Queen s University. Department of Economics. Instructor: Kevin Andrew

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1 Figure 1: 1b) GDP Queen s University Department of Economics Instructor: Kevin Andrew Econ 320: Math Assignment Solutions 1. (10 Marks) On the course website is provided an Excel file containing quarterly data on Real GDP and Real Consumption for Canada. (a) See Spreadsheet (b) See Spreadsheet (c) See Spreadsheet 1

2 Figure 2: 1b) Consumption Figure 3: 1c) GDP 2

3 Figure 4: 1c) Consumption 2. (10 Marks) For this question you will need to use your % deviations data from Question 1c) (a) Consumption is less volatile than output since the ratio of σc σ y than 1. is less (b) See Spreadsheet The correlation between the two series is This is positive and close to 1. Therefore, the two series have strong positive co-movement. (c) See Spreadsheet The autocorrelation of y is and the autocorrelation of c is Therefore, they are both very persistent, however GDP is more persistent than consumption. (d) An HP filtered series would be less volatile. This is because the trend evolves over time, meaning the deviations from trend are smaller. The linear filter leads to a constant trend over time. This is why there are persistent deviations from trend in the graphs to 1c). This leads to a higher standard deviation. 3. (3 Marks) Find the derivative for the following functions: (a) dy = 3x2 (b) dq dp = 2p 1 2 (c) dq dp = χηpη 1 3

4 4. (3 Marks) Differentiate the following with respect to x using the product rule. It is helpful to break the functions up into f(x) = g(x)h(x) as a first step. (a) g(x) = 8x 2 1, g (x) = 16x. h(x) = 3x 1 and h (x) = 3. Therefore: (b) (c) = g (x)h(x) + h (x)g(x) = 16x(3x 1) + 3(8x 2 1) = 1 3x 2 = 2αψx γ βγψx γ 1 = 72x 2 16x 3 5. (3 Marks) Differentiate the following with respect to x using the quotient rule. It is helpful to break the functions up into f(x) = g(x)/h(x) as a first step. (a) We can set up the problem as follows: (b) (c) g(x) = x 2 + 3, g (x) = 2x h(x) = x h (x) = 1 = g (x)h(x) h (x)g(x) [h(x)] 2 = 1 3 x 2 This is the same as the answer to question 4b). This shouldn t be surprising as they are the same function. 16x(x + 3) 16x 16x(x + 2) = (x + 3) 2 = (x + 3) 2 = (γ 1)αδxγ + ɛαγx γ 1 δβ (δx + ɛ) 2 6. (3 Marks) Use the chain rule to find the derivatives with respect to x. It is often helpful to break the functions up into f(x) = g(h(x)) where z = h(x) so that f (x) = g (z)h (x) = g (h(x))h (x). (a) g(z) = z 3 g (z) = 3z 2 z = h(x) = 3x 2 12 h (x) = 6x = 18xz2 = 18x(3x 2 12) 2 4

5 (b) (c) = 24(12x + 1) 3 = λθ(θx + κ)λ 1 7. (5 Marks) The Keynesian Consumption Function is given by: C = a+by. With a > 0 and b (0, 1). (a) The marginal propensity to consume is given by: dc dy = b This represents the extra amount consumed when income goes up by a small amount. The average propensity to consume is given by: C Y = a Y + b This represents the total share of income which is consumed. (b) The income elasticity of consumption is: (c) ε C Y = dc dy Y C = by a + by This is the percentage increase in the amount of consumption when there is a one percent increase in income. Notice that income enters into the formula. If you take the derivative of the elasticity with respect to Y you will notice that the elasticity is increasing in income. That is, richer households consumption responds more in percentage terms to income increases than poorer households. dap C dy (1) = a Y 2 < 0 (2) This means that consumption as a fraction of income declines as income goes up. 8. (3 Marks) Find the partial derivatives f x = f x and f y = f y for the following functions: (a) f f = 2x + 5y x y = 3y2 + 5x (3) 5

6 (b) f x = x2 + 1 x 2 y f y = 1 x2 xy 2 (4) (c) For the first function: For the second function: f(1, 2) x f(1, 2) x = = 12 (5) = = 1 (6) We interpret the partial derivative as follows. It is the change in the function, f, at point (1,2) due to a marginal increase in x, holding the value of y fixed: f(1, 2) x = lim 0 f(1 +, 2) f(1, 2) (7) 9. 3 Marks Find the total differential: (a) (b) (c) dz = z z + x y dy = (6x + y) + ( 6y2 + x)dy (8) dy = y x y x 2 2 = x 2 (x 1 + x 2 ) 2 1 x 1 (x 1 + x 2 ) 2 2 (9) dy = αx α 1 1 x β βx α 1 x β (10) Marks The utility function for a consumer takes the form: (a) U(c 1, c 2 ) = α ln c 1 + (1 α) ln c 2 (11) U c 1 = α c 1 U = 1 α (12) c 2 c 2 (b) du = α c 1 dc α c 2 dc 2 (13) 6

