Tutorial 4

A consumer lives for two periods. He has incomes 𝒀𝟏 and 𝒀𝟐 in the two different periods, and he can borrow or lend at an interest rate 𝒓. He maximises the lifetime utility function 𝑼 = 𝒖(𝑪𝟏)+ 𝒖(𝑪𝟐)/ (𝟏+ 𝝆) Write down the consumer's budget constraint and substitute this into the lifetime utility function in order to find the first-order condition for consumption. CONFUSED ABOUT DIFFERENTIATION

!!!!! The budget constraint is C1 + C2/ (1+r) = Y1 + Y2/ (1+r) Re-arranging this for C2 yields C2 = (1+r)Y1 + Y2 - (1+r)C1 Substituting into the lifetime utility function and maximising with respect to C (More working out on paper) U'(C1)/ ((U'(C2)/(1+p))=1+r

Matrixes question

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More matrixes

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Assume that the real interest rate and the subjective rate of discount are both zero. A person is 20 years old and expects to live to 80 - working for an after-tax salary of £50,000 per year until the age of 60, then retired with a state pension of £5,000 per year from age 60 until death (and no pension from work - only whatever savings they have). To simplify, we assume that the person acts as if future incomes and the length of life were known with certainty. What is his optimal level of consumption and how much do they save?

For the first order condition: u'(Ct)/ u'(Ct+1) = (1+r)/ (1+p), we see that if r=p, real consumption must be constant over time, so we have C=((50kx40)+(5kx20))/ 60 = 2100k/60 = 35k Optimal consumption is thus 35 000 per year in all years - the person will save £15 000 per year while they work, and dissave £30 000 per year while retired

How might the existence of consumer durables reduce the accuracy of year-to-year changes in consumer spending as a gauge of year-to-year changes in consumption?

If a consumer buys a dishwasher this year, then it shows up in this year's consumer spending, but assuming the dishwasher lasts for several years, most of the consumption benefits will be accrued in later years (when the only expenditures will be on electricity and soap). Thus consumer spending average over long periods should reflect consumption, but over short periods the trends could diverge somewhat.

Economists often distinguish 'permanent' from 'temporary' income. Explain what these terms mean. Which one of these measures do you think is more strongly related to consumption?

In general, we can think of current income as including two components: one reflects long-term or permanent factors such as the level of skills and human capital, while the other reflects temporary or transitory factors (like being out of the labour force due to human capital investments, fertility, job loss, and the like). For any given household, permanent income changes slowly over time. When it does, it is because of unpredictable events, like the way in which new technologies affect the price and quantity demanded of one's skills (for better or worse), along with unexpected promotions or demotions, a change in the local economy that affects wage levels, and other factors. Consumption should be strongly related to permanent income, and ideally not related at all to temporary income (which should be smoothed away through saving and borrowing).

Suppose you are concerned about inequality and its effects on wellbeing. Is it more important to look at data on income inequality, or consumption inequality? Why? Which of these approaches do you think is more common in practice?

It is more common to look at income inequality, but consumption inequality should be more important. Households prefer a smooth to a variable consumption flow. Hence, households would choose consumption to be a constant fraction of their permanent or lifetime income, not current income. Because current income can be highly volatile from one year to the next, it may give a partial snapshot of people's living standards. The extent to which households can achieve a smooth consumption flow depends on the tools they have to move resources over time and stats of nature. Savings can be used to absorb certain income shocks and can be accumulated for such a purpose. Other tools for consumption may include access to credit and insurance markets, and interpersonal and government transfers. The ability to move resources across time and states explains why consumption may not track income. Consumption may exceed current income as a consumer is borrowing (permanent income is above current income, as in the case of a medical student taking out loans in the expectation of higher future earnings) or it may be below current income because the consume is saving (and the doctor is now repaying medical school loans). Large wealth effects (people spend more as the value of their assets rise) can also have a considerable influence on consumption independently of income. It is then possible for the income distribution to reveal no changes in well-being even though the underlying consumption distribution is shifting in response to wealth effects. Consumption may vary from income for other reasons as well. For example, consumption falls below current earnings and wages because of taxes paid, and above them because of government transfers - a different form of consumption smoothing especially relevant for households at the bottom of the distribution. Even if full smoothing is not feasible, perhaps because of borrowing restrictions and imperfections in insurance and credit markets, some consumption smoothing would still occur.

Do you think people's savings decisions are optimal in the economists' sense? Why or why not? Do you think it might still be helpful to model people's savings behaviour as if they were rational?

Lack of cognitive ability and lack of will prevent people from making optimal decisions.

