LUBS 3505 Advanced Macroeconomics 2024-2025 Lecture 2 Consumption PDF

LEEDS UNIVERSITY BUSINESS SCHOOL Advanced Macroeconomics LUBS 3505 2024-2025 Lecture 2. Consumption Theory, Keynes and PIH Bibliog Brief intro/motivation Today, Consumpt...

LEEDS UNIVERSITY BUSINESS SCHOOL Advanced Macroeconomics LUBS 3505 2024-2025 Lecture 2. Consumption Theory, Keynes and PIH Bibliog Brief intro/motivation Today, Consumption includes “Expenditure … on goods and services. … durable … and non-durable products” (C+S’15, p.11) Conflicting explanations and policy implications Keynesian Permanent Income consumption Hypothesis (PIH) optimal, theory New-Keynesian Behavioural Largest component of UK’s GDP. For data on other economies, see Table 1.1 (C+S’15, p.3) In last decades, consumption-led cycles, (e.g. USA, UK or Spain) Policy implications for fiscal policy are very important Figure 1.1 Components of GDP in the UK as a % of GDP between 1948 and 2010 Source: UK Office for National Statistics Outline - Keynesian Consumption - Permanent Income Hypothesis (PIH) - Overview - Assumptions “You said we did” - Optimal Consumption Lagrange refreshing materials - PIH Predictions - Summary Keynesian Consumption. IS curve So far, we have formulated consumption as: Where, 𝐶=Aggregate consumption expenditure. 𝑐0 =autonomous consumption, 𝑐1 =marginal propensity to (1) 𝐶 = 𝑐0 + 𝑐1 𝑦 − 𝑇 consume and it is between 0 and 0 < 𝑐1 < 1. 𝑦=income or output. 𝑇=Total taxes net of transfers Exogenous Endogenous, depending component on income level This means that consumption depends on current income (𝒚). However, in the 1950s, Friedman, Modigliani and Brumberg noted that this representation has several problems. According to eq (1), Consumption… … should be very volatile, change every time current income (𝒚) changes, eg. when 𝒚 = 𝟎, then 𝑪 = 𝒄𝟎. However, consumption is very stable. Does not differentiate between temporal from permanent income change (e.g. one-off bonus) and a permanent change (e.g. pay-rise). But we know that they have very different effects on consumption. There is no forward-looking behaviour of individual, consumption depends on today’s income only. Permament Income Hypothesis (PIH). Overview  Friedman, argued that instead, Individuals prefer to smooth consumption and are forward-looking, therefore… Y, C Savings Consumption De-savings 𝑐 ∗ =% life-time income Borrow Income Start Time Promotion Retirement work Individuals decide how much to consume over their life-time, Considering not only their current but also future income. Choose to consume a proportion of their life-time income or some “average income individuals expect to earn over their lifes”. And use borrowing and savings to balance consumption over their live-time. Friedman referred to this life-time income as permanent income, thus, Permanent Income Hypothesis (PIH). Modigiliani and Brumberg presented a similar model based on life-cycle. PIH. Assumptions 1. Representative individual/household/family. Problematic heterogenous agents. 