7 (c) Setting du = 0 and manipulating yields: dc 2 du=0 = α c 2 (14) dc 1 1 α c 1 Since in the assignment I asked for dc1 dc 2 du=0, this is acceptable as well Marks We are given the following function: This is a demand equation. q t = p α t (15) (a) We can use the following formula to calculate the elasticity: ε q p = dq p dp q = αpα 1 t p t p α t = α (16) Interpret this as follows: α is the percentage change in demand in response to a 1% change in price. It is constant for all levels of the price. (b) Taking natural logs of both sides of the equation at both times t and t + 1 gives: ln q t = α ln p t ln q t+1 = α ln p t+1 (17) Using the definition of the growth rate we get the following expression: ln q t+1 ln q t = α(ln p t+1 ln p t ) (18) g q αg p (19) Where we take into account that the relationship between log differences and growth rates is an approximation. (c) We can write the logarithm of the steady state quantity as: We can subtract this from ln q t to get: ln q = α ln p (20) ln q t ln q = α(ln p t ln p) (21) ˆq t = αˆp t where ˆq t = ln q t q (22) It is clear that the parameter α is important for determining the magnitude of the effects of price changes on quantities. This makes sense given its interpretation as the price elasticity of demand. 7

8 Marks A firm has the following total cost function: The demand function is given by: C = 1 3 Q3 7Q Q + 50 (23) Q = 100 P (24) (a) First rewrite the demand function as P = 100 Q and remember that total revenue is the price multiplied by the quantity sold. R(Q) = (100 Q)Q = 100Q Q 2 (25) (b) The profits are just the total revenues minus the total costs: Π(Q) = R(Q) C(Q) = 1 3 Q3 + 6Q 2 11Q 50 (26) (c) The FOC is: dπ dq = Q2 + 12Q 11 = 0 (27) Q 2 12Q + 11 = 0 (28) (Q 11)(Q 1) = 0 Q = 11 Q = 1 (29) We need to check which of the two solutions maximizes profits. Π(11) = Π(1) = (30) Therefore, the profit maximizing quantity is Q = 11 (d) The SOC is: d 2 Π = 2Q + 12 (31) dq2 Notice that at Q = 1 this is positive, so this is not a maximizer. However, it is negative at Q = 11. We could check which values of Q the second derivative is negative for: d 2 Π dq 2 < 0 Q > 6 (32) We can see why our solution turned out to be Q = 11 and not Q = 1. This is why it is always important to check the second derivative, even if there is only one solution Marks A consumer has the utility function given in Question 10. The price of goods one and two are p 1 and p 2 respectively. As well, income is given by y. 8

9 Figure 5: Solution to Consumers Problem (a) The budget line is given by: p 1 c 1 + p 2 c 2 = y (33) Perhaps for plotting you would like to see it in y = mx + b form: c 2 = y p 2 p 1 p 2 c 1 (34) (b) See Graph Utility is increasing to the northeast. The consumer will try and attain the highest level of utility subject to the budget constraint. Thus, the tangency point is the constrained optimum. The two red indifference curves are not optimal. The lower one represents a situation where the consumer could do better while the higher one is unaffordable. Note that at the tangency point the slope of the budget line must equal the marginal rate of substitution. We could just use this fact to derive our solution, but we will be comforted by the fact the Lagrangian method attains this solution and can be applied to more complicated cases. (c) The constrained maximization problem is of the form: max c 1,c 2 α ln c 1 + (1 α) ln c 2 (35) s.t. p 1 c 1 + p 2 c 2 = y (36) 9

10 The Lagrangian is: Λ = α ln c 1 + (1 α) ln c 2 + λ[y p 1 c 1 p 2 c 2 ] (37) (d) This closely resembles my notes on constrained optimization. Λ = α λp 1 = 0 c 1 c 1 (38) Λ = 1 α λp 2 = 0 c 2 c 2 (39) Λ λ = y p 1c 1 p 2 c 2 = 0 (40) Refer to my notes for more of the algebra, but the solution is: c 1 (p 1, p 2, y) = αy p 1 c 2 (p 1, p 2, y) = (1 α)y p 2 λ(p 1, p 2, y) = 1 y (41) (e) The indirect utility function was also found in my notes. V (p 1, p 2, y) = α ln αy (1 α)y + (1 α) ln (42) p 2 p 2 = ln y + α ln α p 1 + (1 α) ln (1 α) p 2 (43) Taking the derivative of this function with respect to y we get: V (p 1, p 2, y) y = 1 y = λ(p 1, p 2, y) (44) Thus, we can interpret the Lagrange multiplier in this case as the marginal utility of a little extra income at the optimum. In general, Lagrange multipliers represent the value of relaxing the constraint a little, in this case by giving the consumer a little more income Marks Suppose the demand for liquidity (i.e. money) is given by: L t = AY α t e βit (45) Where Y denotes income and i is the nominal interest rate. A, α, β are parameters which are greater than 0. Suppose there exist trend values of income and the nominal interest rate, Ȳt and ī. Notice that the trend is time dependent for income, but not for interest rates. This is because income is non-stationary and the nominal interest rate is stationary. This just means that income is on average growing over time while the interest rate fluctuates around a single value ī. 10