A consumer lives for two periods. She maximises the lifetime utility function subject to a budget constraint (note: 𝑨𝟏 is initial wealth) 𝑼 = 𝐥𝐧(𝑪𝟏)+ 𝐥𝐧(𝑪𝟐)/ (𝟏+𝝆) & 𝑪𝟏 + 𝑪𝟐/(𝟏+𝒓) = 𝑨𝟏 + 𝒀𝟏 + 𝒀𝟐/ (𝟏+𝒓) (a) What is the first-order condition for consumption in this case? (b) Use the first-order condition to substitute for 𝑪𝟐 in the budget constraint, and solve for 𝑪𝟏. c) 𝝆 = 𝟎 and investigate how 𝒀𝟏 𝓵 , 𝑨𝟏, 𝒀𝟐 𝓵 , and 𝒓 affect 𝑪𝟏. Interpret the results. (d) Suppose that 𝑨𝟏 = 𝒀𝟏 𝓵 = 𝒀𝟐 𝓵 = 𝟏𝟎𝟎 and 𝝆 = 𝒓 = 𝟎. What is consumption in period 1? (e) Suppose that 𝑨𝟏 = 𝒀𝟏 𝓵 = 𝒀𝟐 𝓵 = 𝟏𝟎𝟎 and 𝒓 = 𝟎, and 𝝆 = 𝟎. 𝟏𝟎. What is consumption in period 1? Explain the difference compared with the previous case.

On paper

Q8. Given 𝐀 = [ 𝟐 𝟑 𝟓 𝟏 −𝟑 𝟏 𝟏 −𝟏 𝟐 ], 𝐁 = [ 𝟏 𝟐 𝟑 𝟕 −𝟓 𝟑 𝟏 𝟒 −𝟏 ], find AB.

On paper

Transpose the following matrices:

On paper

Imagine drawing a graph where the horizontal axis had a person's age, and the vertical axis was a measure of the real value of their income or consumption. Draw a rough sketch of what you expect the "income" line to look like, as well as what you expect the "consumption" line to look like. Give an intuitive explanation.

People want their consumption to be relatively constant over time (hence the flat blue line), but in order to achieve this, they need to save while working and dissave while retired (and while being educated)

4. How might government policy responses to income inequality and consumption inequality be different?

The distinction between temporary and persistent shifts in the wage or income distribution has important policy and welfare implications. Policies aimed at reducing inequality under the two scenarios are very different. In the first case, it is probably necessary to reduce in the endowments of human capital, whilst in the latter it may be sufficient to improve the access to smoothing mechanisms. In theory, permanent shocks are harder to absorb and insure and are thus more likely to be reflected in substantial changes in consumption and welfare. In contrast, temporary shocks are easier to smooth through borrowing or running down accumulated assets. Hence, if all changes in income inequality were of a transitory nature, we could expect no large changes in consumption inequality.

The chart below shows global real rates since 1980. What has the basic pattern been? What explanation would you give for the trend? Why do you think the Advanced and Emerging economies have been so similar? (These issues will be discussed further in the lectures in week 5, so feel free to postpone this question to next week!)

There is an overall decline in real rates from around 6% in 1980 to something more than 0% or 1% in recent years. In terms of the model we are discussing here, the decline in r could be caused by either a decline in p (e.g., as populations become more patient on average, possibly due to a greater education or life expectancy), or a decline in g (possibly due to slower technological progress) or some combination of the two. As for the similarity in the trend between advanced and emerging economies, this is basically a result of a 'no arbitrage' condition. Since money can move around, if the rates were much higher in one part of the world than another, the supply of savings would increase in the high-yield region (pushing rates down there) and decrease in the low-yield region (pushing rates up there) until they were roughly equal.

The figure below (reproduced from last week's tutorial sheet) shows that investment fluctuates much more than consumption. Both series are highly correlated with GDP. Are these observations consistent with the theories of investment and consumption presented in Chapters 3 and 4?

Yes. Consider investment. Because income changes tend to be fairly persistent, an increase in GDP increases expected future demand, and firms will invest so as to increase production capacity (the accelerator effect). As a consequence, investment fluctuates a lot. Considering consumption, because of the persistence in income changes, it is reasonable that consumption reacts quite strongly to GDP but fluctuates less than investment (or GDP itself)

Q3. Calculate the following and comment on your findings in light of your answer to the previous question. (a) Assume that private consumption is 60% of GDP and private consumption increases 3%. By how much (in percentage terms) will aggregate demand change? (b) Assume that private investment is 20% of GDP and private investment increases 5%. By how much (in percentage terms) will aggregate demand change?

a) AD will increase 1.8% b) AD will increase 1% Because consumption is typically a much larger share of GDP than investment, even proportionately smaller changes can be larger in absolute terms. In practice, both components contribute roughly equally to GDP fluctuations - the greater magnitude of consumption is offset by the greater volatility of investment.

