Chapter Consider an economy described by the following equations: Y = 5,000 G = 1,000

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1 Chapter C evel Qestions. Imagine that the prodction of fishing lres is governed by the prodction fnction: y.7 where y represents the nmber of lres created per hor and represents the nmber of workers employed per hor. a. Does this prodction fnction exhibit constant, decreasing, or increasing retrns to labor? How can yo tell? This prodction fnction exhibits decreasing retrns to labor. The first worker hired prodces nit of otpt while the second worker hired prodces.6 nits of otpt. By definition, this is simply saying that the first derivative of the prodction fnction with respect to labor is positive (the more workers hired, the more otpt) and the second derivative is negative (as more workers are hired the additional otpt prodced is less). In fact, this is the case: dy..7 > 0 d dy.. < 0 d b. Imagine that each fishing lre sells for $ and that each worker is paid $6 per hor. How mch labor is hired by this firm? What is the total otpt prodced by this firm? This firm maximizes its profits when its MP is eqal to the real wage. In this case the real wage is.5 and the MP etting these eqal and solving for gives 0.9 and sbstitting this vale for back into the prodction fnction gives Y.05. c. Imagine now that the government charges a tax on wages of 0% (similar to the strctre of the social secrity taxes paid by employers). With this new tax in place, how mch labor is hired by this firm? What is the total otpt prodced by this firm? What is the effect on employment given an increase in wage taxes? Profits.7-6 (+τ) where τ.0. Maximizing this profit fnction gives: (+τ) 0 or. 6.8/(6 (+τ)). When τ.0,.5. A 0% labor tax decreases labor hired from 0.9 to.5 nits (a 7.% decrease from the original level of labor hired). Otpt will fall to from.05 (or a fall of 9.9%).. Consider an economy described by the following eqations: Y 5,000 G,000 T,000 C (Y-T) I,000 50r a. In this economy, compte private savings, pblic savings, and national savings. National avings Y C G 5000 ( ( )) Private avings Y T C ( ( )) 750 Pblic avings T G b. Find the eqilibrim interest rate (r) r ths 50r 50 and r 5. c. Now sppose that G rises to,50. Compte private, pblic, and national savings. What is the new eqilibrim interest rate? National avings Y C G 5000 ( ( ))

2 Private avings Y T C ( ( )) 750 Pblic avings T G r ths r 0. d. Now sppose that the government increases taxes and government prchases by eqal amonts (both G and T eqal,50). What happens to the interest rate and investment in response to this balanced bdget change? Does yor answer depend on the marginal propensity to consme? Why? National avings Y C G 5000 ( ( )) Private avings Y T C ( ( )) Pblic avings T G r ths r 6.5 Yes, my answer does depend pon the MPC becase an increase in taxes redces my consmption by an amont dependent pon the MPC (which in trn increases savings). B evel Qestions. se the prodction fnction Y K..7 to answer the following qestions: a. olve for the eqilibrim real wage for an economy with this prodction fnction. (Yor answer will be a fnction of K and ). In class, we discssed the relationship between the real wage and the marginal prodctivity of labor; as a matter of fact, we claimed that the real wage mst eqal the marginal prodctivity of labor. The marginal prodctivity of labor is: Y.. W MP.7K P b. olve for the eqilibrim real rental rate of capital for an economy with this prodction fnction. The rental wage of capital occrs when the real payments to capital owners eqals the marginal prodctivity of capital. Again: Y.7.7 r MPK.K K P. c. What fraction of total income do workers receive in this economy? Capital owners? Workers receive 70% of income and capital owners receive 0% of income. (ee appendix to Chapter ). This is fairly easy to see once yo realize that each worker is paid the MP and, given workers, then total payments to workers mst be MP.7K. -..7K..7.7Y. Total payments to workers amont to 70% of total otpt. The same techniqe works for payments to capital owners. d. ppose that immigration raises the labor force by 0%. What happens to total otpt (in percent)? What happens to the real rental rate of capital (in percent)? What happens to the real wage (in percent)? Careflly explain why the changes in the wage and rental rate of capital move in the direction yo find. Otpt prior to the increase was Y K..7. Otpt after was Y K. (.).7. The percentage change is therefore K. (.).7 / K so otpt grew by 6.8%. By the same process, the rental rate of capital also grows by 6.8%. The real wage declines by.7k. (.) -. /.7K / so the real wage fell by.8%

3 . Consider an economy described by the following eqations: Y 000 C (Y T) G 500 T 500 In or society, a nmber of different types of investment occr. For instance, residential investment, inventory investment, and bsiness investment are all different types of ses for real goods. Consider a society with two types of investment: residential and bsiness. An eqation for each are given below: I r 50 0r I B 50 0r where I r represents residential investment and I B represents bsiness investment. Remember that total investment is eqal to I r + I B. a. On the plot below, graph national savings and total investment demand. (6) r Investment Demand National avings b. What is the eqilibrim interest rate in this economy? What is the eqilibrim level of investment? (0) avings eqals investment when: r. r -7.5 and I 50. c. In March of 00, President Bsh signed into law a tax bill that gives tax credits to bsinesses which generates increased bsiness investment demand. How will this policy impact the eqilibrim interest rate? What will happen to the total qantity each of bsiness investment and residential investment? () Bsiness investment demand increases while residential investment stays the same leading to an increase in total investment demand. If total investment demand increases then r rises. The amont of savings does not change (it remains 50) and given a greater amont of bsiness investment demand, bsiness investment grows at the expense of residential investment. d. One flaw with the classical model introdced above is that it doesn t take into consideration the fact that interest rates are positively correlated with government revenes (e.g. dring periods of high interest rates, late tax payers pay a greater penalty than dring low interest rates). If interest rates positively inflence tax revenes, plot the reslting savings and investment crves on the diagram below: (6) If higher interest rates lead to more taxes and greater taxes redce consmption less consmption increases savings, then higher interest rates lead to more savings. Graphically, this means that the savings crve is pward sloping.

