To Create or to Redistribute? That is the Question

Demetris Koursaros; Christos Savva; Nektarios Michail; Niki Papadopoulou

doi:10.1515/bejm-2023-0191

Article

To Create or to Redistribute? That is the Question

Demetris Koursaros , Christos Savva , Nektarios Michail and Niki Papadopoulou

Published/Copyright: November 27, 2024

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From the journal The B.E. Journal of Macroeconomics Volume 24 Issue 2

Abstract

This study delves into the persistent low GDP growth observed in the post-crisis period, despite aggressive financial easing measures. It explores how economic agents allocate available funds, either through investments in new capital creation or the acquisition of existing assets for capital gains (asset redistribution). The former approach bolsters total income and employment, while the latter reshapes wealth distribution among agents. Through a combination of theoretical insights and empirical evidence, we argue that during economic downturns, investors often find it more lucrative, and lenders consider it safer, to finance the repurchasing of existing assets rather than investing in new ones. This trend not only worsens recessions but also hampers the recovery process by limiting entrepreneurs’ access to funding and altering the economy’s capital structure. Moreover, given that asset redistribution tends to benefit the wealthy, a surge in inequality fosters a cycle of income redistribution, further exacerbating economic downturns.

Keywords: investment; business cycles; asset redistribution; inequality; capital misallocation

JEL Classification: E52; E25; E32; E21

Corresponding author: Demetris Koursaros, PhD, Department of Finance, Accounting and Management Science, Cyprus University of Technology, 30 Archibishop Kyprianos av, 3036, Lemesos, Cyprus, E-mail: demetris.koursaros@cut.ac.cy

Appendix

A The DSGE Model

A.1 Matching Technology

Given the household’s wealth, the Entrepreneurs (Investors) can maintain up to N ̃ t N ̃ t R lines of credit. Each period there are N _t N t R active credit lines and thus the Entrepreneurs (Investors) can search for N ̃ t − N t ( N ̃ t R − N t R ) new lines of credit. On the other side we have the banks that search for Γ_t lines of credit with customers. They allocate a fraction s t E of those for Entrepreneurs and the rest s t R = 1 − s t E to Investors. There are two matching functions, with constant returns to scale that determine the new credit line matches for Entrepreneurs and Investors with the lenders.

The matching function that determines the number of credit lines opened for Entrepreneurs over each period is

(25) M t E = A Γ N ̃ t − N t α Γ s t E Γ t 1 − α Γ

where N ̃ t − N t is the number of credit lines the Entrepreneurs can apply for and s t E Γ t the number of lines of credit the bank is offering to Entrepreneurs.

The matching function for loans to the Investors takes a similar form

(26) M t R = A Γ N ̃ t R − N t R α Γ 1 − s t E Γ t 1 − α Γ

where N ̃ t R − N t R is the maximum number of credit lines the Investor can apply for and 1 − s t E Γ t the corresponding number the bank assigns to them.

The probability a line of credit assigned to Entrepreneurs to find a match is

(27) ρ t E = A Γ s t E Γ t N ̃ t − N t − α Γ

while the probability a line of credit for the Investors to find a match is

(28) ρ t R = A Γ s t R Γ t N ̃ t R − N t R − α Γ .

Lastly, the probability for an Entrepreneur to successfully establish a credit line with the bank is

(29) q t E = A Γ s t E Γ t N ̃ t − N t 1 − α Γ

while the probability for the Investor to open a credit line with the bank is

(30) q t R = A Γ s t R Γ t N ̃ t R − N t R 1 − α Γ .

The Entrepreneurs (Investors) can optimally adjust the amount of credit lines to search for N ̃ t − N t N ̃ t R − N t R by acquiring housing which can be used as collateral for securing loans. There is no shortage of ideas in this economy and thus the number of differentiated products are restricted only by credit constraints.^[20] Banks also optimally adjust the number of credit lines to search for in the two different sectors, the production sector and the asset redistribution sector.

A.2 Common Households

Common households, denoted by “C”, are non-Ricardian and enter period t with D _t−1 units of deposits that earn a gross real return R t − 1 d . During period t common households purchase consumption C t C . In the market for deposits, since the Central bank determines the interest rate R t d on deposits in this model and the banks determine the amount of loans and thus the amount of deposits they need, the common households simply provide as many deposits as the banks need^[21] D _t.

Furthermore, households receive wage income N _t w _t L _t by selling labor to Entrepreneurs, where L _t denotes their labor effort, w _t is the wage rate and N _t is number of employed workers. At the end of period t they also receive dividends T _t from the final goods-producing firms’ profits and banks and face a budget constraint, which is defined as follows

(31) C t C + D t = N t w t L t + R t − 1 d D t − 1 + T t .

Common households get utility indicated by U(.) from consumption and disutility from labor indicated by G . . Hence, the common household, subject to the budget constraint in equation (31), chooses C t C , L t for all t to maximize its expected lifetime utility function defined as follows

(32) max C t C , L t t = 0 ∞ E t ∑ t = 0 ∞ β t U C t C − N t G L t

where 0 < β < 1 is the discount rate. By denoting λ t C the Lagrange multiplier on the budget constraint, the common household’s first-order condition (FOC) with respect to C t C is U ′ C t C = λ t C as usual. The FOC for labor hours L _t, is given by

(33) G ′ L t U C t C = w t

A.3 Household with Entrepreneurs

To avoid dealing with a “heterogenous-agent” model challenges, we assume for simplicity that Entrepreneurs pool their incomes together to determine their consumption and Housing decisions while the diversification decision is dealt on their own as in Section 4.1. The reason Entrepreneurs deal with diversification on their own is because income pooling would eliminate idiosyncratic risk. On the first hand, attaching all decisions to the large households eliminates the need to diversify due to the law of large number. On the other hand, assuming each Entrepreneur is a single household increases the model’s perplexity. Thus, without loss of generality we use a “hybrid” specification where the diversification decision is curried by the individual, while consumption and housing is dealt at a household level. For the specifics of the effect of avoiding income pooling check Section F.3 in Appendix and for diversification with more than one asset at a time check Section F.1 of the Appendix.

There is a continuum of identical Entrepreneurs in this household with the following characteristics. The Entrepreneurs choose the amount of consumption C t E and housing H t E to hold. Housing is also used as collateral to apply for credit lines albeit there is a utility cost for maintenance δ H H t E every period.

