Download Investment - The Profs

 F ir s t - C la s s U n iv e r s it y T u t o r s
Investment In economics, Investment is defined as an increase in the capital stock. A capital good is any good that produces other goods, such as ploughs for ploughing, trucks for hauling, cranes for heavy lifting, robotic arms for building cars and diggers for digging. Investment is therefore hugely important for increasing the productivity of our workforce. Imagine a world in which we did not have any capital goods. There are three types of investment: 1) Business Fixed Investment Business spending on capital goods 2) Residential Investment Consumers purchasing a new house 3) Inventory “Investment” When a firm does not sell its stock and effectively “buys” the surplus Important Note: Saving is not Investing In everyday language we often say that I’m making an investment with my money, or I’m investing my money in the bank. This is not how we mean it in economics. In economics, we call these actions saving. How Does a Firm Decide How Much to Invest? We are interested in the first the first type of investment. Investment by firms on capital goods. Firms buy more capital to gain from the increased productivity they bring. For example, a construction firm will buy cranes because this will far outweigh the costs of hiring the amount of labour needed to carry building materials up 20+ flights of stairs! Therefore, investment is an important way of remaining profitable. The Investment decision depends upon: 1) The Present Discounted Value of the Additional Expected Profits it Generates (i.e. it’s profitability) 2) The Cost 1 F ir s t - C la s s U n iv e r s it y T u t o r s
The Two Period Investment Model Assumptions: Two periods: current and future. After the second period, the firm shuts down and sells any remaining capital equipment because the owner (from the two period consumption model) is dead. There price of all output is 1, meaning that the value of the output is the amount of amount, so we do not have to deal with prices. Current Capital, K1, is given, but future, K2, needs to be decided by the firm. The first step is to calulate the present value of the firm’s profits In economics, all firms aim to maximise profits. Now that we have two time periods, the firm will want to maximise profits across both. Therefore, we must calculate the present value of current and future profits. Remember: you must never, ever directly compare two periods without discounting. 𝜋! = 𝑌! − 𝑤! 𝑁! − 𝐼 Our current profit is the value of our output, Y, minus the total wages paid, wN minus the money spent on investment, I. 𝜋! = 𝑌! − 𝑤! 𝑁! + (1 − 𝑑)𝐾! Our future profit is similar, except the firms does not invest in the future period because it is liquidated at the end of the second period, so instead it sells any remaining capital that has not been depreciated. Quick Maths: (1 – d)K2 is the amount of capital left after depreciation. Say that you have 100 machines (K=100) and the depreciation rate is 10% (d=0.1) meaning that one in ten machines breaks, or, more realistically, all machines lose one tenth of their value as they are now second hand. Without thinking too hard, we can see that the answer should be 90 machines because we had 100 and we lost 10%, of this number. Sub in for K and d: 100 − 0.1(100) 𝐾 − 𝑑𝐾 In other words, we have 90% of our machines remaining and so factorising out K we are left with: 0.9 ×100 (1 − 0.1)100 (1 − 𝑑)𝐾 2 F ir s t - C la s s U n iv e r s it y T u t o r s
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑂𝑢𝑡𝑝𝑢𝑡: 𝑌! = 𝑧! 𝐹 𝐾! , 𝑁! 𝐹𝑢𝑡𝑢𝑟𝑒 𝑂𝑢𝑡𝑝𝑢𝑡 𝑌! = 𝑧! 𝐹 𝐾! , 𝑁! These are our standard production functions, one in each period, where total output is a function of both capital and labour, and also Total Factor Productivity. How much capital does the firm have in the second period? It depends on how much the old capital depreciated, and how much new capital was added (this is investment because investment is defined as an increase in the capital stock) 𝐾! = 1 − 𝑑 𝐾! + 𝐼 We are going to rearrange this in the form 𝐼 = 𝐾! − 1 − 𝑑 𝐾! because we want our profit function in terms of labour and capital when we differentiate. Finally, putting all this information together, which means substituting in for both profits, then for both production functions and also for investment, we can write down our profit function: 𝜋!
𝑉 = 𝜋! + 1+𝑟
𝑌! − 𝑤! 𝑁! + (1 − 𝑑)𝐾!