4 r avings pply when interest rates impact taxes Investment Demand National avings e. Contrast the effect of the Bsh tax credit program on interest rates and total investment when interest rates effect tax revenes and when they don t. nder which scenario do interest rates change more? Why? (8) r avings pply when interest rates impact taxes Investment Demand National avings The Bsh tax credit increases investment demand. If savings positively depends pon interest rates, then the increase in demand raises rates and encorages savings. Becase more savings is flowing into the market, there is less opportnity for interest rates to rise. When savings does not depend pon interest rates the increase in demand jst pshes rates p no mitigating increase in savings. Ths rates rise more when savings does not depend pon interest rates. A evel Qestions 5. Althogh the Cobb-Doglas Prodction fnction is widely sed by economists to describe the prodction process, certainly some process are not described by this process. For instance, consider the eontief prodction fnction which is defined as Y min(ak, b) where a and b are coefficients and the fnction min indicates that Y is eqal to the minimm of the two argments in parenthesis. An example of this prodction process might be the nmber of pipe fittings prodced in an hor. Given a 0 and b 0, hiring one nit of labor and no pipe wrenches (capital), one wold prodce Y min(0 0,0 ) min(0,0) 0 (becase the minimm of zero or ten is zero). ikewise, bying a wrench and no labor also prodces zero otpt: Y min(0,0 0) 0. Hiring more wrenches than workers also prodces only the amont that the workers can do (in other words hiring the extra wrench does not increase otpt). a. Does the eontief prodction fnction exhibit increasing, decreasing, or constant retrns to scale?

5 The eontief prodction fnction exhibits constant retrns to scale. Given any set of inpts, dobling them will lead to an exact dobling of otpt. b. What type of retrns to labor does the eontief prodction fnction exhibit? Given a fixed amont of capital, an increase in labor by one nit increases otpt by a constant amont as long as b<ak. If b>ak then an increase in labor by one nit doesn t change otpt at all in other words an increase in labor by one nit increases otpt by a constant amont (zero). Ths, in a sense, there is constant retrns to labor. c. Imagine that a 0, b 0, w 5, r 0, and P 0. Can yo find the profit maximizing level of labor and capital to hire? The answer is an infinite amont of labor and capital will prodce an infinite amont of profits. Consider the profit fnction: Π 0 min(0 K, 0 ) K. Obviosly, hiring either labor and no capital or no labor and capital will prodce a negative profit so the minimm hiring of inpts a firm owner wold embark with are K and. If capital and labor both eqal, profits will be Hiring nits each of capital and labor will doble profits to 70 ths increasing labor and capital to infinity will prodce infinite profits. (becase there is constant retrns to scale and constant increasing costs). 6. In most prodction activities, a large nmber of inpts are reqired. This problem asks yo to consider a prodction fnction that incldes two types of workers: skilled and nskilled. The prodction fnction is: Y K where is the nmber of nskilled workers and is the nmber of skilled workers. a. If nskilled workers get paid w, skilled workers get paid w, capital owners are paid r, and the firm receives price P for each of its goods it prodces, write the firm s profit fnction. Π PK rk b. olve for the eqilibrim real wage for nskilled workers. What happens to wages nskilled workers earn when capital increases? Why? dπ d PK 0 w P K Increase in capital raises nskilled wages becase more capital increases the marginal prodctivity of the nskilled. c. olve for the eqilibrim real wage for skilled workers. What happens to wages skilled workers earn when the nmber of nskilled workers increase? Why? dπ d PK s w 0 P K killed worker s wages rise as more nskilled workers are hired becase more nskilled workers raise skilled MP. d. Imagine that the firm sells its prodcts for $00 each, owns nit of capital, and pays a w $0 per hor and a w $.5 per hor. How many nskilled workers does this firm hire? How many skilled workers does this firm hire? How mch total otpt is created?

6 Both of the above Marginal Prodcts of abor are sed to determine how mch skilled and nskilled labor are hired. After plgging in w $.5, w $0, K, and P 00, the MP s are:.. 5 sing the first eqation, I determine that:.. I will then sbstitte this expression for into the second eqation and find:. 5. From here we solve for and discover that 00. Plgging 00 into..8 either of the above MPs gives nskilled workers are hired and 60 skilled workers are hired. e. ome activists arge for a more eqal wage distribtion in the nited tates at a living wage. Imagine that the living wage in the above economy was $.50 per hor and Congress passed a law forcing firms to pay each worker, whether nskilled or skilled, $.50 per hor. Describe the effects of this legislation on the economy. What happens to total prodction? What happens to the nmber of nskilled and skilled workers hired? WHY? (0) Higher W means fewer will be hired. As fewer are hired, the marginal prodct of falls (remember the answer to part b). ince the MP of is lower, the firm does not want to hire as many workers so it lays off both the nskilled (becase the nskilled cost more) and skilled (becase they are less prodctive). With fewer of both the skilled and nskilled, this firm redces prodction.