2. Rep. individual aims for smooth or stable consumption: Utility function grows on consumption But at diminishing rate, ‘cos rep indiv wants stable consumption Formally: 𝑢 𝑐𝑡 = 𝑙𝑛 𝑐𝑡 Where, 𝑢 is the utility, 𝑐𝑡 is consumption in period 𝑡 This 𝑢 𝑐𝑡 denotes consumption smoothing because: 𝜕𝑢 1 = > 0, 𝜕𝑐 𝐶𝑡 𝜕2 𝑢 1 and =− < 0, 𝜕𝑐 2 𝑐𝑡2 Graphically: 𝑢(𝑐𝑡 ) 𝑢 𝑐𝑡 = 𝑙𝑛 𝑐𝑡 𝑐𝑡 PIH. Assumptions 3. Rep. individual is forward looking. Hence, decide how much to consume by maximising present value of expected utility over life-time, 𝑽𝑬𝒕 : Lives ∞ periods: 1 1 𝑉𝑡𝐸 = 𝑢 𝑐𝑡 + 𝐸 𝑢(𝑐𝑡+1 ) + 𝐸 𝑢(𝑐𝑡+2 )+⋯ (2) 1+𝜌 1 1+𝜌 2 𝝆=discount rate to calculate todays value of future consumption (in utility terms). Also known as consumers rate of time preference, or impatience. Equation (2) can also be written in compact form as: ∞ 1 𝐸 𝑉𝑡𝐸 =෍ 𝑖 𝑢(𝑐𝑡+𝑖 ) 1+𝜌 𝑖=0 Lives 𝟐 periods: 1 How does 𝝆 measures impatience? 𝑉𝑡𝐸 = 𝑢 𝑐𝑡 + 𝐸 𝑢(𝑐𝑡+1 ) (2’) 1+𝜌 1 larger 𝝆 smaller value of future consumption (in utility Using 𝑢 𝑐𝑡 = 𝑙𝑛 𝑐𝑡 : terms)More impatient 1 𝐸 (2’’) 𝑉𝑡𝐸 = 𝑙𝑛 𝑐𝑡 + 1 𝑙𝑛 (𝑐𝑡+1 ) 1+𝜌 PIH. Assumptions cont’ 4. Rep individual makes consumption decisions, considering her life-time income (𝚿𝒕𝑬 ): Lives ∞ periods: 11 𝐸 𝐸 1 Ψ𝑡𝐸 = 1 + 𝑟 𝐴𝑡−1 ++𝑦σ𝑡 ∞+ 𝑖=0 1+𝑟 𝑦 𝑦 𝑡+𝑖 1 𝑡+1 + 𝐸 𝑦𝑡+2 + ⋯ (3) 1+𝑟 𝑖 1+𝑟 2 ∞ or 1 𝐸 Ψ𝑡𝐸 = 1 + 𝑟 𝐴𝑡−1 + ෍ 𝑖 𝑦𝑡+𝑖 1+𝑟 𝑖=0 Lives 𝟐 periods: 𝐸 1 𝐸 Ψ𝑡 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 1 𝑦𝑡+1 1+𝑟 Where, 𝐴𝑡−1 stands for assets from previous period, 𝒓 is interest rate, hence, 1 + 𝑟 𝐴𝑡−1 =value 𝐸 assets plus interest paid for holding them. 𝑦𝑡+𝑖 is expected (post-tax) income in period 𝑡 + 𝑖. 5. No-inheritance. All income is consumed within an individual’s life, hence, over a life- time, all consumption needs to be equal to all income. Budget constraint of rep. indiv.: ∞ ∞ 1 𝐸 1 𝐸 (4) For ∞ periods: ෍ 𝑐𝑡+𝑖 = 1 + 𝑟 𝐴𝑡−1 + ෍ 𝑦𝑡+𝑖 1+𝑟 𝑖 1+𝑟 𝑖 𝑖=0 𝑖=0 1 𝐸 1 𝐸 (4’) For 𝟐 periods: 𝑐𝑡 + 1 𝑐𝑡+1 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 1 𝑦𝑡+1 1+𝑟 1+𝑟 6. Rep. individual is able to borrow without constraints  ’everyone is credit-worthy all the time’. Problematic, eg. Unemployed? PIH. Optimal consumption path, 2 periods case Hence, Individual utility maximizing problem for two periods* can be written as: 1 For a refresher, see video in 𝐸 𝑀𝑎𝑥 𝑉𝑡𝐸 = 𝑙𝑛 𝑐𝑡 + 𝑙𝑛 (𝑐𝑡+1 ) (2’’) Other readings 𝐸 1+𝜌 1 𝑐𝑡 , 𝑐𝑡+1 1 𝐸 1 𝐸 s.t 𝑐𝑡 + 𝑐𝑡+1 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 𝑦𝑡+1 (4’) 1+𝑟 1+𝑟 Hence, we construct the relevant Lagrangian: 1 𝐸 1 𝐸 1 𝐸 ℒ = 𝑙𝑛 𝑐𝑡 + 1 𝑙𝑛(𝑐𝑡+1 ) −λ 𝑐𝑡 + 𝑐𝑡+1 − 1 + 𝑟 𝐴𝑡−1 − 𝑦𝑡 − 𝑦𝑡+1 1+𝜌 1+𝑟 1+𝑟 Take first derivatives: 𝜕ℒ 1 … and equate to 0 to find FOC: (5) = −λ =0 𝜕𝑐𝑡 𝑐𝑡 𝜕ℒ 1 1 1 = − λ =0 (6) 𝐸 𝐸 1 + 𝜌 𝑐𝑡+1 1+𝑟 𝜕𝑐𝑡+1 𝜕ℒ 1 𝐸 1 𝐸 = − 𝑐𝑡 + 𝑐𝑡+1 − 1 + 𝑟 𝐴𝑡−1 − 𝑦𝑡 − 𝑦𝑡+1 =0 (7) 𝜕λ 1+𝑟 1+𝑟 *For exam purposes you are only expected to know the 2 period case! (solved here & in Seminar) PIH. Optimal consumption path, 2 periods case 1 (5’) Solve (5) for λ: =λ 𝑐𝑡 1 1 1 1+𝑟 1 Solve (6) for λ: 𝐸 =λ 1+𝑟 ; (6′) 1 + 𝜌 𝑐𝑡+1 𝐸 =λ 1 + 𝜌 𝑐𝑡+1 1 𝐸 1 𝐸 Regroup (7): 𝑐𝑡 + 𝑐𝑡+1 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 𝑦𝑡+1 (7′) 1+𝑟 1+𝑟 Equate λ from (5’) and (6’) to find: 1 1 1 = 1+𝑟 (8) 𝑐𝑡 1+𝜌 𝐸 𝑐𝑡+1 in the optimum, we consume today, up to the point that Marginal utility of an extra Present value of Marginal utility of a pound saved pound in consumption today (in utility terms) today to be consumed tomorrow Regrouping (8)… 1+𝜌 𝐸 Eulers equation: dynamic optimality condition, relates 𝑐𝑡 = 𝑐 (8’) 𝐸 1 + 𝑟 𝑡+1 optimal levels of 𝑐𝑡 and 𝑐𝑡+1 over different periods PIH. Optimal consumption path, 2 periods case 1+𝜌 𝐸 𝑐𝑡 = 𝑐 (8’) Euler equation 1 + 𝑟 𝑡+1 Euler’s equation tells us that optimal 𝒄𝒕 compares to 𝒄𝑬𝒕+𝟏 depending on degree of patience, 𝝆, and interest rates, 𝒓. There are three possibilities: 𝝆 = 𝒓  𝟏 + 𝝆 = 𝟏 + 𝒓  𝒄𝒕 = 𝒄𝑬𝒕+𝟏 : Constant consumption over time 𝝆 > 𝒓  𝟏 + 𝝆 > 𝟏 + 𝒓  𝒄𝒕 > 𝒄𝑬𝒕+𝟏 : Impatient consumer, consumption falls over time. 𝝆 < 𝒓  𝟏 + 𝝆 < 𝟏 + 𝒓  𝒄𝒕 < 𝒄𝑬𝒕+𝟏 : Patient consumer, consumption grows over time. We will solve for the case when 𝝆 = 𝒓: For case 𝜌 ≠ 𝑟, see Seminar 1 1 1+𝜌 𝐸 𝑐𝑡 = 𝑐 (8’) 1 + 𝑟 𝑡+1 PIH. Optimal consumption path, 2 periods case From (8’), we know: 𝐸 𝑐𝑡+1 = 𝑐𝑡 From (7’), budget restriction: 1 𝐸 1 𝐸 𝑐𝑡 + 𝑐𝑡+1 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 𝑦𝑡+1 1+𝑟 1+𝑟 𝐸 Plug 𝑐𝑡+1 from (8’) into (7’): 1 1 𝐸 𝑐𝑡 + 𝑐𝑡 = 1 + 𝑟 𝐴𝑡−1 + 𝑦𝑡 + 𝑦𝑡+1 1+𝑟 1+𝑟 Recall from (2), = Ψ𝑡𝐸 1 Hence: 1+ 𝑐𝑡 = Ψ𝑡𝐸 1+𝑟 2+𝑟 Adding square bracket: 𝑐𝑡 = Ψ𝑡𝐸 1+𝑟 Isolate for 𝑐𝑡 : 1+𝑟 𝐸 𝑐𝑡 = Ψ 2+𝑟 𝑡 Plug into (8’): 𝐸 1+𝑟 𝐸 𝑐𝑡+1 = Ψ 2+𝑟 𝑡 PIH. Optimal consumption path, 2 periods case Optimal intertemporal consumption 𝒄𝒕 = 𝒄𝑬𝒕+𝟏 , when 𝝆 = 𝒓: 1+𝑟 𝐸 1+𝑟 𝐸 𝑐𝑡 = Ψ 𝐸 𝑐𝑡+1 = Ψ 2+𝑟 𝑡 2+𝑟 𝑡 1+𝑟 𝑐𝑡 is proportion 2+𝑟 < 𝟏of life-time expected income (Ψ𝑡𝐸 ). 𝑐𝑡 only changes when expected life-time income (Ψ𝑡𝐸 ) changes. Contrary to Keynesian Consumption, where what matters is current income, 𝒚, see eq(1) If you look at textbook, ∞ periods solution (recall for exam only 2 periods): 𝑟 𝑐𝑡 = Ψ𝑡𝐸 1+𝑟 Same applies: 1+𝑟 𝑐𝑡 is proportion < 𝟏of life-time expected income (Ψ𝑡𝐸 ). 2+𝑟 𝑐𝑡 only changes when expected life-time income (Ψ𝑡𝐸 ) changes. PIH Predictions. Anticipated changes How does consumption change when income changes (𝑦𝑡 )? It depends on whether the changes is anticipated or unanticipated. Anticipated: Change is known before it occurs, eg. My application for a better-paid job that starts in 3 months is accepted... 