A 2 period dynamic general equilibrium model

Transcription

1 A 2 period dynamic general equilibrium model Suppose that there are H households who live two periods They are endowed with E 1 units of labor in period 1 and E 2 units of labor in period 2, which they supply inelastically Their intertemporal preferences are given by UC ( 1 + (1 β UC ( 2 In the economy there is also a large number of firms that have access to the following production function Y = AL+ BK + DF( K, L where A, B, and D are strictly positive constants, F ( is a constant-returns-to-scale production function, and F(0, L = F( K,0 = 0 The output Y produced by firms can be used for consumption or for investment All markets are perfectly competitive The economy starts out without capital Capital depreciates completely in production HH problem Max UC ( 1 + (1 β UC ( 2 subject to C1 + C2 /(1 + r = w1e1 + w2e2 /(1 + r Why isn t the capital income that households may earn (if they save in the second period part of the right-hand-side of the budget constraint? /( /(1 Solution characterized by U '( C = (1 β (1 + r U '( C and C + C + r = w E + w E + r Note that these are 2 equations in 2 unknowns (the 2 consumption levels for each household as households take wages and interest rates as given FIRM problem Max AL + BK + DF( K, L wl R K in each of the two periods ( t = 1, 2 t t t t t t t Market clearing conditions Labor market L d t = E in each of the two periods ( t = 1, 2 t Market for loans S = I where saving is the part of first period income that households do not consume S = H w E C and investment becomes the capital stock available for production in ( 1 1 the second period I = K2 1

2 No arbitrage between buying and expecting to rent capital goods R + r 2 1 If this condition was violated, there would be profits to me made from bank-financing the purchase of capital goods in the first period (at 1+ r and renting them out to firm to produce in the second period (at R 2 In class we also used that, in an equilibrium where there is investment and savings, R2 1+ r Notice that it this condition was violated, ie if R2 < 1+ r, then firms would never want to invest (ie bank-financing the purchase of capital goods in the first period because they expect capital goods to rent at a cheaper price ( R 2 than the cost of the loans need to finance their investment (1+ r Collecting all equations we get that, in an equilibrium with investment, the following 6 conditions must be satisfied: (i w1 = A (here it has been used that in the first period there is only labor for production; ie there is no capital FK ( 2, EH 2 (ii w2 = A+ D (notice that here the second-period labor market condition L2 has been used as the amount of labor used by firms has been set equal to labor supply F( K2, E2H (iii 1+ r = B+ D K (iv K2 = H( w1e1 C1 (v U '( C1 = (1 β (1 + r U '( C2 (vi C + C /(1 + r = w E + w E /(1 + r These are 6 conditions in 6 unknowns, wages in the 2 periods, consumption in the 2 periods, the interest rate, and capital in period 2 (which is also equal to investment and savings The model without decreasing returns and no labor supplied in period 2 D=0 and E 2 = 0 Notice that in this case households have NO labor income in the second period If the marginal utility of the first unit of consumption (in the second period is very high 2

3 which we usually assume to be the case this implies that households will have positive savings and there will therefore be positive investment in equilibrium As a result, we know that R2 = 1+ r which combined with the production function Y = AL+ BK, remember that we set D equal to zero in this example, implies B = 1+ r Therefore, the equations defining the equilibrium in this case become: (i w1 = A (ii 1+ r = B (iii U '( C1 = (1 β (1 + r U '( C2 (iv C + C /(1 + r = AE = 1 1 (v K H( AE C The only remaining thing is to solve the household maximization problem in (iii and (iv explicitly by assuming a utility functions For example, assume that utility is CES (constant elasticity of substitution, then: (iii 1/ σ 1/ σ 1 β r C2 1 2 /(1 1 ( C (1 (1 ( (iv C C r AE σ C 1 r C (1 (1 + σ /(1 + = 1 = + = ((1 β ( = [with (iii] ( β C C r r AE Question: Calculate HH savings Show that, when σ = 1, the HH saving is independent of the interest rate Show also that when σ > 1, then HH saving is increasing in interest rates and that when σ < 1, the HH savings is decreasing in interest rates Using 1+ r = B in the equations above yields σ AE1 AE1 ((1 β B C1 = and C σ σ 1 2 = 1 + (1 σ σ 1 β B 1 + (1 β B Question: What happens to investment, savings, and the capital stock in period 2 if the marginal product of capital in the second period increases? What happens to consumption in the 2 periods? What happens to investment, savings, and the capital stock in period 2 if the marginal product of labor in the first period increases? What happens to consumption in the 2 periods? What made this case relatively easy were two things: 1 The equilibrium interest rate was independent of the capital stock in the second period (because of the no-decreasing returns assumption As a result, it was easy to find the equilibrium interest rate in this example Compare Figure 1 with Figure 2 to see this (Figures are at the end of this document In Figure 1, which is the 3