Entrepreneurs, at the end of period t choose C t E , H t E to maximize their expected lifetime utility function from consumption

(34) max C t E , Q t E , H t E t = 0 ∞ E t ∑ t = 0 ∞ β t U C t E + G H H t E − δ H H t E ,

subject to the constraint imposed by the budget^[22]

(35) C t E + P t h H t E − H t − 1 E = ∫ N t Ψ i t E d i .

where Ψ t E is the average cash flow expected by firms^[23]

(36) Ψ i t E = N t R N t Q t E V i , t + 1 − Q t E P t I + N t − N t R N t V i , t

The aggregate utility can be the same or different with the individual utility function in (2). The representative Entrepreneur has N _t active credit lines with the bank and thus owns N _t firms as the budget constraint (35) confirms. A number N t R of those firms are partially owned by the Entrepreneur and the rest N t − N t R are fully owned. Equation (36) implies that the exante diversification cash flow for the Entrepreneurs is uncertain. With a probability of N t R N t the ith firm is partially owned, distributing a fraction Q t E of V _i,t to the Entrepreneur and 1 − Q t E fraction to Investors in exchange for 1 − Q t E P t I cash. With a probability N t − N t R N t , the ith firm is fully owned by the Entrepreneur who is entitled to the full value V _i,t.

Over each period, the Entrepreneurs have N _t credit lines active and thus N _t operating firms. Therefore, the law of motion for the number of credit lines or firms in the following period is defined as follows

(37) N t + 1 = 1 − p d , t E N t + N ̃ t − N t q t E

As before, p d , t E is the probability that a firm defaults on its obligations to the bank and in this event the credit line is broken and the following period p d , t E N t credit lines disappear in equilibrium. The probability to default is therefore defined as follows:

(38) p d , t E ≡ Pr V i , t ≤ 0 = Pr V i , t − V ̄ t S t ≤ − V t ̄ S t = Φ − V t ̄ S t

where Φ is the cdf of the standard normal. The Entrepreneurs can search for N ̃ t − N t new credit lines, each one having a probability of q t E to find a match with the bank. The number N ̃ t depends on the funding-in-advance constraint, which is

(39) N ̃ t R t E w t L t ≤ m E E t P t + 1 h H t E

where N ̃ t denotes the maximum number of firms that can apply for a credit line given the collateral of the household, R t E denotes the gross interest rate of the loan and m ^E denotes the loan-to-value ratio.

Denoting with λ t E and μ t E the Lagrange multipliers of constraints (35) and (39) respectively, the first-order condition for consumption is given by

(40) λ t E = U c C t E

for housing is given by

(41) G H ′ H t E + β E t λ t + 1 E P t + 1 h + μ t E m E E t P t + 1 h = λ t E P t h

and the envelope condition^[24] for N ̃ t is given by

(42) β q t E E t λ t + 1 E Ψ t + 1 E = μ t E R t E w t L t .

By combining (41) and (42), then the following condition can be derived

(43) G H ′ H t E + β E t λ t + 1 E P t + 1 h + β m E q t E R t E w t L t E t P t + 1 h E t λ t + 1 E Ψ t + 1 E = λ t E P t h

stating that the entrepreneurs accumulate housing up to the point the marginal cost of housing which is the RHS of (43) is equal to its marginal benefit arising from (i) the marginal utility from housing, (ii) the value of the housing in the following period and (iii) the value of the additional credit lines that can be opened in the following period by increasing the amount of current collateral.

A.4 Household of Investors

The Investors, denoted by “R” (standing for redistribution), also pool their income together in the same way as the Entrepreneurs. As before, the Investors pick their asset portfolio on their own but pool their income to determine their consumption and housing for the same reason as the Entrepreneurial household.

Investors may default if the realized cash flow from the asset is less than the cost incurred to acquire it. Therefore, the probability of an Investor’s activity to default is defined as follows

(44) p d , t R ≡ Pr V t ≤ R t R P t I = Pr V t − V ̄ t S t ≤ R t R P t I − V ̄ t S t = Φ R t R P t I − V ̄ t S t .

An the end of period t, Investors maximize utility in a similar way as in the previous section.

(45) max C t , Q t R , H t R , N ̃ t R t = 0 ∞ E t ∑ t = 0 ∞ β R t U C t R + G H H t R − δ H H t R

subject to the budget constraint

(46) C t R + P t h H t R − H t − 1 R = N t R Ψ t R .

where Ψ t R is defined in equation (8). We assume that the Investors are risk neutral or fully diversified and thus the utility function U C t R is linear in simulation which is consistent with the utility of the individual investor in (8). Assuming that U C t R is concave only affects the relative importance of consumption versus housing.

Each period they have N t R credit lines open and thus N t R operating firms. Therefore, the law of motion for the amount of credit lines or firms in the following period is defined as follows

(47) N t + 1 R = 1 − p d , t R N t R + N ̃ t R − N t R q t R

where p d , t R is the probability that an Investor defaults on their obligations to the bank and the credit line is broken and thus, the following period p d , t R N t number of credit lines disappear. The Investor can search for N ̃ t R − N t R new credit lines, each one having a probability of q t R to find a match with the bank. N ̃ t R is the maximum number of firms they can partially acquire each period and depends on the amount of collateral they posses.

The funding-in-advance constraint, that signifies the maximum amount of credit lines to apply for depends on the available collateral in the following period (housing) and also on the amount owed to the bank next period and is defined as follows

(48) N ̃ t R R t R Q t R P t I ≤ m R E t P t + 1 h H t R

where N ̃ t R denotes the maximum credit lines for which they can apply, given the collateral of the household, R t R denotes the gross interest rate on the loan and m ^R denotes the loan-to-value ratio. Therefore, N ̃ t is such that the total amount that can be owed to the bank in the following period N ̃ t R R t R Q t R P t I must be no greater than the loan-to-value ratio m ^R times the value of housing in the following period.

Denoting with λ t R and μ t R the Lagrange multipliers of constraints (46) and (48) respectively, then the first-order condition for consumption is given by

(49) U ′ C t R = λ t R

for housing is given by

(50) G H ′ H t R − δ H + β E t λ t + 1 R P t + 1 h + μ t R m R E t P t + 1 h = λ t R P t h

and the envelope condition for N ̃ t R is given by

(51) β E t λ t + 1 R q t R Ψ t + 1 R = μ t R R t R Q t R P t I .

By eliminating μ t R by plugging (51) into (50) the following condition is derived

(52) G H ′ H t R − δ H + β E t λ t + 1 R P t + 1 h + β m R q t R R t R Q t R P t I E t λ t + 1 R E t P t + 1 h Ψ t + 1 R = λ t R P t h .

This equation has a similar interpretation as equation (43) since both household specifications are more or less symmetric.

A.5 Banks

In reality there are multiple channels from which credit can flow but since in the model we seek simplicity we assume there is only a single lender, a bank.^[25] The bank uses its deposits D _t to fully allocate to loans, with R t d denoting the borrowing rate. For this task, the bank is required first to open credit lines with potential customers, a process which is subject to search frictions. When a match is formulated and a credit line is open, the bank funds any amount required for the customer’s project. In the event that the customer defaults on the loan, the line of credit breaks and the customer must return to the state of searching for a new line of credit with a bank. Both Entrepreneurs and Investors in need of credit search for banks to open a credit line, implying that there are constant returns to scale matching functions which determine the allocation of credit lines to the two parties each period. The bank searches for lines of credit among Entrepreneurs and Investors. If the total amount of lines of credit is Γ_t, the bank opens s t E share of the total for the Entrepreneurs and s t R = 1 − s t E share for the Investors. For an extension of the partial equilibrium model in Section 4.1 with banks, check Appendix G.