11 (a) Taking natural logs first we see that: ln L t = ln A + α ln Y t βi t (46) Taking the derivative of this with respect to ln Y t we see that the elasticity is: ε L Y = dln L t dln Y t = α (47) (b) We use the natural logarithm from above to find: ε L i = dln L t di t = β (48) Where the tilde denotes that it is a semi-elasticity. The reason we use a semi-elasticity here is that interest rates are already quoted in percentages, so we don t need the natural logarithm to make this conversion. (c) Note that we can write the logarithm of the trend value as: ln L t = ln A + α ln Ȳt βī (49) Subtracting this from the logarithm of L t above we get: ln L t ln L t = α(ln Y t ln Ȳt) β(i t ī) (50) ˆL t = αŷt βî t (51) (d) We use the following formula for a first order expansion around ( x, ȳ): f(x, y) f( x, ȳ) + f x ( x, ȳ)(x x) + f y ( x, ȳ)(y ȳ) (52) For our function it is helpful to compute a couple partial derivatives: L t = αay α 1 L t e βit t = βayt α e βit (53) Y t i t Therefore the expansion is: L t L t + αaȳ t α 1 e βī (Y t Ȳt) βaȳ t α e βī (i t ī) (54) L t L t L t α Y t Ȳt Ȳ t β(i t ī) (55) Recall that the percentage deviations of L t and Y t can be approximated as ln L t ln L t. Then we can rewrite: ln L t ln L t = α(ln Y t ln Ȳt) β(i t ī) (56) 11

12 Marks Consider the following variant of the cobweb supply demand model presented in class. The inverse demand function is given by: P t = γ δd t (57) Supply depends on the expected price, P e t. It is given by: Expectations are adaptive. That is: S t = βp e t α (58) P e t = (1 η)p e t 1 + ηp t 1 (59) That is, expectations are a combination of yesterdays forecast and yesterdays price with η (0, 1). (a) The model in class had expectations of the form Pt e = P t 1 These backward looking expectations are just a special case of the adaptive expectations with η = 1. (b) Setting D t = S t gives: γ δ P t δ = βp t e α (60) P t = γ δ(βpt e α) (61) (c) Notice that we can solve the above equation for: P e t = γ + αδ P t βδ (62) Substitute P e t 1 into the RHS of the expectations equation This gives: γ + αδ P t βδ = (1 η) γ + αδ P t 1 βδ + ηp t 1 (63) Rearranging, we get: P t = η(γ + αδ) + (1 η ηβδp t 1 ) (64) This takes the form suggested with ξ = η(γ + αδ) and θ = (1 η ηβδ). (d) Setting P t = P t 1 = P We get: P (1 η ηβδ) P = η(γ + αδ) (65) P = γ + αδ 1 + βδ (66) 12

13 (e) Notice that we can rearrange to get: γ + αδ = P (1 + βδ) (67) Inserting this into our difference equation gives: P t = η P (1 + βδ) + (1 η(1 + βδ)p t 1 (68) Subtract P from each side. P t P = η(1 + βδ 1) P + (1 η(1 + βδ))p t 1 (69) P t P = (1 η(1 + βδ))(p t 1 P ) (70) This takes the required form with φ = (1 η(1 + βδ)). Notice that θ = φ. (f) We can replace P t 1 P on the RHS with P t 1 P = φ(p t 2 P ). This gives: P t P = φ 2 (P t 2 P ) (71) We can keep doing this recursively for P t 3, P t 4,..., P 0 until we are left with: P t P = φ t (P 0 P ) (72) P t = P + (1 η(1 + βδ)) t (P 0 P ) (73) In order for this to converge to its steady state, P we require that the absolute value of φ is less than 1. That is: 1 η(1 + βδ) < 1 (74) If φ (0, 1) then the price converges monotonically. This will be the case if: If 0 < η(1 + βδ) < 1 (75) 1 < η(1 + βδ) < 2 (76) Then the price converges in an oscillatory manner. For any other parameter values the difference equation is unstable. 13