Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET 1

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1 Macroeconomics I, UPF Professor Antonio Ciccone SOLUTIONS PROBLEM SET (from Romer Advanced Macroeconomics Chapter 1) Basic properties of growth rates which will be used over and over again. Use the fact that the growth rate of a variable equals the time derivative of its log to show: (a) The growth rate of the product of two variables equals the sum of their growth rates. Z (t) X (t) Y (t) Ż(t) d ln Z(t) d ln[x(t) Y (t)] d ln X(t) d ln Y (t) Z(t) + Ẋ(t) X(t) + Ẏ (t) Y (t) (b) The growth rate of the ratio of two variables equals the difference of their growth rates. Z (t) X(t) Y (t) Ż(t) d ln Z(t) Z(t) X(t) d[ln Y (t) ] d[ln X(t)+( 1) ln Y (t)] d[ln X(t)] d[ln Y (t)] Ẋ(t) X(t) Ẏ (t) Y (t) 1.2 (from Romer Advanced Macroeconomics Chapter 1) Implications of change in the growth rate for the evolution of a variable. Suppose that the growth rate of some variable X is constant and equal to a 0 from time 0 to time t 1 ; drops to 0 at time t 1 ; rises gradually from 0 to a from time t 1 to t 2 ; and its constant and equal to a after time t 2. (a) Sketch a graph of the growth rate of X as a function of time. 1

2 Figure 1 (Note that the graph between t 1 and t 2 may as well be a straight line or convex shaped) (b) Sketch a graph of ln X as a function of time. Figure Properties of constant returns functions. The function F (K, L) is homogenous of degree d if F (λk, λl) λ d F (K, L) for all λ 0. If d 1, the function is said to be linear homogenous (or, alternatively, subject to constant returns to scale if it is a production function). (a) (TBG) If production factors are paid their margial products, all output is paid out to production factors. Show that if F (K, L) linear homogeneous, then F (K, L) K + L (this result is called Euler s theorem). What does this imply about the profits of a perfectly competitive firm producing subject to constant returns to scale? F (λk, λl) λ d F (K, L) ; d 1 K + K + [L F,1)] L L [L F L,1)] K + [L F L,1)] L L F L,1) L L F 1 L, 1) 1 L K + [ L F 1 L, 1) ( K L 2 ) + F L, 1)] L F 1 L, 1) K + F 1 L, 1) ( K) + F L, 1) L K [F 1 L, 1) F 1 L, 1)] + L F L, 1) F (K, L) A firm with a constant returns to scale production function acting in a perfectly competitive environment has zero economic (or pure) profit, i.e. every input is paid exactly its marginal contribution to the output, implying that income is exhausted entirely. There are no profits from speculation, chance or the exploration of for example market inequalities. One might refer to this as fair factor prices. 2

3 (b) Constant returns means that all measures of productivity are independent of the scale of production. Show that if F (K, L) is linear homogenous, then and are homogenous of degree 0. Suppose now that a firm producing subject to constant returns to scale doubles its capital and its labor. What happens to th marginal product of capital and the marginal product of labor as a result? F 1 (K, L) L F,1) ( L LF K 1 L, 1) 1 L F 1 L, 1) F 1 (λk, λl) F 1 ( λk λl, 1) F 1 L, 1) λ 0 F 1 (K, L) F 2 (K, L) K F(1, L K ) ( ) KF 2 1, L 1 K K F 2 ( ) 1, L K ( ) ( ) F 2 (λk, λl) F 2 1, λl λk F2 1, L K λ 0 F 2 (K, L) alternatively: F (K+,L) F (K,L) lim (K+ ) K F (λk,λl) F (λ(k+ ),λl) F (λk,λl) lim λ(k+ ) λk λf (K+,L) λf (K,L) F (K+,L) F (K,L) lim λ((k+ ) K) lim (K+ ) K F (K,L+ ) F (K,L) lim (L+ ) L F (λk,λl) F (λk,λ(l+ )) F (λk,λl) lim λ(l+ ) λl λf (K,L+ ) λf (K,L) F (K,L+ ) F (K,L) lim λ((l+ ) L) lim (L+ ) L Due to these results, a firm that chooses λ 2 does not influence its marginal product of capital and its marginal product of labor at all. There is no reason that an additional employee equiped with the same amount of capital as all other employees would be less or more productive than any earlier hired employee. 3