𝜕𝑐𝑡 𝜕𝑐𝑡 Y, C PIH: >0 =0 𝜕Ψ𝐸 𝑡 𝜕𝑦𝑡 Income Path Consumption, PIH Path Consumption, Keynes Borrow “Excess 𝜕𝑐𝑡 Keynes: > 0 sensitivity” 𝜕𝑦𝑡 News of greater Actual rise Time future income o Once change is known, this new information is incorporated to budget restriction and optimal consumption changes accordingly, using borrowing. o Hence, when the change occurs, actual rise, it has no impact. o Contrary to Keynes predictions on Consumption that suggests: PIH Predictions. Un-anticipated changes Unanticipated: Change in income takes place at same time that it is announced. Permanent  income changes forever, Temporary  one-off change in income, eg. pay-rise e.g. one-off bonus Y, C Y, C Income Path Consumption, PIH Path Consumption, Keynes Path Consumption, PIH Income Path Consumption, Keynes News & Time News & change Time change Permanent income change (solid grey): o New information is incorporated to budget restriction and optimal consumption changes at the same time than income, there is no need for borrowing. o No difference with Keynesian consumption (cos time of news and change coincide) Temporary income change (dashed grey): o Temporary or one-off improvement in income, extra income is spread over the entire future consumption, hence, consumption rise very modestly. o Contrary to Keynes, where we observe a peak. Again, “excess sensitivity”. Summary We have presented two explanation of consumption:  Keynes’ consumption function, which depends on current income Keynes (𝒚): Problems: (1) 𝐶 = 𝑐0 + 𝑐1 𝑦 − 𝑇 Implies very volatile Consumption Does not differentiate temporal /permanent No forward-looking behaviour  Permanent Income Hypothesis (PIH) proposed by Friedman: 𝐸 Optimal intertemporal consumption 𝑐𝑡 = 𝑐𝑡+1 , when 𝜌 = 𝑟: 1+𝑟 𝐸 𝑐𝑡 is proportion of life-time expected income (Ψ𝑡𝐸 ), 𝑐𝑡 = Ψ 2+𝑟 𝑡 rather than current income. consumption under… PIH Keynes Anticipated Permanent Δ𝑐𝑡 > 0 when 𝚫𝚿𝒕𝑬 known, 𝜕𝑐𝑡 Consumption, only changes, > 0, before Keynes predicts 𝜕Ψ𝐸 𝑡 Δ𝑐𝑡 > 0 when current Unanticipated income changes 𝚫𝒚𝒕 > 𝟎. Permanent Δ𝑐𝑡 > 0 when 𝚫𝚿𝒕𝑬 known. 𝜕𝑐𝑡 But same time than Keynes predicts When: >0 𝜕𝑦𝑡 Temporary Δ𝑐𝑡 > 0 when 𝚫𝚿𝒕𝑬 known, SMALLER than permanent Bibliography: Basic readings Carlin and Soskice (2015) Macroeconomics. Institutions, Instability and the Financial System. Ch.1 sections 1.2.1 (pp.11-12), 1.2.5 (pp.20-23), 1.2.7 (pp.32-34 Forward looking IS intuition) Appendix 1.4.2 (pp.37-38) Or Carlin and Soskice (2006) Macroeconomics. Imperfections, Institutions and Policies. Ch.7 sections 1.1 (pp.206-209), 1.3 (pp.214-215), 1.4 (pp.215-216). See Minerva/Content/Basic reading NOT AVAILABLE ELECTRONICALLY @library Advanced readings Carlin and Soskice (2015) Macroeconomics. Institutions, Instability and the Financial System. Ch.1 sections 1.2.7 (pp.32-34 Forward looking IS intuition) Carlin and Soskice (2006) Macroeconomics. Imperfections, Institutions and Policies. Ch.7 sections 1.2 (pp.209-214) and 1.6 (pp.218-220). Ch15 Section 2 (pp.574-578) Romer D. (2012) Advanced Macroeconomics. 4e. Ch.8.1 & 8.4 Jappelli T. and Pistaferri L. (2010) The consumption response of to income changes. Annual Review of Economics. 2:1 pp.479-506 https://www.annualreviews.org/doi/pdf/10.1146/annurev.economics.050708.142933 Other readings: Galanchuk A. “How to maximize utility using Lagrange” Youtube.com. https://www.youtube.com/watch?v=YNNEjUqqj0c Back