A.5.1 Bank Headquarters

At the start of the period, the bank is committed to funding firm and Investor operations with an existing line of credit. Its decision has to do with the number of new lines of credit to offer Γ_t and the way those lines are going to be allocated to Entrepreneurs s t E and Investors 1 − s t E .

The law of motion for the number of credit lines to Entrepreneurs is defined as follows

(53) N t + 1 = 1 − p d , t E N t + ρ t E s t E Γ t

stating that credit lines in the following period include those for firms that have not defaulted in the previous period, as well as those which are newly created. New credit lines depend on the number of lines the bank allocates for Entrepreneurs s t E Γ t and the corresponding probability each line to match, ρ t E .

Similarly the law of motion for the lines of credit to Investors is defined as follows

(54) N t + 1 R = 1 − p d , t R N t R + ρ t R 1 − s t E Γ t .

In the end, the balance sheet of the bank is the following

(55) N t w t L t + N t R Q t R P t I + κ t E s t E Γ t + κ t R s t R Γ t = D t + Ω t

stating that the amount of lending and the cost to post new potential credit lines κ t E s t E Γ t and κ t R s t R Γ t are financed from deposits D _t and initial reserves Ω_t.^[26]

Hence, subject to the constraints based on the Entrepreneurs and Investors law of motions in equations (53) and (54) and the bank’s balance sheet in equation (55), the bank seeks to maximize the following objective

(56) J B t = max Γ t , s t E t = 0 ∞ N t R t E − p d , t E R L E w t L t + N t R R t R − p d , t R R L R Q t R P t I − R t d D t + E t β J B t + 1

by choosing the number of new lines of credit Γ_t ^[27] and the way they will be allocated to Entrepreneurs s t E and where it earns R t E minus the losses from default R L E that occur to a share p d , t E of all firms and R t R − p d , t R R L R from the N t R credit lines maintained with the Investors. In the former case the bank earns this return for the N _t credit lines maintained with Entrepreneurs and the loan amount to each firm is w _t L _t and the firm is free to ask for any amount as long as a credit line is open. For the latter term, the Investors borrow Q t R P t I amount to purchase firms, where P t I is the price offered for the firm and Q t R the share of the firm the Investor seeks to acquire.

The first-order condition with respect to Γ_t is

(57) R t d κ t E s t E + κ t R s t R = s t R ρ t R E t β d J B t + 1 d N t + 1 R + s t E ρ t E E t β d J B t + 1 d N t + 1

and with respect to s t E is

(58) R t d κ t E − κ t R = ρ t E E t β d J B t + 1 d N t + 1 − ρ t R E t β d J B t + 1 d N t + 1 R .

To interpret the way the allocation of credit lines between Entrepreneurs and Investors works, assume just for this example that the cost of opening credit lines is equal for each type of borrower and thus κ t E = κ t R . Therefore, equation (58) implies ^[28]

(59) ρ t E E t β d J B t + 1 d N t + 1 = ρ t R E t β d J B t + 1 d N t + 1 R .

Note that ρ t E and ρ t R are decreasing functions of s t E and s t R respectively (check equations (27) and (28)). If one sector promises higher return on the marginal credit line than the other, (for example if d J B t + 1 d N t + 1 > d J B t + 1 d N t + 1 R ), then the portion of the vacant credit lines s t E assigned to Entrepreneurs increases while the portion s t R going to Investors decreases. As more vacant credit lines are assigned to Entrepreneurs, the probability of each one to find a match drops which pushes ρ t E downwards ( ρ t R upwards) until equation (59) is satisfied.

To get the credit line posting conditions, the two envelope conditions are derived by taking derivatives of objective in equation (56) with respect to N _t and N t R , as follows

(60) d J B t d N t = R t E − p d , t E R L E − R t d w t L t + 1 − p d , t E E t β d J B t + 1 d N t + 1 E

and

(61) d J B t d N t R = R t R − p d , t R R L R − R t d Q t R P t I + 1 − p d , t R E t β d J B t + 1 d N t + 1 R

Lemma 3.

Given the objective function (56) and the associated first order conditions (57) and (58) along with the envelope conditions (60) and (61), the line of credit creation conditions for Entrepreneurs and Investors respectively are:

(62) κ t E R t d ρ t E = E t β R t + 1 E − p d , t + 1 E R L E − R t + 1 d w t + 1 L t + 1 + E t β 1 − p d , t + 1 E E t κ t + 1 E R t + 1 d ρ t + 1 E

and

(63) κ t R R t d ρ t R = E t β R t + 1 R − p d , t + 1 R R L R − R t + 1 d Q t + 1 R P t + 1 I + E t β 1 − p d , t + 1 R E t κ t + 1 R R t + 1 d ρ t + 1 R

The proof is in the Appendix. As the interest rate R t + 1 E is increasing in p d , t + 1 E R L E , condition (62) (and similarly (63)) imply that on the first hand, higher probability of default may benefit the bank by enabling it to charge a higher risk premium, but on the other hand it decreases the likelihood of keeping the credit line open in the future which captures the dilemma creditors face when making loans.

The profit of the bank in the current period is as follows

(64) Ω ̃ t = N t R t E − p d , t E R L E w t L t + N t R R t R − p d , t R R L R Q t R P t I − R t d D t

In the end the reserves and thus the initial capital in the following period is defined by

(65) Ω t + 1 = 1 − ω Ω ̃ t + 1 − δ B Ω t

where the bank pays ω fraction of the current earnings as dividends to the owners, while there is also a cost to maintaining bank capital and thus a fraction δ _B of capital/reserves is lost.

A.5.2 Interest Rates

The following proposition introduces the interest rates charged to Entrepreneurs and Investors from the banks.

Proposition 4.

If the bank splits the surplus of a matched credit line using Nash-bargaining where ζ is the portion going to its customers, then the interest rate to Entrepreneurs is:

(66) R t E = 1 − ζ R t V w t L t + ζ p d , t E R L E + ζ R t d − ζ q t E ρ t E R t d κ t E w t L t

and the interest rate to Investors is:

(67) R t R = 1 − ζ V t P t I + ζ p d , t R R L R + ζ R t d − ζ q t R ρ t R R t d κ t R Q t R P t I

The derivation of the above is in the Appendix. Equation (66) states that the interest rate charged to Entrepreneurs increases along with the expected revenue from the firm relative to its cost R t V w t L t . There is also a risk premium captured by the risk of default 1 − p d , t E and the loss given default R L E . Higher deposit rate R t d leads to a higher rate charged to Entrepreneurs. The last term depends on the market tightness for credit lines. If there are fewer applications for credit lines than the banks are offering to Entrepreneurs, it becomes more likely for a bank to fail to find a match and thus more likely for the bank to have to carry the cost of vacant credit lines κ t E next period. Therefore, the bank’s bargaining power deteriorates and thus charges a lower rate. As the bank needs to pay this cost from deposits, the borrowing rate is also relevant in this last term of equation (66). Equation (67) has a similar interpretation as in equation (66).

