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1. Consider the simple endogenous growth model where the only inputs into produ...
May 9, 2024
1 Solution
a
To derive the equilibrium growth rate with decreasing returns to scale in knowledge production, we set up the production function with knowledge (K) and labor (L) as inputs. Assuming a Cobb-Douglas production function: Y=KαL1−αY = K^{\alpha}L^{1-\alpha}, where 0 < \alpha < 1 represents decreasing returns to scale in knowledge production. The growth rate of knowledge (gKg_K) is determined by the rate of investment in knowledge relative to the depreciation of knowledge. The equilibrium growth rate is where the investment in knowledge equals depreciation
b
Diagram Illustration: The diagram would typically show the production function with the knowledge stock on the x-axis and output on the y-axis, illustrating the diminishing marginal returns to knowledge. The equilibrium growth rate is where the slope of the production function equals the depreciation rate
1 Answer
The equilibrium growth rate in the model with decreasing returns to scale in knowledge production is determined by the balance between investment in knowledge and its depreciation, and it can be illustrated with a production function diagram showing diminishing marginal returns.
2 Solution
a
With constant returns to knowledge in the R&D sector, the production function can be written as Y=AKLY = AKL, where A represents a technology parameter and K and L are inputs into production. The growth rate of knowledge (gKg_K) is now determined by the productivity of the R&D sector (A) and the amount of resources allocated to R&D. With constant returns, the equilibrium growth rate of knowledge is proportional to the R&D effort
2 Answer
The equilibrium growth rate of knowledge with constant returns to knowledge in the R&D sector is proportional to the R&D effort and the productivity of the R&D sector.
3 Solution
a
An increase in the R&D share implies allocating more resources to the production of knowledge. In scenario (a) with decreasing returns to scale, the additional R&D share leads to a smaller increase in the growth rate due to diminishing marginal returns. In scenario (b) with constant returns to scale, the increase in the R&D share leads to a proportionate increase in the growth rate of knowledge
3 Answer
An increase in the R&D share has different implications depending on the returns to scale in knowledge production: it leads to diminishing increases in growth rate under decreasing returns, while under constant returns, it leads to a proportionate increase in the growth rate of knowledge.
4 Solution
a
Distortions that affect the share of resources allocated to R&D include market failures such as externalities, where the social return of R&D exceeds the private return, leading to underinvestment in R&D. Other distortions include imperfect competition, informational asymmetries, and government policies that either do not adequately support R&D or over-regulate it
4 Answer
Typical distortions affecting the share of resources allocated to R&D include market failures like externalities, imperfect competition, informational asymmetries, and suboptimal government policies.
Key Concept
Equilibrium Growth Rate in Endogenous Growth Models
Explanation
The equilibrium growth rate in endogenous growth models is determined by the returns to scale in knowledge production and the allocation of resources to R&D.
2(a) Solution
a
Ricardian Equivalence: The Ricardian equivalence proposition suggests that it does not matter whether a government finances its spending with debt or taxes because the outcome on the total level of demand in the economy will be the same
b
Model Illustration: Using the simple two-period model, we can show that individuals anticipate future taxes that will be needed to pay off government debt. They increase their savings to pay for future taxes, which offsets the increase in demand from government spending. The Ramsey model would similarly show that optimal consumption paths are unaffected by the timing of taxes
2(a) Answer
The Ricardian equivalence suggests that fiscal policy through debt financing does not affect the overall level of demand, as individuals adjust their savings in anticipation of future taxes needed to repay the debt.
Key Concept
Ricardian Equivalence
Explanation
Ricardian equivalence posits that the method of government financing (debt vs. taxes) does not influence aggregate demand, as individuals adjust their behavior in anticipation of future fiscal obligations.
2(b) Solution
a
Reasons for Failure: Ricardian equivalence may fail due to several reasons such as the presence of liquidity constraints, intergenerational considerations, and myopia among consumers
b
Explanation: Liquidity constraints prevent some consumers from increasing their savings. Intergenerational considerations mean that current taxpayers may not feel obliged to save for future taxes that will be paid by future generations. Myopia leads consumers to not fully account for future tax liabilities
2(b) Answer
Ricardian equivalence may fail due to liquidity constraints, intergenerational considerations, and consumer myopia, which prevent the offsetting behavior assumed by the theory.
Key Concept
Failure of Ricardian Equivalence
Explanation
The failure of Ricardian equivalence is attributed to practical issues such as liquidity constraints, intergenerational transfer of debt, and myopic behavior of consumers, which prevent the theoretical offsetting behavior.
2(c) Solution
a
Fiscal Gaps: Fiscal gaps refer to the difference between the present value of all projected future government expenditures and the present value of all projected future tax revenues
b
Fiscal Sustainability: Fiscal gaps are used to assess the long-term fiscal sustainability of a country. A large fiscal gap indicates that the government may not be able to meet its future obligations without policy adjustments
2(c) Answer
Fiscal gaps measure the long-term fiscal sustainability of a country by comparing the present value of future expenditures and tax revenues, indicating the need for policy adjustments if the gap is large.