𝑉 = 𝑌! − 𝑤! 𝑁! − 𝐾! − 1 − 𝑑 𝐾! + 1+𝑟
Doesn’t notation make this entire model so much simpler to learn? No? You don’t agree? How strange… The firm must decide on the levels of N1, N2, and K2 that maximise profits (not K1, which we assumed as given). In order to do that we optimise, meaning we partially differentiate with respect to each variable we have to determine and set equal to zero. Remember: when partially differentiating, everything that is not the variable we are interested in is treated as a constant! The first two differentials are the usual conditions for choosing the optimal amount of labour1: 𝛿𝑉
= 0: 𝑀𝑃𝑁! = 𝑤! 𝛿𝑁!
𝛿𝑉
= 0: 𝑀𝑃𝑁! = 𝑤! 𝛿𝑁!
I have therefore not explained how I got these answers, but you should try yourself as a practise before looking at the next one, which is significantly more challenging. 1 Hire until the quantity are which the cost of employing one more worker, w, equals the extra output they produce, MPN 3 F ir s t - C la s s U n iv e r s it y T u t o r s
𝛿𝑉
𝑀𝑃𝐾! + (1 − 𝑑)
= 0: − 1 +
= 0 𝛿𝐾!
1+𝑟
?!?!?!?!?!?!! How on earth did you get that? I’ll break it down into each partial differentiation of K2 𝛿
−𝐾! = −1 𝛿𝐾!
𝛿
𝑌 = 𝑀𝑃𝐾! 𝛿𝐾! !
By definition, the amount of extra output you get when you add an additional unit of capital is the marginal product of capital. This is exactly what the above equation says. 𝛿
(1 − 𝑑)𝐾! = (1 − 𝑑) 𝛿𝐾!
Putting all three of these together, along with (1 + r) we treat at a constant attached to our last two terms, we get the optimum quantity of K2 is at the level of output when: 𝑀𝑃𝐾! + (1 − 𝑑)
= 0 1+𝑟
𝑀𝑃𝐾! + (1 − 𝑑) = 1 + 𝑟 −1 +
𝑀𝑃𝐾! + (1 − 𝑑) = 1 + 𝑟 𝑀𝑃𝐾! − 𝑑 = 𝑟 Great! We have derived our optimal condition for how much to invest. But what an earth does it tell us? Trying to get some intuition for the above equation (is like trying to make a funny economics joke) Q: Why do firms invest in capital goods? A: To increase profits A Second Q: When do they stop buying machines A Second A: When they stop adding to profits. This is when the Marginal Cost of Investment equals the Marginal Benefit of investment The Marginal Cost of Investment is the cost of an additional unit of investment, which we are going to assume is equal to 1. Why? Because one additional unit of investment will reduce the present value of profits by one. 𝑀𝐶 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 = 1 Now we are assuming that this is constant. 4 F ir s t - C la s s U n iv e r s it y T u t o r s
However, the Marginal Benefit of Investment is going to decrease as we increase the number of machines in the second period because the Law of Diminishing Returns says so.2 The benefit from one additional unit of investment will be: 1) The additional output of one extra unit of capital in the future, this is MPK2 (note: this is an increase in future productivity only) 2) Adds an additional (1 – d) units of capital to be sold at liquidation 𝑀𝑃𝐾! + (1 − 𝑑)
𝑀𝐵 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 = 1+𝑟
3) Any benefit from investment will only occur next period because we do not get the machine until the future period, therefore, we must discount At equilibrium: MB[Investment] = MC[Investment] 𝑀𝑃𝐾! + (1 − 𝑑)
= 1 1+𝑟
And again we get: 𝑀𝑃𝐾! − 𝑑 = 𝑟 “MPK2 – d” is the net marginal product of capital, meaning the benefit that we get from one extra unit of investment, after taking into account depreciation. “r” is the opportunity cost of capital, meaning it is the next best alternative to investing in capital, the interest we would have received had we put the money into a bank instead. 2 Having chosen the optimal level of labour, adding more of one factor will result in less and less additional output and therefore, less benefit to the firm. 5 F ir s t - C la s s U n iv e r s it y T u t o r s
That didn’t really make anything clearer at all… Can you say it in words please KEY: The representative consumer owns the firm and so indirectly owns the capital stock. The representative consumer receives profits from the firms and so wants the firms to choose levels of investment that maximise profits. So the consumer can either put their savings into a bank account or invest in capital Now, MPK2 – d is the net marginal product of capital because it is the return (think: money) that the consumer will ultimately receive when the firm adds one additional unit of capital. And r is the opportunity cost of capital because it is the alternative to investing in capital, it is the return (think: money) that the consumer receives from saving in a bank account instead of investing in more capital. So it would not be optimal for the firm (on behalf of the consumer) to invest beyond the point where MPK2 – d = r because this is when MPK2 – d < r meaning that the benefit from an extra unit invested in capital is less than the consumer could have got from just saving their money into a bank account. Equally, the firm would not want to invest at less that this point, when MPK2 – d > r because now the return from investing in capital is higher than the interest rate that the bank pays out. 6 F ir s t - C la s s U n iv e r s it y T u t o r s