Due to the search and matching framework employed for the allocation of credit lines, both Entrepreneurs and Investors are not simply seeking to maximize profits but they are targeting the surplus^[29] from the match with a bank. Thus, maximizing (74) is not appropriate as R t E is not the true cost of borrowing. The intermediate firm determines the price p t x that maximizes the surplus from an active credit line with the bank by allocating a ζ portion to the Entrepreneurs and 1 − ζ to the Investors and thus the optimal strategy of the firm is to optimize this surplus. Whichever the firm’s pricing behavior is, the bank is going to set the interest rate to such a level that the firm eventually receives a ζ portion of the surplus. Therefore, the optimal strategy of the firm is to maximize that surplus. A more detailed discussion on this can be found in Trigari (2009) and Koursaros (2017).

Assume that the relative price of the intermediate firm is p i t x P t and the demand x _it. The following proposition determines the “true” cost of borrowing:

Proposition 5.

Maximizing the surplus S U R t E of an active credit line with the bank is equivalent to maximizing the objective:

(68) max p i t x S U R t E = max p i t x p i t x P t x i t − R t S , E w t L i t

subject to the demand, eq. (71), the production function, eq. (73) and the real cost of borrowed funds R t S , E ≡ p d , t E R L E + R t d .

Given the previous section, the proof of the above is almost trivial.^[30] The above proposition states that the “true” cost of borrowing is not the interest rate R t E per se, but R t S , E ≡ p d , t E R L E + R t d instead, which is actually a part of the wage. Intuitively, this is no other than the interbank rate R t d (which coincides with the deposit rate in this model) plus a measure of the risk premium as p d , t E is the probability of default and R L E is the loss-given-default rate.

Similarly the Investor maximizes the surplus of the match and thus the “true” cost of borrowing is R t S , R and not the interest rate R t R . The following proposition summarizes this:

Proposition 6.

If V ̄ t is the expected firm valuation, maximizing the surplus S U R t R of an active credit line with the bank is equivalent to maximizing the objective:

(69) max Q t R S U R t R = max Q t R Q t R V ̄ t − R t S , R P t I

subject to the inverse demand, eq. (5), where the cost of borrowing is R t S , R ≡ p d , t R R L R + R t d .

The proof is similar with the Entrepreneur’s case.^[31] The “true” cost of borrowed which is the interbank (or deposit) rate plus a measure of the risk premium composed of the probability of default p d , t R and the loss-given-default return to the bank R L R .

The way the banks interact with Entrepreneurs and Investors can better visualized in another partial equilibrium section in Appendix G.

A.6 Intermediate Goods-Producing Firms

There are N _t intermediate firms generated and managed by Entrepreneurs, each producing a differentiated product x _it and thus the aggregate intermediate good X _t is as follows

(70) X t = A t x ∫ N t x i t θ x − 1 θ x d i θ x θ x − 1 .

The elasticity of substitution is θ _x and A t x = N t ξ − 1 θ x − 1 is a variety effect.^[32] The usual cost minimization problem determines the demand for each good defined as follows

(71) x i t = A t x θ x − 1 p i t x P ̃ t x − θ x X t

where p i t x is the price of each intermediate firm. The price of the aggregate good is as follows

(72) P ̃ t x = 1 A t x ∫ N t p i t x 1 − θ x d i 1 1 − θ x .

Each firm produces using the labor effort of a single worker L _it according to the following production function

(73) x i t = z t L i t

The cost of creating a firm can only be funded by loans and is defined as w _t L _it, where w _t is the wage of each worker. Furthermore, there is idiosyncratic uncertainty associated with intermediate firms, captured by the random cost φ _i.^[33]

The Entrepreneur’s current profit from the project is

(74) V i t = p i t x P t x i t − R t E w t L i t − φ i

where R t E is the interest rate charged by the bank and φ _i is an IID random shock and φ i ∼ N φ ̄ , S t 2 . Instead of maximizing eq. (74) the firm maximizes the surplus which is objective (68) subject to the demand (71) and the production function, eq. (73).

The corresponding first-order condition from maximizing the surplus in (68) is as follows:

(75) P t x p i t x P ̃ t x = θ x θ x − 1 R t S , E w t z t .

where P t x ≡ P ̃ t x P t is the relative price. Since firms are all ex ante identical, from equation (72) one can get p i t x P ̃ t x = N t ξ and therefore the aggregate relative price is

(76) P t x = θ x θ x − 1 R t S , E w t z t N t ξ

which shows that the relative price is a markup over the real marginal cost.

In addition, plug (76) and also R t S , E ≡ p d , t E R L E + R t d in the interest rate equation for the Entrepreneurs (66) to get

(77) R t E = θ x − ζ θ x − 1 p d , t E R L E + R t d − ζ q t E ρ t E R t d κ t E w t L t

This expression is both intuitive and realistic. The interest rate is proportional to the interbank rate^[34] R t d plus a risk premium which is captured by the probability of default p d , t E and the loss-given-default R L E . The second and last term of (77) is an adjustment for how easy it is for the bank to find a match for a credit line.

Using (77) in (74) the value of each firm is thus

(78) V i t = ζ θ x − 1 R t S , E + ζ q t E ρ t E R t d κ t E w t L t w t L t − φ i

A.7 Final Goods-Producing Firms

The objective of the final goods-producing firms is to maximize profits subject the demand for each firm derived from a cost minimization problem of the household is defined as follows

(79) y j t = p j t P t − θ Y t

where the aggregate price level is

(80) P t = ∫ 0 1 p j t 1 − θ d j 1 1 − θ

while the aggregate output is

(81) Y t = ∫ 0 1 y j t θ − 1 θ d j θ θ − 1

with θ being the elasticity of substitution.

The production function is as follows

(82) y j t = z t f X i t

where z t f is productivity and X _it the amount of the aggregate intermediate good used by the firm.

The firm also faces nominal rigidities (Calvo pricing), where with probability γ is not able to adjust its price every period.

Therefore, the objective of the firm is to maximize the profits Π_t defined as follows

(83) Π t = max p t t = 0 ∞ ∑ k = 0 ∞ γ k E t Q t , t + k p t P t + k y t + k p t − P t + k x z t f y t + k p t .

The first-order condition is

(84) θ − 1 Λ t 1 = θ Λ t 2

where

(85) Λ t 1 = p t P t ∑ k = 0 ∞ β k γ k E t λ t + k P t P t + k 1 − θ Y t + k

and

(86) Λ t 2 = ∑ k = 0 ∞ β k γ k E t λ t + k P t + k x z t + k f P t P t + k − θ Y t + k .

