Econ Review Set 2 - Answers

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1 Econ 4808 Review Set 2 - Answers EQUILIBRIUM ANALYSIS 1. De ne the concept of equilibrium within the con nes of an economic model. Provide an example of an economic equilibrium. Economic models contain variables. The variables that one is trying to explain within the con nes of the model are called endogenous variables. The variables whose values are determined outside of the model are call exogenous variables. In static equilibrium models, if for given values of the exogenous variables, there is no tendency for the values of the endogenous variables to change the model is in equilibrium. In dynamic equilibrium models, if for given values (or given paths) of the exogenous variables, there is no tendency for the values (or the paths) of the endogenous variables to change the model is in equilibrium. For example, in a model to explain equilibrium quantities and prices for some commodities, the equilibrium condition is that supply equals demand for all of the commodities. In a model to explain the behavior of a consumer facing constraints, the individual is in equilibrium if she is doing the best she can given her constraints, in which case she does not want to change her behavior (that is purchasing the most preferred bundle of goods that she can a ord by spending all of her income). A competitive rm that is maximizing its pro ts is in equilibrium because it cannot do better, so it has no incentive to change its behavior. 2. De ne, in a few sentences, the concept of static equilibrium within the con nes of an economic model. An economic model is in static equilibrium if, for given values of the exogenous variables, the endogenous variables have no tendency to change values. Note that one can t use the word equilibrium to de ne equilibrium. 3. De ne, in a few sentences, the concept of a comparative static prediction. A comparative static prediction is a prediction about what happens to the equilibrium value of an endogenous variable in a model when the value of one of the exogenous variables in the model changes (the value of all of the other exogenous variables remaining constant). 1

2 4. Consider the market for some good. Assume that x d 5 3p x s p x d x s ; where x d is the quantity demanded, x d the quantity supplied and p is the price of the good. (A) Determine the equilibrium price and quantity. (B) Graph the demand function and the supply function with price (p) on the vertical axis. Identify the slope and vertical intercept for each function. Identify the equilibrium price and quantity on the graph. (A) Substitute the demand and supply function into the equilibrium condition 5 3p p 7 7p p e 1 Substitute this into either the supply or demand function to determine that x e 8. Does this model make sense? Not if x is a desirable commodity. How about if x is garbage? Then you pay to have it picked up. The disposal service purchases the garbage from you at a negative price (it gets paid). (B) To graph the demand and supply functions with price (p) on the vertical axis, you need to rewrite the demand and the supply equations expressing price (p) in terms of quantity (x d or x s ). Solving the demand equation for p, we obtain the inverse demand function p xd Now graph this function. It has a vertical intercept of 5 3 and a slope of 1 3. Solving the supply equation for p, we obtain the inverse supply function p xs Graph this function. It has a vertical intercept of 3 and a slope of

3 5. Consider two goods - good 1 and good 2. Determine the equilibrium prices and quantites of the two goods if x d 1 5 3p 1 p 2 x s p 1 x d 1 x s 1 and x d p 2 x s p 2 + p 1 x d 2 x s 2 In equilibrium: 5 3p 1 p p 1 and 15 3p p 2 + p 1 Simplifying: 4 7p 1 p 2 0 and 13 7p 2 p 1 0 Express p 2 in terms of p 1 from the rst equation: p 2 4 7p 1 ; then substitute with it in the second equation: and solve for p 1 : 13 7 (4 7p 1 ) p p 1 p p 1 0 p e

4 So p e 1 5. Plugging this into 4 7p 1 p 2 0 and solving for p 2 one gets: p 2 0 p e So, p e To check whether you ve got p e 2 right, you can plug pe 1 5 condition 13 7p 2 p 1 0 and solve for p 2 : into the other equilibrium So, we did p () 5 7p p 2 p e (7) 29 Now determine the equilibrium quantities of x 1 and x 2 by plugging p e 1 5 and pe 2 29 into the demand functions: and 5 29 x e x e To do one nal check on the math, let s calculate the equilibrium quantities x 1 and x 2 by plugging p e 1 5 and pe 2 29 into the supply functions: 5 x e x e So, everything seem checks. 6. Determine the price and quantity if x d a bp x s c + dp x d x s a; b; c; d; > 0 and (ad bc) > 0 4