4 1.4 Suppose that the production function F (K, L) is subject to constant returns to scale to capital and labor. Suppose also that the production function is subject to decreasing returns to capital 2 F (K,L) MP K 2 0, and that the production function is continuosly differentiable (an important property of these functions is explained in (a) below). (a) Decreasing returns to capital imply that marginal productivity of capital falls as the firm employs more capital per worker (by definition) and that marginal productivity of labor increases (less obvious). Show that this implies that the marginal product of labor is 2 F (K,L) MP L an increasing function of the stock of capital: 0. (Hints: First show [ that the fact that 0 is homogenous of degree 0 implies that δ 2 F (K, L) /δk 2] K + [ δ 2 F (K, L) /δkδl ] L 0 Then use the fact that for continuosly differentiable functions 2 F (K,L) 2 F (K,L) ). PART I 2 F (K,L) 2 K + 2 F (K,L) L [ ] K + [ ] L F1(K,L) K + F1(K,L) L F1,1) L K + F1,1) L L ( F K ) 11 L 1 L K + F ) ( ) 11 L K L L PART II MP L 2 F (K,L) 2 F (K,L) from PART I K L 2 F (K,L) 2 ( K L 0 ) MP K 0 (b) Implications of (a) above for labor demand. Assume that a perfectly competitive firm endowed with a capital stock K produces with the production function above. Show that the labor demand of this firm is strictly increasing in its capital stock. 4

5 If we increase the capital stock K, (a) tells us that the marginal product of labor, MP L, increases. Thus, on the labor market we have MP L w which will increase demand for labor as long as the wage rate remains constant. Formally, we can derive the effect of a change in the capital stock on the optimal labour input by deriving the labour demand condition w.r.t. captial. w F L, L) deriving both sides w.r.t. capital. 0 F LK, L) + F LL, L) 0 F LK, L) + F LL, L) Rearranging gives us F LK,L) F LL,L) 0 As the cross-derivative is positive and the second derivative is negative (see above), the fraction multiplied by (-1) is positive. Note that here the implicit function theorem is at work: For a perf. competitive firm the labour demand condition has to hold at any time. Thus if K gets changed, the firm has to adjust L such that the equality still holds. This means, L is an implicit function of K and when deriving the whole expression w.r.t. K, we get as an inner derivative the effect of a K-change on L. 1.5 Consider a household who values consumption C and leisure F according to the following utility function: ( ) U (C, F ) αc 1 + (1 α) F 1 1 where 0 and 1 α 0. Let C be the numeraire (i.e. set its price equal to 1). (a) What is an elasticity? And the elasticity of substitution? Show the relationship between the parameter 0 and the elasticity of substitution between consumption and free time. Recall that the elsaticity of substitution is defined as: ln( F C ) ln w ( F C ) w where F and C are the utility maximizing demand for leisure and consumption and w is the real wage. I.e. the elasticity of substitution is the percentage change in the demand for leisure relative to consumption in response to a one-percent increase in the relative price of leisure (note that w can be interpreted as the price of buying leisure as the opportunity cost of leisure is equal to the wage that could be earned working). w F C 5