Recursively, the two expressions become

(87) Λ t 1 = p t * P t λ t Y t + β γ p t * P t E t P t + 1 p t + 1 * P t P t + 1 1 − θ Λ t + 1 1

and

(88) Λ t 2 = λ t P t x z t f Y t + β γ E t P t P t + 1 − θ Λ t + 1 2

In the end, the relative price p t * P t of the price adjusting firms is

(89) 1 = 1 − γ p t * P t 1 − θ + γ P t − 1 P t 1 − θ

A.8 Monetary Authority and Government

The central bank uses standard monetary policy via a common interest rate rule to adjust the federal funds rate i t f f

(90) i t f f i f f = i t − 1 f f i f f ρ m P t P t − 1 g π Y t Y g y 1 − ρ m

where the federal funds rate is equal to the deposit rate in equilibrium, i.e. i t f f = R t d . Possible inflation stabilization alternatives are out of our scope but such exercises can be found in De Fiore, Teles, and Tristani (2011).

The government’s budget constraint is:

(91) G t = Π t + ω Ω ̃ t

where G _t is government spending. To isolate the 3 types of agents we already defined, we assume that the final good firm’s profits and the dividends by banks are distributed to the government. In calibration the dividends paid by banks are set to zero. The introduction of final good firm’s purpose is to introduce price stickiness ala Calvo. However, the way the profits are split between agents is hard to determine and thus we distribute those profits to the government. The government in this model represents not only the government but the rest of the agents other than the 3 introduced thus far.

B Parametrization

Table 4 presents plausible values for the model’s parameters along with their description. The probability of default for Entrepreneurs is the average for new firms from 1 to 3 years old coming from the Business Dynamic Statistic (BDS) and spans the period from 1977 to 2014. This determines the interest rates for the decision problems R ^S,E and R ^S,R. After the 90s, bank prime rate has be fluctuating between 3.25 and 9.5 %. The cost of credit is determined such that the interest rates to Entrepreneurs is 3 % which is close to the rate charged for collateralized loans. The interest rate for asset redistribution is then determined at 10 %. The loss given default for both entrepreneurs and hedge funds is the same ( R L E = R L R ) and set to 10 %. The bargaining parameter ζ is set such that the gross interest rate is at low levels.

Table 4:

Theoretical model parametrization.

Variable	Value	Description	Variable	Value	Description
N	1.00	Number of firms	κ ^E	0.04	Cost credit line E
N ^R	0.80	Firms owned by R	κ ^R	0.30	Cost credit line R
p d E	0.10	Prob. default E loan	ρ _m	0.85	Interest rate inertia par.
p d R	0.18	Prob default R loan	γ	0.85	Calvo prob.
R ^d	1.01	Gross deposit rate	R L E	0.10	Bank LGD (same for R)
R ^E	1.03	Rate to E	β	0.99	Discount factor
R ^R	1.10	Rate to hedge funds	m ^E	0.7	LTV ratio (same for R)
R ^S,E	1.02	Cost of funding E	q ^E	0.73	Prob. E get loan
R ^S,R	1.03	Cost of funding R	q ^R	0.96	Prob. R get loan
α _Γ	0.70	Matching function elast.	ζ	0.80	Bargaining par.
A _Γ	0.79	Matching function const.	θ	11.00	Elast. of subst. firms
ϕ	3.12	Risk aversion par.	ρ ^E	0.85	Prob. bank lends E
S _t	1.65	Risk std	ρ ^R	0.64	Prob. bank lends R
l	2.00	Disutility of labor param.	g _y	0.01	Interest rate rule gdp resp.
θ _x	7.46	Elasticity of subst. int.firms	g _π	1.10	Interest rate rule infl. Resp
a ^H	0.70	Housing elast.	N ̃ E	1.14	Max credit lines E
A ^H	1.22	Housing utility par.	N ̃ R	0.95	Max credit lines R

The letter E corresponds to entrepreneurs while HF to hedge fund.

The profit of the bank is calibrated to zero to avoid the ad-hoc decision of which agent should receive the bank profits. The number of firms N in steady state is set to 1 and the number of those firms partially owned by entrepreneurs is N ^R = 0.8. Usually, transactions classified as wealth redistribution can be much higher, but the redistribution of wealth from one Investor to another is ignored as all Investors are grouped into one big household for simplicity.^[35] The rest of the parameters are set to plausible values.

We set the default probability for new firms at 10 %. Although one-year-old firms tend to default more, typically around 23 %, data on 4 to 10-year-old companies presents a default rate of about 10 %. Given our intent for a more conservative estimate, we adopt the lower rate of 10 %. Consequently, the default probability for Investors rests at 18 %, still under the 23 % benchmark.

Determining the scale of asset-redistributing sectors compared to the productive economy poses a challenge. Hence, we calibrate the latter’s profits to be roughly double those of the former. Our estimate is grounded in post-crisis corporate profits, where non-financial firms amassed approximately $1.2 trillion compared to the $800 billion from financial firms. Considering monetary institutions’ profits at around $200 billion, implying asset redistributors could garner nearly half the profit of non-financial companies.

These calculations inform the values for risk aversion ϕ and the idiosyncratic volatility S aligning Investors’ profits at around half. However, asset redistribution isn’t purely a diversification tactic; it often stems from differences in beliefs, particularly heightened during recessions.

The calibration also implies that housing value to income is P E H E + H R N Ψ E + N R Ψ R = 144 %. In the data housing is on average around 310 % of mean income while more recently (after 2015) is above 500 %.

Labor’s share of income becomes a bit low in this model, 25 %. The reason is because it is assumed for simplicity that the financial sectors such as banking and investing does not employ labor.

C Interest Rate Determination

The interest rate charged by the bank to each type of customer is derived from a Nash bargaining where the underlying rate is the one that splits the surplus from the match to the two parties. In this section, the Bellman equations necessary to derive the surplus from a match is determined, along with the Nash Bargaining problem.

C.1 Surplus

For the surplus determination we assume that all participants discount the future in the same way.^[36] Given the maximization problem in the previous section, the value of a vacant credit line to Entrepreneurs for the bank is

(92) V t E = − R t d κ t B + ρ t E E t β J t + 1 E + 1 − ρ t E E t β V t + 1 E

where V t E = 0 for every t and the value of an active credit line to the bank is basically the envelope condition (60)

(93) J t E = R t E − p d , t E R L E − R t Γ w t L t + 1 − p d , t E E t β J t + 1 E

where p d , t E is the probability of default which depends on the state of the economy and is explicitly determined in a following section. Upon default, the bank is still covered (receives R t E ) but forces the borrower to liquidate the collateral, a move that affects the bank’s reputation (or carries legal costs) and results in a constant loss of R L E .^[37]

The value of a vacant credit line for Investors to the bank is

(94) V t R = − R t d κ t B + E t β ρ t + 1 R J t + 1 R + 1 − ρ t R E t β V t + 1 R

where V t R = 0 for every t. The value of an active credit line with an Investor to the bank is

(95) J t R = R t R − p d , t R R L R − R t d Q t R P t I + 1 − p d , t R E t β J t + 1 R

where p d , t R is the probability of default for the Investor. The probabilities of default for both projects are obviously correlated but the bank cannot account for the correlation of defaults of every pair of assets in the economy and, accordingly, has no reaction. The value of a credit line to Entrepreneur is

(96) Ξ t E = R t V − R t E w t L t + 1 − p d , t E E t β Ξ t + 1 E + p d , t E E t β Ξ ̃ t + 1 E

where R t V is just the revenue of the firm, while the value of an Entrepreneur’s idea seeking a credit line is

(97) Ξ ̃ t E = q t E E t β Ξ t + 1 E + 1 − q t E E t β Ξ ̃ t + 1 E .