5 Now graph the demand function and supply function with price (p) on the vertical axis. Identify the slope and vertical intercept for each function. Identify the equilibrium price and quantity. See you lecture notes. 7. Determine the equilibrium price and quantity if x d 8 p 2 x s p 2 2 x d x s Assume the equilibrium price is positive. In equilibrium 8 p 2 p p 2 p p 5 So the equilibrium price is p e 5. the equilibrium quantity is Plugging this into the demand or supply function, x e 8 p Assume a world of only two goods: beer (B) and nuts (N). Further assume that these two goods are typically consumed together (complements). In addition, assume that B d 50 + T P B P N B s 25 where B d the demand for beer B s the supply of beer T temperature in degrees Fahrenheit P B the price of beer P N the price of nuts 5

6 Assume T is exogenous. What are the endogenous variables in this model? What is the economic relationship between beer and nuts? Determine the equilibrium price of beer? If you were to criticize this model, what would you say. How would you change the model to address the criticism? T is exogenous. B d and P B are endogenous. I base that conjecture on the fact the the model speci es an equation to explain B d. B s is exogenous, since it s value is being set by assumption. What about P N? If one assumes that it is endogenous, there are more unknowns than equations and one can only solve for one endogenous variable, P B, as a function of another endogenous variable, P N. So, assume for the moment P N is exogenous. When B d B s : Solving this for P B one obtains: 50 + T P B P N 25 P B 25 + T P N However, given that beer and nuts are close compliments, it does not make much sense to assume that the equilibrium price of beer can change without a ecting the equilibrium price of nuts. This model needs to be generalized and include N d ; N s and P N as endogenous variables. This can be done by adding three equations to the model: a demand function for nuts, a supply function for nuts and N d N s. Compare this question to the next one that includes the market for nuts. 9. Assume the following simple Keynesian macroeconomic model: Y C + I 0 + G 0 C :75 + :25Y 2 (1) I 0 G 0 0 (2) Solve for the equilibrium levels of income and consumption. preferred? Which will society end up at? Which equilibrium is Substitute with C :75 + :25Y 2 ; I 0 G 0 0 in the rst equation Y :75 + :25Y 2 (3) and solve for the equilibrium Y 6

7 :25Y 2 Y + :75 0 (4) Y e 1 and Y e 3 (5) Plug Y e into the second equation to solve for the equilibrium C C e :75 + :25 (1) 2 1 and C e :75 + :25 (3) 2 3 (6) The equilibrium with Y e C e 3 is preferred over the equilibrium with Y e C e Assume the following simple Keynesian macroeconomic model: Y C + I 0 + G 0 C 3 + :025Y 2 (7) I 0 G 0 1 (8) Solve for the equilibrium levels of income and consumption. Substitute with C 3 + :025Y 2 and I 0 G 0 1 into the rst equation Y 3 + :025Y (9) and solve for the equilibrium Y using the quadratic formula :025Y 2 Y (10) Y e 34: 142 and Y e 5: (11) Plug Y e into the second equation to solve for the equilibrium C C e 3 + :025 (34: 142) 2 32: 142 and C e 3 + :025 (5: 857 9) 2 3: (12) So there are two equilibriua (Y e 34: 142 ; C e 32: 142) and (Y e 5: 857 9; C e 3: 857 9) : 11. Assume the following simple Keynesian macroeconomic model: 7