6 maximizing utility means to find F and C such that - in Equilibrium - the Marginal Rate of Substitution between leisure and consumption is equal to the relative price (with the price of consumption normalized to 1). w u F u C w [ 1 1 α C 1 +(1 α) F 1 [ α C 1 +(1 α) F 1 ] +1 1 (1 α) 1 F 1 ] +1 1 α 1 C( 1 ) 1 α α ( ) 1 C F F C ( ) 1 α α w ( F C ) w w ( F C ) ( ) w 1 ( ) 1 α α w ( 1 α α ) w Hence is the elasticity of substitution between consumption and free time. I.e. an increase of the real wage by 1% leads to a decrease of the demand for leisure relative to the demand for consumption by %. In general the price elasticity describes the percentage change in demand for a good due to a 1% change of the price of that good, whereas the elasticity of substitution may be described as the percentage change in relative demand between two goods due to a change of their relative price. (b) What makes Cobb-Douglas utility and production function so special? Show that when 1, then U (F, C) α ln C+(1 α) ln F [and hence that the utility function is of the Cobb-Douglas form U (C, F ) C α F 1 α. (Hint: Write the utility function as ( αc 1 +(1 α)f 1 ln U (C, F ) ln 1 and apply l Hospital s rule.) ) [ ln U (C, F ) ln α C 1 +(1 α) F 1 1 ] simply plugging in 1 yields: ln U (C, F ) ln which is not defined. Thus we look at the limit: lim [ln U (C, F )] lim 1 1 [ ln α C 1 +(1 α) F 1 1 ] 6

7 l Hospital lim αc +(1 α)f [ ] ( 1 αe ) ln C 1 ln C 1 ) ln F 2 +(1 α)e( ln F ( 1 [α 1 ln C 1+(1 α) 1 ln F 1] 1 ) α ln C + (1 α) ln F As shown in (a) the elasticity of substitution equals 1 in the case of a Cobb- Douglas function, i.e. the demand of an input factor relative to the second input factor decreases by 1% if it s price relative to the price of the second input factor increases by 1%. (c) (TBG) Can labor supply be independent of the real wage? Suppose that the total time endowment of the households is E. What is the household s budget constraint? Determine the household s labor supply as a function of the time endowment E and the real wage w (the money wage relative to the money price of consumption). Show that the supply of labour can be independent of the real wage, when 1. Why is the labor supply inelastic with respect to the real wage in this last case? [Hint: think about income and substitution effects] In the last decades, real wages have risen strongly but labor supply has not. What does this tell you about the utility function of households? from utility maximization, we know: (see (a)) F C ( ) 1 α α w E L w L ( ) 1 α α w E L(E, w) ( 1 α α ) w given the budget constraint: F E 1 w C (E,w) w E (( 1 α α ) w ( ) 1 α +1 +1) 2 α ( + 1) w 0 }{{} pivotal 0; 0 1 0; 1 0; 1 The increase in the wage gives two possible incentives, as far as labor supply is concerned. One would be to increase labor supply because payment for every unit of work supplied is higher. The other would be to work less because the same level of consumption can be obtained with less effort in terms of supplied 7

8 labor. In the case of the Cobb Douglas function both effects, the income and the substitution effect of an increase in the wage cancel out (property of additive log-utility functions), which is why labor supply is inelastic to changes in the real wage. If it is empirically true that real wages have risen but labor supply has not, we can conclude that the (approximate) Cobb Douglas utility function is a good estimate of the true utility function of households as far as their preferences concerning leisure and consumption are concerned. (d) Consider the case where 1 in the household s utility function. Suppose that there is a perfectly competitive firm endowed with a capital stock K that produces consumption goods using a constant-return-to-scale production function F (K, L). Suppose also that the production function is subject to decreasing returns to capital, 2 F (K,L) MP K 2 0, and that the production function is continuously differentiable. Find labor supply, labor demand, and the labor market equilibrium both graphically and analytically. What happens to employment and real wages in this economy as the capital stock K of the firm increases (hint: recall 1.4 (a))? 1 U(F, C) C α F 1 α Labour supply From 1.5 (c), we know: L S (w) E ( 1 α α ) w E ( E 1 α α ) w 0 1 α +1 α + α α α E (note: Labour supply is inelastic w.r.t. the real wage) Labour demand Firm: max L [F (K, L) wl rk] max L (F L, 1) w ) rk L F L, 1) w + L F 1 L, 1) ( K L 2 )! 0 8