Similarly, the value of a credit line to the Investor is

(98) Ξ t R = Q t R V ̄ t − R t R Q t R P t I + 1 − p d , t R E t β Ξ t + 1 R + p d , t E E t β Ξ ̃ t + 1 R

and the value of searching for a credit line to the Investor is

(99) Ξ ̃ t R = q t R E t β Ξ t + 1 R + 1 − q t R E t β Ξ ̃ t + 1 R

C.2 Surplus Maximization and Interest Rate

In order for the bank to determine the interest rates charged for loans to Entrepreneurs and Investors, it solves the usual bargaining problem that splits the surplus from an existing credit line, allocating ζ portion to the Entrepreneurs and 1 − ζ to itself. Specifically, the bank solves

(100) max R t E t = 0 ∞ Ξ t E − Ξ ̃ t E ζ J t E 1 − ζ .

The first-order condition is

(101) ζ J t E = 1 − ζ Ξ t E − Ξ ̃ t E .

By plugging in the above equations (93), (96), (97) and (92), at the equilibrium the gross rate to Entrepreneurs is the following

(102) R t E = 1 − ζ R t V w t L t + ζ p d , t E R L E + ζ R t d − ζ q t E ρ t E R t d κ t E w t L t

For the derivation of the interest rate charged to Investors one can use the first-order condition for the Nash-bargaining problem between the bank and the Investor in order to get

(103) ζ J t R = 1 − ζ Ξ t R − Ξ ̃ t R

and by plugging in equations (95), (98), (99) and (94), one can derive that the gross rate for the Investors is as follows

(104) R t R = 1 − ζ V t P t I + ζ p d , t R R L R + ζ R t d − ζ q t R ρ t R R t d κ t R Q t R P t I

D Lemma 2

By using equation (58) to eliminate ρ t R E t β d J B t + 1 d N t + 1 R from equation (57), it can be implied that

(105) κ t R R t d = ρ t R E t β d J B t + 1 d N t + 1 R

signifying that at the optimal level, the cost of searching for a credit line must be equal to the expected benefit from opening it, which depends on the probability ρ t R . Similarly, by using equation (58) to eliminate ρ t E E t β d J B t + 1 d N t + 1 from equation (57) it can be implied that

(106) κ t E R t d = ρ t E E t β d J B t + 1 d N t + 1 .

By leading equation (60) a period in advance and substituting in equation (105) and the same equation (105) a period in advance, one can get the line of credit creation condition for the Entrepreneurs

(107) κ t E R t d ρ t E = E t β R t + 1 E − p d , t + 1 E R L E − R t + 1 d w t + 1 L t + 1 + E t β 1 − p d , t + 1 E E t κ t + 1 E R t + 1 d ρ t + 1 E

Using the same steps, the line of credit creation condition for the Investor credit lines is as follows

(108) κ t R R t d ρ t R = E t β R t + 1 R − p d , t + 1 R R L R − R t + 1 d Q t + 1 R P t + 1 I + E t β 1 − p d , t + 1 R E t κ t + 1 R R t + 1 d ρ t + 1 R

E Proof of Propositions (Partial Equilibrium)

This section provides the proofs for the closed form of the partial equilibrium model

E.1 Proposition 1

From equation (9)

(109) d Q R d V ̄ = 1 − R 2 R ϕ S 2 < 0

as R > 1 because it is the gross interest rate charged by the bank.

For the next result plug in equations (8)–(10). This implies that

(110) Ψ R = 1 − R V ̄ + R ϕ S 2 2 R ϕ S 2 1 − R V ̄ + R ϕ S 2 2 = 1 − R V ̄ + R ϕ S 2 2 4 R ϕ S 2

Differentiate the above and use equation (9) to get

(111) d Ψ R d V ̄ = Q R 1 − R < 0

For d Ψ E d V ̄ differentiate equation (36)

(112) d Ψ E d V ̄ = 1 − R − 1 2 2 R 2 ϕ S t 2 V ̄ t = 1 − R − 1 R R − 1 V ̄ t 2 R ϕ S t 2 > 0

because for both terms, it is required that R − 1 R < 1 and R − 1 V ̄ t 2 R ϕ S t 2 < 1 for Q ^R to be positive as (9) indicates.

When the interest rate R ^R increases then obviously

d Ψ R d R R = d V ̄ 1 − R R + R R ϕ S 2 2 4 R R ϕ S 2 d R R = − ( 1 − Q R ) V ̄ − Q R P I + 1 − Q R V ̄ = − Q R P I < 0

For the risk aversion ϕ (or the risk S)

(113) d Ψ R d ϕ = 1 2 d Q R d ϕ 1 − R t R V ̄ + R R ϕ S t 2 + Q R 2 R R S 2

where

(114) d Q R d ϕ = − 1 − R R V ̄ 2 R R ϕ 2 S 2 > 0

Plug the above in and after some algebra

d Ψ R d ϕ = − 1 2 1 − R R V ̄ ϕ Q R + 1 2 Q R R R S 2 = 1 2 R R − 1 V ̄ ϕ Q R + 1 2 Q R R R S 2 > 0

For the Entrepreneur cash flow

(115) d Ψ E d ϕ = d 1 − Q t R V ̄ t + Q t R P t I d ϕ = d Q t R d ϕ P t I − V ̄ t + d P t I d ϕ Q t R < 0

Since d P t I d ϕ = − S 2 2 < 0 and P t I − V ̄ t = ϕ Q t R N t R − N t S t 2 < 0 .

E.2 Proof of Proposition 2

Rearrange eq. (17) to get:

s E 1 − s E α Γ b R b E α Γ = R − R d − Φ − V ̄ S R L R Γ − R d − Φ R − 1 V ̄ − R ϕ S 2 2 S R L

Differentiate the above with respect to V ̄ given that s ^E is an implicit function of V ̄ . That is

b R b E α Γ α Γ 1 − s E s E 1 − α Γ 1 1 − s E 2 d s E d V ̄ = f − V ̄ S R L S R Γ − R d − Φ R − 1 V ̄ − R ϕ S 2 2 S R L + f R − 1 V ̄ − R ϕ S 2 2 S R L R − 1 2 S R Γ − R d − Φ R − 1 V ̄ − R ϕ S 2 2 S R L 2

where d Φ x d x = f x where f . is the pdf of a standard normal. Every single term in the above expression is positive thus d s E d V ̄ > 0 .