8 Y C + I + G + X 0 M C a + b Y T 0 with a > 0; b 2 (0; 1) (13) G gy with g 2 (0; 1) (14) M M 0 my with m 2 (0; 1) (15) I iy with i 2 (0; 1) () a bt 0 + X 0 M 0 > 0 (17) b + m + i + g < 1 (18) where Y national income C consumption of domestically produced goods G government expenditures I investment X 0 the exogenous level of exports T 0 the exogenous level of taxes M consumption of imported goods M 0 the exogenous level of imports A) What is the marginal propensity to import (aka the marginal propensity to consume imported goods)? B) In this model, are imports superior goods? C) Solve for the equilibrium level of income. D) Determine what would happen to the equilibrium level of imports (i.e., would they increase or decrease and by how much) if the marginal propensity to invest increases by a small amount. A) Marginal propensity to import is how much imports (M) change when income (Y ) is increased by one unit, that is M this model M M 0 my )). In this model, it is M Y imports decrease as income increases. (or also the slope of the import function (in m, which is negative. That is, B) Because imports decrease as income increases, imports are an inferior good, not a superior good. C) Substitute with the C; G; M and I functions in the function for Y Y a + b Y T 0 + iy + gy + X 0 M 0 my a + by bt 0 + iy + gy + X 0 M 0 + my (b + i + g + m) Y + a bt 0 + X 0 M 0 8

9 which implies that so Y (b + i + g + m) Y + a bt 0 + X 0 M 0 Y (b + i + g + m) Y a bt 0 + X 0 M 0 (1 b i g m) Y a bt 0 + X 0 M 0 Y e a bt 0 + X 0 M 0 1 b i g m D) To determine what happens to the equilibrium level of imports when the marginal propensity to invest, i, increases, one has to rst determine the equilibrium level of imports. This is accomplished by substituting Y e into the import demand function M e M 0 my e (19) a bt M X 0 M 0 m (20) 1 b i g m M 0 m a bt 0 + X 0 M 0 (21) (1 b i g m) What happens to this when i increases? 1st Approach: The denominator (1 b i g m) decreases ) the ratio m(a bt 0 +X 0 M 0 ) (1 b i g m) increases and the di erence M 0 m(a bt 0 +X 0 M 0 ) (1 b i g m) decreases. So, if i increases, then M e dercreases. In words, if the marginal propensity to invest increases, equilibrium investments increase and therefore equilibrium income increases, causing imports to decrease because in this model they are an inferior good. 2nd Approach: Take the partial derivative of M e with respect to i (treat everything apart from i as M 0 m(a bt 0 +X 0 M 0 ) (1 b i g m a bt 0 + X 0 M 0 0 (1 b i g m) m a bt 0 + X 0 M 0 (1 b i g m) [0] (1 b i g m) (1 b i g m) 2 m a bt 0 + X 0 M 0 (1 b i g m) 2 < 0 (1 b i g m) 2 m a bt 0 + X 0 M 0 ( 1) The negative derivative impies that if i increases, then M e will decrease. In particular, if i increases by a small amount, then M e will decrease by m(a bt 0 +X 0 M 0 ). (1 b i g m) 2 9

10 12. Assume the following simple Keynesian macroeconomic model: Y C + I + G + X 0 M C a + b Y T 0 with a > 0; b 2 (0; 1) (22) G gy with g 2 (0; 1) (23) M M 0 my with m 2 (0; 1) (24) I iy with i 2 (0; 1) (25) a bt 0 + X 0 M 0 > 0 (26) b + m + i + g < 1 (27) where Y national income C consumption of domestically produced goods G government expenditures I investment X 0 the exogenous level of exports T 0 the exogenous level of taxes M consumption of imported goods M 0 the exogenous level of imports A) What is the marginal propensity to import? B) In this world, are imports superior goods? C) Solve for the equilibrium level of income. D) Determine what would happen to the equilibrium level of imports (i.e., would they increase or decrease and by how much) if the marginal propensity to consume domestic goods increases by a small amount. E) What further restrictions might you impose on the parameters that would be su - cient for the equilibrium level of imports to always decrease when the marginal propensity to consume domestic goods increases? A) - C) are the same as in Problem 11. D) To determine what happens to the equilibrium level of imports when the marginal propensity consume domestic goods, b, increases, one has to rst determine the equilibrium level of imports, M e. In part (D) of Problem 11 we determined that M e M 0 m a bt 0 + X 0 M 0 (1 b i g m) What happens to M e when b increases? This is di cult to tell by eyeballing because b appears in both the numerator and the denominator. taking the partial derivative of M e with respect to b: (28) You can answer the question by 10