4 general case, the equilibrium savings/investment will depend on the equilibrium interest rate and vice versa In Figure 2, the equilibrium interest rate is independent of equilibrium savings/investment (because, this is the technical terminology, investment is perfectly elastic at B-1, which means that firms are indifferent regarding the investment volume when the interest rate is equal to B- 1 Consider also Figure 3, which corresponds to the Solow model, where the equilibrium is simple to find because savings is independent of the interest rate 2 Households do not earn future labor income As a result, savings did not depend on future labor income and because future wages depend on future capital stocks the future capital stock The general case with a CES utility function and a Cobb-Douglas function F( CES utility and FKL (, 1 = K L, which implies that 1 Y = AL+ BK + DK L The HH problem is solved as above to yield: C AE + w E = and C 1 + (1 β (1 + r σ σ 1 ( AE + w E ((1 β B σ = σ σ (1 β (1 + r The future wage and the interest rate now depend on the future capital stock: FK ( 2, EH 2 w2 = A+ D = A+ (1 D( K2 / E2H L 1 + r = B+ D ( K / E H The equilibrium savings=investment requires that ( K = H AE C Substituting what we got for consumption K H AE AE + w E = 1 σ σ (1 β (1 + r Substituting what we got for future factor prices AE + ( A + (1 D( K / E H E K = H AE 1 + (1 β ( B+ DD( K2 / E2H σ 1 σ 1 4

5 Which can be written as K 1 HA( E1 + E2 + (1 D( E2H K2 2 HAE1 σ 1 σ 1 1 (1 β ( B DD( K2 / E2H = + + There is only 1 unknown in this equation, the future capital stock So we have 1 equation (very complicated though in 1 unknown This equation is hard to solve in general But there are special cases that are easier In particular the case were σ = 1, which yields that savings is independent of the interest rate (a key simplification also present in the Solow model In this case, we get that K HA( E E (1 D( E H K = HAE 1 + (1 β 1 DEH 2 HAE1 + E = 1 (1 ( ( K K HAE 1 + (1 β 1 + (1 β The left-hand side of this equation is increasing in the capital stock The right-hand side is a constant Assume that parameters are such that the right-hand side constant is positive In this case there is a unique interaction point with positive capital between the two sides, which defines the equilibrium capital stock What does the assumption that the right-hand side of the equation above is positive amount to? Basically, it amounts to the assumption that HH want to save Why might they not want to save? Because in this model they always receive some labor income in the second period (even if there is no capital! If this income was very high, HH might want to borrow against future income rather than save Question: What happens to consumption and savings if D falls (D is the productivity of future capital and labor What happens to investment? And what happens to interest rates and the future wage? 5

6 Figure 1 (Real Interest Rate INVESTMENT SAVINGS Investment, Savings The general case Investment and savings are functions of interest rates (As an aside: Remember that savings may in theory decrease with interest rates Figure 2 (Real Interest Rate SAVINGS B-1 INVESTMENT Investment, Savings B is the MPK in the second period The investment demand curve looks like this for the following reasons If 1+r>B (ie r>b-1, firms do not want to invest because the expected rental price of capital in the second period is so low that they prefer to rent tomorrow rather than invest (buy today Hence, investment is zero If 1+r<B, investment is infinity, as firms expect to earn the difference 6

7 between the rental price of capital and the user cost of capital on each unit of investment When 1+r=B, firms are indifferent regarding the investment volume as the user cost of capita is the same as the expected rental price of capital (for them it is the same to buy now ie invest or rent later Figure 3 (Real Interest Rate INVESTMENT SAVINGS Investment, Savings This is what the situation is like in the Solow model (and why the Solow model is relatively easy Also in the model described here when the elasticity of intertemporal substitution is equal to 1 7