9 F (K, L) w L + F 1 (K, L) ( K) 0 L D F (K,L) F1(K,L) K w (note: L decreases as w increases) Equilibrium L S L D α E F (K,L) F1(K,L) K w w F (K,L) F1(K,L) K α E Increasing the capital stock ( 0): w 1 α E [ K 2 F (K,L) 2 0 ] 0 Meaning, the wage rate increases, while employment remains fixed because labour supply is inelastic w.r.t. changes in K. Graphically: Figure (from Romer Advanced Macroeconomics Chapter 1) Describe how, if at all, each of the following developments affects the break-even and actual investment lines in our basic diagram of the Solow model: (a) The rate of depreciation falls. Figure 4 The slope of the line representing Break-Even-Investment (BEI) decreases, i.e. BEI becomes less steep. This means that for any k t less investment is required to sustain the initial value. Thus, a decrease in the rate of depreciation increases the BGP-value of k t. (b) The rate of technological progress rises. 9

10 The slope of BEI increases, i.e. the line becomes steeper. Hence, an increase in a leads to an acceleration of A lowering k t and thus at any initial level more investment is required to keep k t constant. k BGP decreases. (c) The production function is Cobb-Douglas, f (k) k α, and capital s share, α, rises The line representing actual ) investment (AI) moves upward i.e. it asigns a greater value of s f ( kt to any value k t. If the capital share rises more investment is needed at any value of capital per effective worker. k BGP increases. (d) Workers exert more effort, so that output per unit of effective labor for a given value of capital per unit of effective labor is higher than before. Same effect as (c). AI moves upward because for any k ) t, s f ( kt is greater. As this does not result in a constant increase of labor effectivity, BEI remains unchanged 1.7 Can an increase in the savings rate end up increasing long-run income but decreasing long-run consumption per worker? Consider a Solow economy without technological progress that is on its balanced growth path. Now suppose there is a permanent increase in the savings rate. Show the evolution of consumption per worker over time (from the time of the increase in the savings rate to the new balanced growth path). In a Solow Economy without technological progress, we have a condition for the BGP: s f (k) (n + δ) k, where : k K L Furthermore C L Y L s Y L y s y f (k) s f (k) f (k) (n + δ) k This means, that in the basic diagramm of the Solow model, consumption per worker is displayed as the difference between the production function and the Break-Even-Line (if the economy is on its BGP). Maximizing consumption per worker, we get: f (k) n + δ which gives the so-called golden-rule value of k that goes along with the maximum consumption level. If an increase of the savings rate might lead to a decrease of long-run consumption per worker 10

11 depends on the question if the economy is initially endowed with a level of capital per worker that is lower, equal to, or higher than the golden rule level. If initially k is smaller than k GR a marginal increase in the savings rate increases both, income and consumption, if k is equal to k GR a marginal increase in s leaves c unaltered. While if k k GR the change in s increases y but decreases c (see illustration). Figure 5 Figure 6 The intuitive explanation says, that for k k GR the increase in f (k) caused by the increase in s does not suffice to maintain k at the higher BGP level, such that consumption needs to be reduced in order to be able to keep k at the higher level. 1.8 Solow model with government. Consider a Solow economy with government. The government taxes households and consumes all of the tax revenue. Households consume a constant fraction c of their disposable income Y T, where T is taxes paid to the government. Show that national savings Y C G (savings by households and the government), where G denotes government consumption, is equal to (1 c) (Y T ). How does, therefore, an increase in taxes affect output per worker in the short and in the long run (assume that there is no technological progress and that the economy is on a balanced growth path when taxes are increased)? Y C G Y c (Y T ) T (Y T ) c (Y T ) (1 c) (Y T ) Effect on output graphically: Figure 7 The introduction of taxes immediately decreases actual investment which is a flow variable. This leads to a situation where Break-Even-Investment is greater than actual investment. Hence, k decreases until it reaches its new Balanced Growth Path value. The stock variable k adjusts slower than investment. Output per worker does not alter in the short run because the share of investment that now goes to taxes is used entirely for government consumption. However, as k decreases in the long-run the missing investment leads to a decrease in output per worker. The new stable equilibrium is in E new. Formally: 11