11 M 0 m(a bt 0 +X 0 M 0 ) (1 b i g (a 0 +X 0 M 0 ) (1 b i g m) m T 0 (1 b i g m) ( 1) a bt 0 + X 0 M 0 # m (1 b i g m) 2 " T 0 (1 b i g m) + a bt 0 + X 0 M 0 # m (1 b i g m) 2 (29) (30) (31) (32) m T 0 (1 i g m) a + X 0 M 0 (1 b i g m) 2 (33) has the same sign as the numerator m T 0 (1 b i g m) + a bt 0 + X 0 M 0. By assumption m > 0, (1 b i g m) > 0 and a bt 0 + X 0 M 0 > 0. There are not enough assumptions in the model to prove that T 0 (1 i g m) a + X 0 M 0 is always positive or negative. Therefore we cannot say whether M e increases or dercreases when b increases. (Note that in empirical applications I would expect the second term, the exogenous level of net expenditures to be greater than the exogenous level of taxes multiplied by the marginal propensity to spend on other stu ). 13. Assume the following simple Keynesian macroeconomic model: Y C + I 0 + G C a + :75 (Y T ) with a > 0 (34) G 10 + gy with g 2 (0; 1) (35) T ty with t 2 (0; 1) (36) (:75t g) > :25 (37) where Y national income C consumption of domestically produced goods G government expenditures I 0 the exogenous level of investment T the level of taxes A) Solve for the equilibrium level of national income. B) Demonstrate that it is always positive. 11

12 C) Determine how the equilibrium level of taxes changes when the income tax rate increases. (Don t worry about trying to simplify your answer.) A) Substitute with the T function in C and then substitute with the C and G functions in the function for Y Y a + :75 (Y ty ) + I gy and solve for the equilibrium Y (1 :75 (1 t) g) Y a + I Y (:75 (1 t) + g) Y + a + I (:25 + :75t g) Y a + I Y e a + I :25 + :75t g B) The numerator of Y e is positive because the exogenous level of investment, I 0, has to be positive and by assumption a > 0. (:75t g) > :25. So, Y e > 0. The denominator is positive because by assumption C) Substitute with Y e into the tax function to nd the equilibrium tax level a + I T e t :25 + :75t g We cannot answer the question by simply "eyeballing" T e. Take te partial derivative of T e with respect to ) (:25+:75t h i a + I (:25+:75t a + I (1) (:25 + :75t g) (:75) t (:25 + :75t g) 2 a + I :25 g (:25 + :75t g) 2 The term a + I is positive because the exogenous level of investment, I 0, has to be positive and by assumption a > 0. The denominator (:25 + :75t g) 2 is positive. has the same sign as the numerator (:25 g). If (:25 g) > 0, > 0 and T e increases when t increases. If (:25 g) < 0, < 0 and T e decreases when t increases. If (:25 g) 0, 0 and T e does not change when t increases. 12

13 14. Assume the following simple Keynesian macroeconomic model: Y C + I + G 0 C a + by with a > 0; b 2 (0; 1) (38) I c + dy with d 2 (0; 1) (39) (b + d) < 1 (40) where Y national income C consumption G government expenditures I investment Solve for the equilibrium levels of income, consumption, and investment assuming a 5; b :5; c 10, and d :3, and G 0 7. Show all of your work. and I functions become With these restrictions on the parameters and exogenous variables the Y; C Y C + I + 7 C 5 + :5Y with a > 0; b 2 (0; 1) (41) I 10 + :3Y with d 2 (0; 1) (42) (b + d) < 1 (43) Substitute the second and third equation into the rst to obtain Solve for Y to obtain Y e Y 5 + :5Y :3Y + 7 (44) Y e 110 Plug this into the consumption function to obtain C e 5 + :5 (110) 60 Plug it into the investment function to obtain Checking I 10 + :3 (110) 43 13