Anna lives for two periods, 1 and 2. Her consumption in the two periods is determined by the tangency point between the indifference curve (𝑰) and the budget line in Figure 4.10 (reproduced below). (a) Is Anna saving or borrowing in period 1? (b) Suppose the real interest rate falls. Draw the new budget line so that it is tangent to the indifference curve 𝑰̃. How is Anna's consumption in period 1 affected by the decrease in the interest rate? (c) What happens to savings/borrowings in period 1? (d) How is Anna's consumption in period 2 affected by the decrease in the interest rate? (e) Explain the results in terms of income and substitution effects

a) ADD DIAGRAM Since consumption exceeds income in period 1, Anna is borrowing in period 1 b) A lower interest rate implies a flatter budget line. It still goes through the endowment point (Y1, Y2), because you can always choose to consume your income. Consumption in period 1 increases to C'. ADD DIAGRAM c) Anna borrows more in period 1 d) In the case drawn here, consumption in period 2 is unchanged. Although Anna borrows more, but the lower interest rate compensates so consumption next period is unchanged. Of course, this is not necessarily the case. e) Since Anna is initially borrowing in period 1, a lower interest rate makes her better off i.e., the income effect is positive. Since the substitution effect of a change like this is always positive, this means that the substitution and income effects go in the same direction, so consumption in period 1 increases. For period 2, substitution and income effects go in opposite directions. On the one hand, lower interest makes it less attractive to save and consume next period i.e., the substitution effect is negative in period 1. On the other hand, a lower interest rate makes this consumer better off i.e., the income effect is still positive. The income effect expresses the impact of increased purchasing power on consumption Substitution effect describes how consumption is impacted by changing relative income and prices

A consumer expects to live forever and he/she has a constant labour income of £30,000 per year, no assets, and a loan of £40,000. The nominal interest rate on the loan is 4% and inflation is 0%. a) What is the consumers' sustainable level of consumption ? b) By how much will the loan increase next year if he/ she consumes this amount? Explain.

a) C = 30 000 - (0.04 - 0) x 40 000 = 30 000 - 1600 = £28 400 b) The outstanding value of the loan is constant

A consumer lives for two periods. His subjective discount rate and the real interest rate are both 5%. (a) What is the relation between consumption in the two periods? (b) Use the lifetime budget constraint to calculate consumption as a function of incomes in the two periods (c) What will consumption be if incomes in both periods are 500? (d) How much will consumption increase if income in the first period increases by 100? Explain the result. (e) How much will consumption increase if income in the second period increases by 100? Explain the result.

a) From the first order condition: u'(Ct)/ u'(Ct+1) = (1+r)/ (1+p), we see that if r=p, consumption mist be constant over time b) C + C/ (1+r) = Y1 + Y2/ (1+r) implies: C=(Y1+Y2/1.05)/ (1+1/ 1.05) = (1.05/2.05)Y1 + (1/2.05)Y2 c) C=500 d) 𝛥𝐶/ 𝛥𝑌1 = 1.05/ 2.05 ≈ 0.51; Consumption will increase by 51 units, which is slightly more than half the increase in income e) 𝛥𝐶/ 𝛥𝑌2 = 1/ 2.05 ≈ 0.49; Consumption will increase by 49 units which is slightly less than half the increase in income. A future increase in income is worth less because it is discounted.

Assume that people are utility maximisers and that there is a competitive economy with a strong financial sector that works roughly as described in the textbook. Now consider each of the pairs of variables listed below. Is it plausible for inequality measured in one of them to be rising while inequality in the other one is falling? a) Total income and permanent income b) Total income and temporary income c) Consumption and permanent income d) Consumption and temporary income

a) Plausible. For example, suppose that permanent income is becoming more equal, for example because of targeted educational improvements which improve outcomes for the least well-educated, but that simultaneously temporary income becomes much more unequal, perhaps because people are more likely to take time out of the workforce to get more education, or to travel, or to have families. If the rise in inequality in temporary income is large enough relative to the decline in inequality of permanent income, then total income inequality can rise even as permanent income inequality falls. b) Plausible. Logic is the same as the previous question, except that now we need to assume for the example that the decline in inequality of permanent income is larger than the rise in inequality of temporary income, so that the inequality of total income and temporary income move in opposite directions. c) Implausible. Consumption of utility maximisers should very closely track permanent income, except in cases where they are constrained, for example because they want to borrow and they can't get a loan. d) Implausible. Not only possibly, in fact, but even likely, because consumption and temporary income should be totally uncorrelated.