F Additional Model Derivations

F.1 Entrepreneurs with a Portfolio of Firm

Suppose the number of firms N _t owned by the household and the number of firms partially sold to third parties N t R are not uncountably infinite. The E then seek to maximize utility by choosing the portion of the firms to keep Q t E given that there is a demand for N t R firms out of the N _t to be partially sold. That is:

(116) max Q t E U N t R Q t E V i t + 1 − Q t E P t I + N t − N t R V i t

Since U . is CARRA, maximizing (116) is the same as maximizing the following

max Q t E N t R Q t E V ̄ t + 1 − Q t E P t I + N t − N t R V ̄ t − N t 2 ϕ 2 N t R N t Q t E + N t − N t R N t 2 S t 2

The FOC implies the optimal share is

(117) Q t E = V ̄ t − P t I − ϕ N t − N t R S t 2 ϕ N t R S t 2

Since Q t E = 1 − Q t R then the demand for the Investors share is

(118) P t I = Q t R ϕ N t R S t 2 + V ̄ t − ϕ N t S t 2

The problem of the Investor is:

max Q t R Q t R V ̄ t − R t R , S P t I

subject to (117). The FOC implies that

Q t R = 1 − R t S , R V ̄ t + R t S , R ϕ N t S t 2 2 R t S , R ϕ N t R S t 2

and thus the price is

P t I = 1 + R t S , R V ̄ t − R t S , R ϕ N t S t 2 2 R t S , R

F.2 Risk Averse Investors

In this section, a version where the Investors are risk averse is considered. The utility function is CARA with risk averse parameter ϕ ^r. The derivation assumes N _t and N t R are not infinite and thus the Entrepreneurs sell a bundle of firms together as above. In case the Entrepreneurs do not consider income pooling when selling their firms, set N t = N t R = 1 below. The profit maximization problem for the portion of the asset to purchase Q t R by Investors becomes

(119) max Q t R t = 0 ∞ Q t R V ̄ t − R t S , R P t I − ϕ r 2 Q t R 2 S t 2

The first order condition after substituting P t I using equation (5) solved for Q t R is

(120) Q t R = 1 − R t S , R V ̄ t + R t S , R ϕ N t S t 2 2 R t S , R ϕ N t R S t 2 + ϕ r S t 2

Plug equation (120) in the demand equation (5) to get

(121) P t I = ϕ r ϕ + 1 + R t S , R V ̄ t − R t S , R N t ϕ + ϕ r S t 2 2 R t S , R

It is evident that Q t R is affected by changes in V ̄ t the same way as in the risk neutral case (ϕ ^r = 0). However, the price is even more volatile when Investors are risk averse by an additional term ϕ r ϕ . Therefore, if Investors are risk averse they provide insurance at a higher cost and thus the result becomes even stronger.

F.3 Income Pooling by Entrepreneurs

The welfare of E with an on-going credit line with a bank evolves according to the following Bellman eq.

V t E , C L H t − 1 E , C L = max H t E , C L , Q t E U C t E , C L + G H H t E , C L − δ H H t E , C L + β 1 − p d , t E V t + 1 E , C L H t E , C L + p d , t E V t + 1 E , N C H t E , C L

subject to the flow budget constraint

(122) C t E , C L + P t h H t E , C L − H t − 1 E , C L = Ψ t E + T t E

where T t E are transfers between Investors that can be either in terms of the real good or in housing units (or both). The LTV constraint is

(123) N ̃ t R t E w t L t ≤ m E E t P t + 1 h H t E , C L

The value from searching for a credit line with the bank is

V t E , N C H t − 1 E = max H t E U C t E , N C + G H H t E , N C − δ H H t E , N C + β N ̃ t R − N t R N ̄ t R − N t R q t E V t + 1 E , C L H t E , N C + 1 − N ̃ t R − N t R N ̄ t R − N t R q t E V t + 1 E , N C H t E , N C

The probability N ̃ t R − N t R N ̄ t R − N t R is the number of credit line applications by the E N ̃ t R − N t R over the total number of credit line applications.^[38] The associated budget constraint is

C t E N + P t h H t E N − H t − 1 E N = T t E F

As the decision to diversify Q t E happens after a share purchase offer from an Investor, and sells only the current period expected profits (without knowing the idiosyncratic shock), then maximizing with respect to Q t E is identical to maximizing U C t E , C L after plugging the constraint (122) in. This is also identical to maximizing (2). Therefore, the decision to diversify, as long as the uncertainty is only on the idiosyncratic component, is unaffected from the rest of the decisions and only depends of the risk aversion parameter of the E.

The first order condition for housing is:

G H ′ H t E , C L − δ H + μ t E , C L m E E t P t + 1 h + E t λ t + 1 E , C L P t + 1 h = λ t E , C L P t h

which is identical to (41). The only difference in the housing decision between a larger household from a single member household is the third term μ t E , C L m E E t P t + 1 h which considers the benefit from relaxing the constraint (122), in case it binds. Thus, it entirely depends on the shadow price μ t E , C L . If the constraint in (123) does not include N ̃ t R which implies the constraint always binds for a single E, then μ t E , C L = 0 . If it binds, then

1 N ̄ t R − N t R q t R V t + 1 R , C L − V t + 1 R , N C = μ t E , C L R t E w t L t

Therefore, μ t E , C L depends on the difference between the value with and without a credit line. This can go from zero to V t + 1 R , C L − V t + 1 R , N C which is the case when E cannot insure against uncertainty in any way. Eq. (42) which corresponds to large households or income-pooling households uses the period ahead cashflow Ψ t + 1 E which is between 0 ≤ Ψ t + 1 E ≤ V t + 1 R , C L − V t + 1 R , N C . The value of μ t E , C L depends on the degree of the insurance between E. Thus the assumption of income pooling basically affects how important housing accumulation is in securing loans.

F.4 Depositors Determine Lending Volume

When loans are set by deposits, the bank solves the same optimization problem in equation (56) with the same constraints, however it only optimally chooses s t E and not the aggregate number of credit lines Γ. In other words, the central bank sets the deposit rate, the depositors set the amount of deposits (by a usual Euler equation) and through equation (55) the bank simply splits this amount between Entrepreneurs and Investors.

Therefore, for this solution the necessary condition for bank optimization is

(124) R t Γ κ t E − κ t R = ρ t E E t β d J B t + 1 d N t + 1 − ρ t R E t β d J B t + 1 d N t + 1 R

along with the two envelope conditions

(125) d J B t d N t = R t E − p d , t E R L E − R t Γ w t L t + 1 − p d , t E E t β d J B t + 1 d N t + 1

and

(126) d J B t d N t R = R t R − p d , t R R L R − R t Γ Q t R P t I + 1 − p d , t R E t β d J B t + 1 d N t + 1 R .

There is also an Euler equation for depositors

(127) 1 = β E t U c C t + 1 C U c C t C R t d

G A Partial Equilibrium with Banks

Assume that there are Entrepreneurs and Investors as in Section 4.1 and trade assets with payoffs as in (4.1). In this example we still use analysis at the steady state i.e. we use a non-dynamic model and thus we remove the t subscript. However, the only difference with Section 4.1 is that we introduce a bank that has Γ credit lines to allocate to fund as many Entrepreneur or Investor projects as possible.