12 Balanced Growth Path: s [f (k) T ] (n + δ) k s f (k) k k T s (n + δ) T k T s s f (k) (n+δ) 0 The denominator is negative, because at the steady state the slope of the AI curve (s*f (k)) is always smaller than the slope of BIE line (n+δ). Moreover, because: y k 0, we have: y T (TBG)(from Romer Advanced Macroeconomics Chapter 1) Consider an economy with technological progress but without popultion growth that is on its balanced growth path. Now suppose there is a one-time jump in the number of workers. (a) At the time of the jump, does output per unit of effective labor rise, fall, or stay the same? Why? At the time of the jump output per unit of effective labor instantly falls, as well as capital per unit of effective labor does. While the capital stock is unable to change instantly and technological progress does not alter other than before k K AL must decrease. The same argument applies for ỹ Y AL. Intuitively, it is less capital per effective worker available and output has to be shared by more effective workers. Formally: k K AL 2 0 ỹ Y AL 2 0 (b) After the initial change (if any) in output per effective labor when the new workers appear, is there any further change in output per unit of effective labor? If so, does it rise or fall? Why? 12

13 The one-time jump of L may be intepreted as a new initial situation for the economy that does not matter in the long run. The decrease in k leads to a situation where actual investment exceeds Break-Even-investment. Intuitively a decrease in k increases MP K and thus leads to MP K R. Firms therefore need to pay less for their capital than the capital yields for them which makes investment lucrative. Hence, the capital stock and thereby k and ỹ increase until the BGP-value of k that was valid before the increase in L is obtained again. k and ỹ end up at the initial situation again. (c) Once the economy has again reached a balanced growth path, is output per unit of effective labor higher, lower, or the same as it was before the new workers appeared? Why? Due to the argumentation in (b) the economy ends up in its initial situation. This result refers to the fact that a jump in L does not alter any growth rate, savings rate or production function but simply effects an initial condition. As one of the basic results of the Solow-Swan Analysis is that initial conditions of K, A and L do not matter, it is obvious that the economy moves to it s balanced growth path again. A comparable situation might be the destruction of capital (e.g. in a war) because this would also alter k without affecting growth rates of K, A and L. Figure (from Romer Advanced Macroeconomics Chapter 1) Consider a Solow economy that is on its balanced growth path. Assume for simplicity that there is no technological progress. Now suppose that the rate of population growth falls. (a) What happens to the balanced growth path values of capital per worker, output per worker and consumption per worker? Sketch the paths of this variables as the economy moves to its new balanced growth path. Figure 9 BGP value of k : s f (k) (n + δ) k s f (k) k n ( ) k + n k n + δ k n k n k s f (k) (n+δ) 0 13

14 Again the denominator is negative, because at the steady state the slope of the AI curve is always smaller than the slope of BIE line. BGP value of y : f k n 0 k 0 f n y n 0 BGP value of c : c BGP f (k) s f (k) (1 s) f (k) c n (1 s) f (k) k n 0 This gives us: Figure 10 (b) (TBG) Describe the effect of the fall in population growth on the path of output (that is, total output, not output per worker). We know that in the steady state, i.e. if the economy is on its BGP, in an economy where there is no technological progress k K L and therefore y Y L are constant. That means, that in this case total output has to grow at the same rate as the equilibrium population. Hence, total output grows at the growth rate n 1 in the initial equilibrium and at the - decreased - growth rate n 2 in the new steady state. Furthermore, we know that y Y L increases if population growth decreases (part (a)). This results from the fact that the population instantly grows slower as the growth rate of the population drops. However, adjustment of the growth rate of total output is gradually as compared to the one time drop of the growth rate of the population, because as y Y L increases total output has to grow at a rate n 2 n n 1 during the adjustment process. 14