Key Concept
Fiscal Gaps
Explanation
Fiscal gaps are critical in assessing whether a country's fiscal policy is sustainable in the long run, as they highlight potential shortfalls in meeting future obligations.
3 Solution
a
Uncovered Interest Parity (UIP) Condition: The UIP condition states that the expected appreciation (or depreciation) of a currency is equal to the interest rate differential between two countries. Mathematically, it can be expressed as: Et[St+1]−St=it−it∗E_t[S_{t+1}] - S_t = i_t - i_t^*, where Et[St+1]E_t[S_{t+1}] is the expected future spot exchange rate, StS_t is the current spot exchange rate, iti_t is the domestic interest rate, and it∗i_t^* is the foreign interest rate
b
Role of Zero Transaction Costs: Zero transaction costs assumption ensures that investors can freely move funds across borders without incurring costs that could prevent arbitrage opportunities. This is crucial for UIP to hold because if there were transaction costs, the interest rate differential would need to be larger than the transaction costs to incentivize investors to engage in currency arbitrage
c
Role of Capital Mobility: Capital mobility allows for the free flow of financial capital across countries. High capital mobility is essential for UIP to hold because it ensures that investors can respond to interest rate differentials without restrictions, such as capital controls, that could prevent the equalization of expected returns across countries
d
Role of Risk-Neutral Speculators: A large number of risk-neutral speculators assumption implies that investors are indifferent to risk when making investment decisions and are only interested in the expected return. This is important for UIP because it assumes that investors will invest in foreign assets with higher expected returns without requiring a risk premium
e
Implications of Relaxing Zero Transaction Costs: If transaction costs are introduced, the UIP condition may not hold because investors would require a higher interest rate differential to compensate for these costs, potentially leading to deviations from the UIP prediction
f
Implications of Relaxing Capital Mobility: With lower capital mobility, due to controls or other barriers, the ability of capital to flow in response to interest rate differentials is hindered. This can lead to persistent interest rate differentials that do not trigger the expected currency adjustments predicted by UIP
g
Implications of Relaxing the Assumption of Risk-Neutral Speculators: If investors are risk-averse, they will demand a risk premium to invest in foreign assets, especially in currencies with higher volatility. This can lead to deviations from UIP as the exchange rate movements will also reflect compensation for risk, not just the interest rate differential
3 Answer
The uncovered interest parity condition is an economic theory that explains the relationship between interest rates and exchange rate expectations. It relies on assumptions of zero transaction costs, high capital mobility, and a large number of risk-neutral speculators to hold. Relaxing these assumptions can lead to deviations from the UIP prediction due to the introduction of transaction costs, capital controls, and risk premiums.
Key Concept
Uncovered Interest Parity (UIP) Condition
Explanation
The UIP condition is a fundamental concept in international finance that relates the expected change in exchange rates to the interest rate differential between two countries. It assumes no transaction costs, perfect capital mobility, and risk-neutral investors. Deviations from these assumptions can lead to the failure of UIP to accurately predict future exchange rate movements.
a Solution
a
Equation Interpretation: The equation st=ft+αEt(dstdt) s_t = f_t + \alpha E_t\left(\frac{ds_t}{dt}\right) indicates that the current exchange rate (st s_t ) is determined by the current fundamental value (ft f_t ) and an adjustment for the expected rate of change in the exchange rate, weighted by a parameter α \alpha . This reflects the idea that the exchange rate not only responds to current fundamentals but also to market expectations about future changes in the exchange rate
a Answer
The equation states that the exchange rate is a function of the fundamental value and the expected change in the exchange rate, adjusted by a parameter. It is expected to hold because market participants consider both current information and expectations about future changes when determining the exchange rate.
Key Concept
Exchange Rate Determination
Explanation
The Krugman target zone model suggests that the exchange rate is influenced by the fundamental value and expectations of future exchange rate movements.
b Solution
b
Random Walk Fundamental: If the fundamental follows a random walk, it implies that the best forecast for tomorrow's fundamental is today's value, meaning Et(ft+1)=ft E_t(f_{t+1}) = f_t . In a free float, without any target zone, there are no expected changes in the exchange rate due to policy interventions, so Et(dstdt)=0 E_t\left(\frac{ds_t}{dt}\right) = 0 . Substituting this into the equation gives st=ft s_t = f_t , indicating that the exchange rate equals the fundamental
b Answer
In the case of a free float and the fundamental following a random walk, st=ft s_t = f_t is a solution to the equation because there are no expected future changes in the exchange rate due to policy interventions.
Key Concept
Random Walk and Free Float
Explanation
When the fundamental value follows a random walk, the exchange rate in a free float system will simply reflect the fundamental value, as there are no expected changes due to interventions.
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