker solves

maxU(x, 1c , s)

subject to

w = x + 1c r

where w is the wage, r is the rental payment, x is the national good, and 1c is the residential land consumption.

Let V denote the indirect utility. M firms produce output using a constant returns to scale production function X = f (1p ,N, s)

where 1p is land used in production, N is the total number of workers in the city, and s is amenity. Normalize the output price to one and denote the unit cost as C(w, r, s). Further, let L denote the total amount of land.

(a) What are the equilibrium conditions in this model?

(b) Suppose both firms and workers prefer high s locations, what would the Roback model predict in terms of relative wages and rents between city 1 and 2 where s1 < s2 ? Draw diagrams and give an example of such amenities.

3) Wage Impact of Immigration

(a) “Immigration will always lead to lower average wages of native workers.” Do you agree with the state- ment? Explain why or why not.

(b) Suppose you regress log wages on log shares of immigrants across different cities and find that a city with a higher immigrant share is associated with higher wage growth. Can we conclude that immigration leads to higher wages? Explain why or why not.

4) McCall Model of Job Search

Consider the continuous time McCall Model ofjob search. An unemployed worker with an infinite-time horizon wishes to maximise the present discounted value of wages. At a moment in time, the worker receives a wage offer w, where w is drawn from a known exogenous distribution F(w). If the worker accepts employment, she ceases search and receives the offered wage.  If the worker rejects employment, she continues searching for employment and receives a flow value of b. Let r denote the discount rate and a denote the job offer arrival rate.

(a) Express the Bellman equation associated with the value of employment N(w) at a wage of w for this worker and the Bellman Equation associated with the value of being unemployed U .

(b) Assume that the worker can search in two different labour markets.  In the first labour market, the distribution of wage offers F(w) is uniform, i.e. w ~ U [0, 4]. In the second labour market the distribution of wage offers, G(w) is also uniform but has w ~ U[1, 3]. Everything else is the same in both markets. Which market would the worker prefer to search and why?

PART B

Answer both questions from this section.

1) McCall Model of Job Search with an On-the-job Search

Consider the continuous time McCall Model with an on-the-job search.  An unemployed worker with an infinite-time horizon wishes to maximise the present discounted value of wages.  At a moment in time, the unemployed worker receives a wage offer w, where w is drawn from a known exogenous distribution F0(w), with an upper bound w¯ . Let r denote the discount rate and a0 denote the job offer arrival rate while unemployed. If the unemployed worker rejects a wage offer, she continues searching for employment and receives a flow value of b where w¯ > b.  If the unemployed worker accepts a wage offer, she receives the offered wage and continues searching on the job while working.

However, the job offer arrival rate and the distribution of wage offer while employed are different from those while unemployed.  Let a1  denote the job offer arrival rate for employed workers and assume that the wage offer for employed workers is always w¯ (i.e. the PDF of wage offer for the employed worker f1 (w) equals one if w = w¯ and zero otherwise).  Since the wage offer for the employed worker is always higher or equal to the current wage, the employed worker always accepts the on-the-job offer. Further, let’s assume that the employed worker who already receives w¯ also accepts the on-the-job offer.

(a) Express the Bellman equation associated with the value of employment N(w) at a wage of w for the worker and the Bellman equation associated with the value of being unemployed U . (Hint: N(w¯) = w¯/r.)

(b) Assume the worker follows a cut-off strategy of accepting wages above or equal to R and rejecting wages below R. Derive the reservation wage R.

(c) Derive the comparative statics of the reservation wage with respect to the job offer arrival rate while em-  ployed, i.e. find dR/da1 . Explain the intuition of the sign of dR/da1 . Hint: 匝max [w _ R, 0] =  R(w) (1 _ F(w))dw.

(d) Now suppose we have exogenous job destruction. Let 入 denote the exogenous job destruction rate which is common for all jobs. Provide an analytical expression of the steady state unemployment rate u. Compare the result with the case in which there is no on-the-job search and explain the difference intuitively.

2) CES Production Function

Assume that firms produce output using high and low skill labor

Yt = At [9htLht(p) +L1t(p)]1/p

where Let  with e e {h,1} is the aggregate labor input of workers of skill e in time t; aE  = 1/(1 _ p) is the elasticity of substitution between skill groups;9et is the relative productivity level of high skilled to low skilled labor.

Within each skill group, workers differ by age

Let = ╱aeYL Yte(n) +L Ote(n) ← 1/n

where a e {Y, O} denotes young and old, and one of the aea  is normalized to one; aA = 1/(1 _ n) is the elasticity of substitution between age groups. Within each age group, workers differ by immigration status

Leat = ╱Nea(6)t+βeat(M)M  tea(6)、1/6

where N denotes natives and M denotes immigrants, and βea(N)t  is normalized to one; aI = 1/(1 _ 6 ) is the elasticity of substitution between immigrants and natives. Assume perfect competition, the labor demand takes the following form

lnWea(s)t = lnAt + lnLt + ln9et + lnaea+ lnβe(s)at

+ ╱  _ ← lnLet + ╱  _ ← lnLeat _ lnSeat

where S e {N,M} is an immigrant status.

(a) Suppose we have an influx of high skill young immigrants. Show that the effects of immigration on the wage of low skill young workers and that on the wage of low skill old workers are the same.

(b) Suppose the order of nesting changes such that firms produce output using young and old labor at the top nest as following,

Yt = At │aYL t+ L t ┐Y(n)O(n) 1/n

Within each age group, workers differ by skill

Lat = ╱ 9htL ht+ L 1a(p)a(p) t ←1/p

Within each skill group, workers differ by immigration status

Laet = ╱N et+ βa(6)aet(M)M eta(6)、1/6

Would an influx of high skill young immigrants have the same effects on the wages of low skill young and low

skill old workers? Explain why or why not.

(c) Now suppose the elasticity of substitution between age groups and that between immigrants and natives

differ by skill groups. In other words, within each skill group, workers differ by age

Let = ╱aeYLe(n)Y(e)t + Le(n)O(e)t ←1/ne

where aAe = 1/(1 _ ne) is the elasticity of substitution between age groups, and nh  nl . Also, within each age group, workers differ by immigration status

Leat = ╱Nea(6e)t+ βeat(M)M  tea(6e)、1/6e

where aIe = 1/(1 _ 6e) is the elasticity of substitution between immigrants and natives, and 6h  6l . In this case, how would you estimate aE , aAe , aIe  and 9ht , aeY , βeat(M)?  Compare to the case in which the elasticity of substitution is restricted to be same for high and low skill labor, what additional difficulties would you face in the estimation?