Economics 5118: Assignment 1 on Simple Models

Economics 5118: Assignment 1 on Simple Models

Due Date: January 25, 2017

1.Have a look at Edward Prescott’s 2004 Nobel Prize lecture “The Transformation of Macroeconomic Policy and Research” (an e-version is on the course website). Based on Sections 1-4 of the paper answer the following questions in your own words:

(a) What were macroeconomic models like before the transformation that Prescott was part of?

(b) What does Prescott consider to be the important business cycle facts that must be modelled?

2. When setting up the Robinson Crusoe model we noted that there were three constraints:

The economy-wide budget constraint: y=c

The production function: y=F(L)

The time constraint: 1 = L + l

(variables are as defined in the notes, notice that I am using ‘l’ to represent leisure).

The notes “solved” Crusoe’s optimization problem two ways. The first substituted the constraints into the objective function while the second combined the constraints into a single constraint and maximized the Lagrangian. As mentioned in the course notes, the Lagrangian could have been set up as follows:

L(c,y,l,L,l1, l2, l3)= U(c,l) + l1[ F(L)-y ] +l2[ 1-L-l ] +l3[ y-c ]

This is then maximized writh respect to: c,y,l,L and the three Lagrange multipliers.

(a) Derive and report the first order conditions for this version of the problem.

(b) (i) In general what does a Lagrange multiplier tell you? (state this in words and keep this interpretation

in mind as you answer (ii) and (iii)).

(ii) What does the first order condition for c tell you? (don't just state the equation: interpret it by

outlining the economic reasoning suggested by the condition)

(iii) What does the first order condition for L suggest is needed for the optimal allocation of time

between L and l ? (state this in words giving the economic intuition -- don't just restate the math)

is the same as that for the unconstrained version of the problem on p.6 of the overheads.

3. In the decentralized version of the Robinson Crusoe model Crusoe the producer must decide how much labour (L) to hire and how much output (y) to produce. Let P be the price of output and W be the wage paid to labour – assume both of these are constants (determined by the “market”). Profits are then revenues from selling output (Py) minus labour costs (WL):

Profit = Py -WL

Say that Crusoe’s only constraint is the production function: y=ALa (A>0 and 0<a<1 are constants)

(a) State the Lagrangian for Crusoe-the-producer’s profit maximization problem. Interpret the lagrange multiplier.

(b) Now set the problem up as an unconstrained maximization problem by substituting the production function into your expression for profits. Solve for the profit maximizing level of output and the profit maximizing quantity of labour demanded

(i.e. L and y as functions of the exogenous variables – W, P and parameters A,a).

of changing exogenous variables and parameters on the optimal outcomes). Specifically:

(i) What is the effect of a rise in W on the solutions for L and y? Does this make sense? Explain.

(ii) What is the effect of a rise in P on the solutions?

(iii) How does a rise in A change the solution? Is this economically sensible?

(iv) Look at your solution in (a). What happens to labour demand if both W and P rise by 10%?

(d) If you plotted your solution for labour demand in (b) against W you would have the labour demand

curve. Your answers in (c) then tell you it's slope as well as how P and A would shift this labour

demand curve. Present your results from (b) and (c) for labour demand in a diagram (W on the

vertical axis and L on the horizontal). Be sure to roughly sketch the labour demand curve; show

the optimal solution given a specific value of W; and show the effect of a rise in W; and also show the effect of a rise in A or P.

4. Say that Crusoe's utility function is: U = ln(B)+(1-b)ln(c)+bln(l) (where c is consumption and l is leisure while B and b are constants with 0<b<1)

(a) What is Crusoe's marginal rate of substitution between consumption and leisure?

(b) If Crusoe's budget constraint is WL + P = P c and his time constraint is T=L+l (T is the total amount

of time -- we set T=1 in the notes). Derive expressions for Crusoe's optimal solutions for c and

l. (Assume P is a constant representing income from other sources)

(c) Give an expression for Crusoe's labour supply function. What is its slope with respect to the wage? (d) Show what happens to labour supply if: (i) b rises – does this make sense? (ii) P rises.

5. The two period utility function in the notes was: U(c1) + b U(c2) with 0<b<1. Replace it with a more specific form where the two "sub-utility" terms (the U(c) terms) are replaced with: U=cs so you have:

(c1)s + b∙(c2)s where 0<s<1

Let the budget constraint be of the same form as in the notes:

Period 1: c1+b1 = b0 + y1 Period 2: c2 = y2+(1+r)b1

where c1,c2 are consumption in periods 1 and 2, y1,y2 are income in periods 1 and 2 (exogenous), b0 are assets the person has at the start of period 1 (left over from period 0), b1 are assets available at the start of period 2.

(a) Give an expression for the marginal utility of period 2 consumption. Does this consumer have

diminishing marginal utility of period 2 consumption? Justify your answer.

(b) Set up the Lagrangian, derive the first order conditions and then use them to obtain the Euler equation

for this specific case.

(c) (i) The parameter b is a discount factor that measures how much less a person values period 2 utility compared to period 1 utility. b is often written instead as: b≡1/(1+r) with r>0 where r is the rate of time preference. In this formulation what does a higher r imply about how the person values future utility compared to present utility? Rewrite your Euler equation from (a) with 1/(1+r) replacing b.

(ii) According to your new Euler equation under what conditions would c1>c2? Explain why this

condition is economically sensible.

6. In the two-period version of the Robinson Crusoe model (the last model in the second set of notes) the second period budget constraint (after combining y2=F(K2) with y2=c2) was: c2=F(K2)

this assumes that Crusoe can’t consume any undepreciated capital left after production. In many models it is possible to consume your leftover capital (e.g. you could sell it to finance consumption).

(a) Write out the second period budget constraint if Crusoe was allowed to consume his

undepreciated period two capital in period 2. Explain the constraint.

(b) In the revised version of the model incorporating the new constraint obtain the first order conditions

and solve for the Euler equation.

(d) You now have seen three versions of the Euler equation.

- version from two-period models of the sort in question 5.

- version from the version where Crusoe can't consume his capital;

- version in (b) where he can consume his capital.

Compare and economically interpret the three versions. (Note especially the similar roles played

by (1+r), FK and FK+(1-d) ).