14 15. Assume the following simple Keynesian macroeconomic model: Y C + I 0 + G 0 C a + by 2 (45) I 0 20; G 0 25; a 5 and b :5 (46) (b + d) < 1 (47) where Y national income C consumption G government expenditures I investment Solve for the equilibrium levels of income. What s going on? With the restrictions on the parameters and the exogenous variables the Y and C functions become Y C C 5 + :5Y 2 (48) Substituting with the consumption function in the income function we get and solving for Y we obtain Y 5 + :5Y (49) :5Y 2 + Y (50) Y 1 p 99 (51) There is no real solution for Y e. Both solutions to the above quadratic equation are imaginary numbers ( 1 + 9: 949 9i; 1 9: 949 9i). So the above model does not have a solution in real numbers.. Assume the following simple Keynesian macroeconomic model: 14

15 Y C + I 0 + G C a + b Y T 0 with a > 0; b 2 (0; 1) (52) G gy with g 2 (0; 1) (53) b + g < 1 (54) a bt 0 + I 0 > 0 (55) where Y national income C consumption of domestically produced goods G government expenditures I 0 the exogenous level of investment T 0 the exogenous level of taxes A) Solve for the equilibrium level of income. Show all your work. B) Determine the equilibrium level of consumption. C) Assuming Y e > 0 what does the theory predict will happen to the equilibrium level of income if the government s marginal propensity to spend increases? D) What happens to the equilibrium level of consumption if the exogenous level of taxes increases? A) Substituting with the C and G functions in Y Y a + b Y T 0 + I 0 + gy and solving for Y Y by + gy + a bt 0 + I 0 Y a bt 0 + I 0 Y e a bt 0 + I 0 B) Substitute Y e into the consumption function to determine the equilibrium level of consumption a bt C e 0 + I 0 a + b T 0 C) g is the marginal propensity to consume on the part of the government. Examining Y e one sees that if g increases the denominator becomes smaller, which causes the equilibrium level of consumption to increase. 15

16 [As an aside, determining what happens to the equilibrium level of consumption when b increases is more di cult because b appears in both the denominator and numerator and the e ects work in opposite directions. To determine the answer we would need to take the partial derivative of C e with respect to g and then what its sign is.] To determine by how much C e will increase if g increases by a small amount, take the h a + b a bt 0 +I 0 T h a + b a bt 0 +I 0 bt " b a bt 0 + I 0 # b a bt 0 + I b a bt 0 + I 0 2 (0) ( 1) b a bt 0 + I 0 2 b a bt 0 + I 0 2 > 0 So, if g increases by a small amount, C e will increase by b(a bt 0 +I 0 ) 2. D) Examining the equation for the equilibrium of consumption, we see that if T 0 increases C e decreases. How 0 a + b a bt 0 +I 0 i T a + b(a bt 0 +I 0 ) bt 0 b ( b) b b 2 b (g 1) b < 0 The denominator is positive by assumption and the numerator is negative because b and g are positive and g < 0 < 0. Your task is to build a simple theory of supply and demand for product x. I want your theory to predict the equilibrium price and quantity of product x. I want your

17 theory to have two exogenous variables. Derive the equilibrium price and quantity from your de nitions and assumptions. In addition, derive three comparative static predictions from your theory. 17. Assume a world of just two goods, x 1 and x 2. Further assume the individual can rank all bundles of these two goods and this ranking can be represented by the utility function u u(x 1 ; x 2 ) a + x 2 1x 2 Explain why this preferences can also be represented by the utility function U U(x1; x2) 2log (x 1 ) + log (x 2 ) Notice that U (x 1 ; x 2 ) log [u (x 1 ; x 2 )]. Because the purpose of a utility funvtion is to represent the ordinal ranking of bundles (x 1 ; x 2 ), the ranking will be preserved if instead of we u (x 1 ; x 2 ) use log [u (x 1 ; x 2 )]. utility. 18. Describe consumer theory in a nutshell. In your description do not use the word 19. De ne the concept of a utility function. For the purposes of the question you can assume a world with only two goods, x 1 and x 2. 17