Consider an economy where production grows exogenously at the rate of 1%. There is no capital stock and no investment so consumption equals production in each period. The subjective discount rate is 3%, the utility function is 𝒖(𝑪𝒕 ) = 𝐥𝐧(𝑪𝒕), and inflation is 5%. (a) What is the real interest rate in this economy? (Hint: Use the first order condition for consumption.) (b) What is the nominal interest rate in this economy? !!!!!!!

a) The first order condition for consumption is: u'(Ct)/ u'(Ct+1) = (1+r)/ (1+p) Since u(Ct)=ln(Ct). u'(Ct)=1/Ct, so we get Ct+1/ Ct = (1+r)/ (1+p) Ct(1+g)/ Ct = (1+r)/ (1+p) 1+g = (1+r)/ (1+p) 1+r = (1+p)(1+g) 1+r = (1.03)(1.01) ≈ 1.04 r ≈ 0.04 b) i = 𝑟 + 𝜋 = 0.04 + 0.05 = 0.09

Now we're going to re-calculate some of the values from the previous question in nominal rather than real terms. Suppose once again that: the real return averages 6% per year, that you're 22 and you put away £1000 in a tax-exempt ISA and you won't touch the money until you're 70. (a) If inflation averages 2% between now and your 70th birthday, how much nominal money would you get from your investment? b) What was the nominal "total interest rate" (in %) that you got over the 48 years? Was this better or worse than the rate that you got in part (b) of the previous question? (i.e. which return would you rather have?) (c) If you want to use up a constant real amount of the money that you saved in part (a) each year after you retire, and you expect to live forever, how much can you consume each year?

a) There is more than one way to solver this. One way is to note that if there is 2% inflation for 48 years, then £1 in today's money is equivalent to 1.02^48 = £2.59 in the money of the future. So the £16 394 of today's money will appear on your balance sheet as £42 412. Another way is to note that the annual return will now be (1.06)(1.02)=1.0812, and since £1000 x (1.0812)^48 = £42, 412 b) i(total) = (42412-1000)/ 1000 x 100% = 4141.2% This is a bigger number, but the real return is of course identical, so it is no better or worse c) If you take 8% per year, then this will preserve the nominal value of your pot of £42412, but the real value of the money will decay by 2% per year along with inflation. To preserve the real value of your pot, you can only collect the real part of your annual return. So in the first year, you can take out 0.06 x 42412 = 2545. And the following year, your pot will be 2% bigger in nominal terms (but have the same value in real terms) and you will be able to withdraw an amount which is nominally 2% bigger. but which actually has precisely the same value.

Around the world, the long-run average annual real return on common stocks has been in the range of 5% - 7%. For the sake of round numbers, we'll assume that the return going forward is 6%. a) Let's say that you're 22 years old and you expect to retire when you're 70 . If you suddenly find £1000 and put it in a tax-free ISA, how much real money will this yield when you're 70? b) Given an annual rate of 6%, what is the real effective total return (in %) over the whole 48 years of your saving c) If you want to use up a constant amount of the money that you saved in part (a) each year after you retire, and you expect to live forever, how much can you consume each year?

a) £1000 x (1.06)^48 = £16 394; of course in the money of the 2060s, the nominal amount will probably be a lot higher, but this is the amount in terms of current money b) T = (16394 - 1000)/ 1000 x 100% = 1539.4% c) Given that the interest rate is 6%, your lump sum of £16 394 will yield 0.06 x £16 394 = £ 983.64 in perpetuity. Rounding this up to £1000, this means that every extra £1000 that you save when you are 22 will yield an extra £1000 per year starting at age 70 to you and your heirs for ever and ever.

According to the theory presented in the textbook, there are four main factors that determine private consumption. For each factor, explain why it affects consumption and the magnitude of the effect.

𝐶 = 𝐶(𝑌, 𝑌^𝑒 , 𝑟, 𝐴) Consumption depends on current income, expected future income, the real interest rate and the level of wealth (assets) For most consumers, the expected stream of incomes in future periods is the most important factor. If expected future labour income increases and the increase is perceived as permanent we expect consumption to increase almost as much as income. Current income has a small effect on consumption if consumers are forward-looking. In order to smooth consumption over time, a forward-looking consumer will only consume the interest rate on a temporary increase in income. Credit-rationed consumers will instead spend any additional income that they get right away. One plus the real interest rate is the price of consumption today in terms of consumption next year. So an increase in the real interest rate has a substitution effect that reduces consumption. For a consumer with positive assets, there is also a positive income effect since the return on his/ her assets increases, but the income effect is negative for consumers who are borrowing. A forward-looking consumer will only consume a proportion corresponding to the real interest rate on the assets, which is 2-3% of the total amount. Again this is because the consumer prefers a smooth flow of consumption over time to an uneven one.