The Optimal Investment Schedule for the Representative Firm… … is an overly complicated way of saying the point at which the firm should invest until. 𝑀𝑃𝐾! − 𝑑 = 𝑟 As the interest rate, r, increases, the firm will find it more expensive to borrow for investment and so will invest less, leading to a smaller K2 but a higher MPK2 (because of the Law of Diminishing Returns) As we just saw, r is also the opportunity cost of capital and so when this increases, the representative consumer will want to put more of their money into a bank account, and so less into future capital. Therefore, the relationship between the interest rate and the optimum level of investment is negative`; interest rate, r MPK2 – d [net marginal product of capital] r1 r2 ID I1 I2 Investment demand, ID As the above diagram shows, when the interest rate falls, it is cheaper to buy more future capital 7 F ir s t - C la s s U n iv e r s it y T u t o r s
What causes the firm to change its investment demand schedule? interest rate, r r1 [MPK2 – d]1 I1 [MPK2 – d]2 Investment demand, ID I2 Anything that increases MPK2 1) éfuture Total Factor Productivity, z2 2) êin the current capital stock, K1 8 F ir s t - C la s s U n iv e r s it y T u t o r s
Investment with Asymmetric Information These investment models all suggest that a firm’s investment decision depends primarily on expected future profits, how much the future capital will add. However, empirical studies did not find this. Investment seems to also be affected by current profits because some firms with highly profitable investment opportunities don’t seem to invest as much as the theory predicts. This is likely to be because firms cannot always receive the loans they require for their investment opportunities. Firms are likely to find it harder to secure these loans if their current profits are low. This could be explained by Asymmetric Information There are many firms: some are will pay back their bank loans, others will not. Banks cannot distinguish between “good” and “bad” firms. Therefore, the bank must charge an default premium which is extra interest to cover the risk of loans not being repaid. This causes the rate of interest at which banks charge for loan repayments, r Loans will be higher than the standard lending rate, r. 𝑟 !"#$% − 𝑟 = 𝑥 Glossary r Loans the interest rate that banks charge on loans to firms for investment r the standard interest rate that banks charge for saving and borrowing x the default premium (the price of risk). This is the extra interest that banks charge due to not being able to tell which firms will repay their loans, and which firms will default, which is costly for the bank Quick Maths There are ten firms, each asking for £100 of investment from the bank. However, half of these firms/investments are “bad” and will not be repaid. The bank usually charges 10% on loans. If it charges this standard rate when there is risk of defaults in the market, it will only receive its ten per cent from half the firms, making its return effectively 5%: • £1000 lend • The five “good” firms each pay back the £100 and the £10 of interest = £550 • The five “bad firms defaulted and paid back nothing • Therefore, the bank lend out £1000 and only received £550 Unless the bank charges a default premium to cover this risk, it will quickly run out of money! When either the risk of defaulting or the level of asymmetric information increases, x, the default premium will have to increase as well, causing the loan rate in increase: 9 F ir s t - C la s s U n iv e r s it y T u t o r s
𝑀𝑃𝐾! − 𝑑 = 𝑟 Loan 𝑀𝑃𝐾! − 𝑑 = 𝑟 + 𝑥 𝑀𝑃𝐾! − 𝑑 − 𝑥 = 𝑟 As the level of asymmetric information increases, banks charge higher default premiums causing investment to become less profitable and so it falls. Graphically, the default premium reduces the net marginal product of capital for any given r. This shifts out investment demand curve inwards and so reduces investment. interest rate, r MPK2 – d – x MPK2 – d – éx r1 Investment I1 I2 demand, ID 10