Set the new labour supply equation equal to labour demand. Thus:

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1 Anser key for Assignment. Question : The demand for and supply of labour (35 points) Part a) From the production function Y AK α ln(n), first derive the marginal product of labour (MPN) and set it equal to, the real age rate. From there re- organise the expression to get the demand for labour (N d ) on the left hand side. Thus: dy AK α d ln(n) AK α dn N dy dn MPN AKα N N d AKα The elasticity of labour demand is: dn d d AKα N d AK α Given K 0,000; A 6.4; α 0.5; and N s, first set N d N s and solve for. Thus: N s N d, then: AKα ( ) / AKα 8 Given the age rate of 8, use the labour demand relationship to find the N d : N d AKα 6.4 (0,000) From the production function, Y is: Y AK α ln(n) ( 6.4 (0,000) 0.5 )ln(8) Part b) Set the ne labour supply equation equal to labour demand. Thus: N s N d AK α Collecting the s: 3 AK α (AK α ) /3 4 With 4, labour demand is no:

2 N d AK α The labour supply curve has become more elastic. Before a age rate of 4 ould yield a supply of 4, no the supply is 8. In the standard diagram, the supply curve has become flatter. s N s N / Wage rate () Labour demand Labour N Part c) Total employment as 8 and a 5% increase ould be 0. Based on the labour supply equation the age rate ould have to be: N s 0 A age rate of 0 hoever ould imply a labour demand of N d AK α , leaving a gap of

3 Part d) Sub part a) To find the level of TFP that ould make the demand for labour equal to the government s target of 0 e need to re-arrange the demand for labour in terms of A. N d AKα A N d 0 0 K α 0, A rise in A to 0 ould shift the labour demand curve to the right by enough to set demand equal to the ne target of 0. Sub part b) We need to solve the labour demand equation for K that ould be consistent ith a demand for labour of 0 and a age rate of 0. N d AKα K N d A α Question : The consumption function (30 points) Part a) Start by noting that in equilibrium that the slope of the utility curve is equal to that of the budget constraint line. Thus: c c (+ r), here the minus signs have been cancelled. Rerite the equation in terms of c. c (+ r)c Next rite out the inter- temporal budget constraint: c + c + r y + a + y + r Use the above relationship to eliminate c in the inter- temporal budget constraint: c + c + c y + a + y + r c + y + a + y + r Part b) 3

4 Start by solving for c. Thus: c + y + a + y r.05 Next use the equilibrium condition beteen the slope of the utility curve and the budget constraint to find c given c: c (+ r)c (+ 0.05)00 05 What is unique about the levels of present and future consumption is that c y + a and c y e are at the no- lending and no- borroing point. Since consumers are consuming all of their income and assets in the present, their savings are zero. Part c) We can use the consumption function to calculate the effect of the increase in a to 5. Thus: c + y + a + y r.05 Given c then: c (+ r)c (+ 0.05) Consumers carry into the future y + a c.5 to support future consumption. The ealth effect is operative here. Part d) Starting again ith the initial conditions except that no r 0% changes the consumption function to: y + a + y r.0 Given c then: c (+ r)c (+ 0.0) Savings is y +a c hich is used to support higher future consumption. Compared ith part b) present consumption is no loer but future consumption is higher as the rise in the interest rate caused consumer to substitute present for 4

5 future consumption. Because e are at the no- lending no- borroing point then the pure substitution effect is operating. Part e) The a term no is a lump- sum transfer from the government hich has to be repaid in the future. The consumption function no becomes: c + y + a + y a(+ r) 05 0(.05) r ( ) 95 As can be seen, the effect of the transfer ashes out. Future consumption is: c (+ r)c (+ 0.05) The Ricardian Equivalence effect is operating here. Question 3: Equilibrium in the goods market (35 points) Part a) Start ith the MPK relationship and set it equal to the user cost of capital: MPK f 75 0K t + r + d P τ k then K t r + d P 0 τ k With r6%, d9%, τ5% and Pk5 e get; K t If the price of capital fell to 75 then: K t If the marginal tax rate rises to 50% then; 5

6 K t Part b) Begin ith the equation for MPK f and substitute in the values for all the variables except r. Thus: K t r + d P 0 τ k r r 0.5 Next use the gross investment identity from the text (equation ) I t K t + ( d)k t With d0.09 and Kt, and substituting in the relationship for Kt+, e get: I t r r When r 6%, It Part c) The definition of national saving is: S Y C G (3+ G) + ( 0.8)Y +00r Setting saving equal to investment and solving for r e get: S (3+ G) + ( 0.8)Y +00r I r I + (3+ G) ( 0.8)Y 00 With Y 0, G.9 and using the value of investment found in part b, then the rate of interest ould be: S I (3+ G) + ( 0.8)Y +00r r r or 6% ( G) ( 0.8)Y Part d) If G falls from.9 to then the rate of interest ould decline to: r ( G( )) ( 0.8)Y

7 The no loer rate of interest increases investment to: I r Using the saving function e get: S (3+ G( )) + ( 0.8)Y +00r The drop in G has croded in investment There are to effects operating on saving: ) the loer rate of interest increases consumption and thus loers saving but ) this is more than offset by the fall in G. Part e) The condition for goods market equilibrium in an open economy becomes: S I + NX (3 + G) + ( 0.8)Y +00r r + NX Saving must no be sufficient to finance both investment and net exports. To find NX hen the orld rate of interest is 7% e need to re- rite the above in term of NX. NX S I (3 + G ) + ( 0.8)Y + ( )r.6 A decline in G to (a drop of 0.9) results in an equal increase in NX. Thus: NX S I (3+ G( ) ) + ( 0.8)Y + ( )r.086 In the case of a small open economy, hen the orld interest rate cannot change, the drop in G results in an equal increase in NX. 7

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