Proposition 7.

Let s ^E and s ^R be the shares of credit lines that maximize (57) for a bank with a fixed number of credit lines ^[39] Γ. Then, as the average firm valuation V ̄ increases, the bank’s share of credit lines for Entrepreneurs s ^E increases d s E d V ̄ > 0 while its share to the Investors s ^R decreases d s R d V ̄ < 0 .

Maximizing bank profits implies that at the optimum, the value of the marginal credit line to the Entrepreneur V ^E must equal the corresponding value to Investor V ^R implying V ^E = V ^R where

(128) V E = ρ E d J B d N = ρ R d J B d N R = V R .

where d J B d N and d J B d N R are the steady state counterparts of eqs. (60) and (61) respectively when β = 1. Substituting (60) and (61) (at steady state) in V ^E and V ^R using also the expressions for the matching probabilities ρ ^E and ρ ^R from (27) and (28) and the default probabilities (38) and (44) the respective values become

(129) V E = b E s E Γ α Γ R R − R d − Φ − V ̄ S R L Φ − V ̄ S w L

and

(130) V R = b R 1 − s E Γ α Γ R R − R d − Φ R R − 1 V ̄ − R R ϕ S 2 2 S R L Φ R R − 1 V ̄ − R R ϕ S 2 2 S Q R P I

It shows that in a recession ( V ̄ decreases), the probability of default for the Entrepreneur Φ − V ̄ S increases while the probability of default for the Investor Φ R R − 1 V ̄ − R R ϕ S 2 2 S decreases. As a result, the bank’s incentive to fund the Entrepreneurs deteriorates while the benefit from allocating the extra credit line to the Investor increases, inducing the bank to increase the share of credit lines 1 − s ^E to Investors (s ^E ↓). This in turn, raises the probability ρ E = b E s E Γ α Γ of a successful credit line match with the Entrepreneurs while ρ R = b R 1 − s E Γ α Γ at the same time decreases, until V ^R equals V ^E. Therefore, there is a tendency for more credit lines to be allocated to Investors in a recession than to Entrepreneurs.

Figure 8 characterizes the above equilibrium credit line allocation by the banks to Entrepreneurs and Investors.^[40] Initially, V ^E is graphed as the downward sloping curve V 1 E which is a function of the share of funds s ^E that go to Entrepreneurs. The upward sloping curve V 1 R is the left hand side of equation V ^R. The equilibrium is where the two curves intersect, which is initially at E ₁. In an expansion, V ̄ increases and therefore the probability for Entrepreneurs to default Φ − V ̄ S decreases as firms become more profitable. Thus the value of the extra dollar loan to Entrepreneurs becomes higher which shifts figure V 1 E upwards to V 2 E in Figure 8. On the other hand, as the need for diversification becomes less important to Entrepreneurs (risk is unchanged), they sell their shares to Investors at a relatively higher price. For the bank the probability of an Investor to default Φ R R − 1 V ̄ − R R ϕ S 2 2 S increases as R ^R > 1 as redistribution becomes less rewarding. This shifts figure V 1 E downwards to V 2 E . The new equilibrium is at E ₂ where more credit lines are allocated by the bank to Entrepreneurs than to Investors. Alternatively, in a recession more funds go to Investors at the expense of Entrepreneurs.

The incentives for the two types of agents competing for funds are in opposition along the business cycle, as documented by the VAR model as well. While the incentives for nearly all other investments coincide along the business cycle, capital investment and asset redistribution one can move in opposite directions. This observation can be exploited by policymakers to boost growth during a recession.

G.1 Endogenous Interest Rates

When the interest rates R ^R and R ^E are determined as in (77) and (67) (where R ^E directly affects V ̄ ), and V ̄ is determined from (78) and is also variable according to the model specification in the previous sections then the result is hard to present in closed form. Figures 9–11 provide some insight to the determination of equilibrium for different values of productivity, risk aversion parameter and idiosyncratic volatility respectively. First, Figure 9 depicts the evolution of the equilibrium as productivity increases. When productivity increases the Entrepreneurs keep a higher share of their firms for themselves which results in a smaller share for Investors, since the urge to diversify deteriorates. This is also evident from the rising price the firms are traded P ^I. In addition, the default probability for loans to Entrepreneurs decreases inducing the opposite effect for the Investors. The amount of credit flowing to the Entrepreneurs is increasing while the amount of credit to the Investors decreases.

In contrast, Figure 10 presents the equilibrium when the Entrepreneurs become more risk averse. This initiates the opposite reaction to the equilibrium because as the Entrepreneurs become more risk averse, the need to diversify is stronger and thus the share going to the Investors increases while the price the firms are traded decreases. The probability of default drops for the Investors and thus the credit portion going to the Entrepreneurs deteriorates while the portion reaching the Investors increases.

The increase in risk works in a similar way as an increase in risk aversion as Figure 11 suggests. Greater risk for the same expected return implies a higher appetite for diversification which benefits the Investors.

H Equilibrium

The expected net cash flow to the Entrepreneurs Ψ t E from each firm created is

(131) Ψ t E = N t R N t Q t E V ̄ t + 1 − Q t E P t I + 1 − N t R N t V ̄ t .

where the first term is the return from selling part of the firm to the Investors. As Investors secure N t R credit lines, they randomly choose N t R firms each period. Therefore, the probability of a firm to be partly purchased by Investors is N t R N t . The second term is the return from the business they fully own.

Bank capital evolves as follows

(132) Ω t + 1 = 1 − ω Ω ̃ t + 1 − δ B Ω t

where initial capital and reserves in the following period Ω_t+1 is the percentage 1 − ω of profits

(133) Ω ̃ t = N t R t E − p d , t E R L E w t L t + N t R R t R − p d , t R R L R Q t R P t I − R t d D t

which are paid as dividends after deducting the percentage δ _B denoting the cost to maintain bank capital.

The value of the investment to the Investors is

(134) Ψ t R = N t R Q t R V ̄ t − R t R P t I

and the number of firms the Investors hold a stake in is N t R ≤ N t .

Market clearing conditions

The aggregate resource constraint is

(135) C t R + C t E + C t C + G t + δ B Ω t + s t E κ t E + s t R κ t R Γ t = Y − p d , t E 1 − R L E N t w t L t − p d , t R 1 − R L R N t R Q t R P t I .

The Government spends according to eq. (91).

The amount of housing is fixed and therefore

(136) H t E + H t R = H ̄ .

There is a continuum of workers with a unit measure and thus the Labor supply is

(137) L t = ∫ 0 1 L i t d i

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Received: 2023-11-17

Accepted: 2024-10-13

Published Online: 2024-11-27

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Articles in the same Issue

https://doi.org/10.1515/bejm-2023-0191

Keywords for this article

investment; business cycles; asset redistribution; inequality; capital misallocation