Economics 100B: Intermediate MacroeconomicsJesse Mora (Summer 2014)

Tuesdays/Thursdays 9:00 AM–12:30 PM

August 4, 2014

Outline

I. Admin: Turn in Problem Set 1; Midterm 1 on Thursday

II. Today’s Models: Understand differences in income over time andacross countries

III. Model: Solow Growth Model

⇒ Capital Accumulation (Ch. 8)

⇒ Population Growth (Ch. 8)

⇒ Technological Change (Ch. 9)

IV. Conclusion: Savings is key!

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Real GDP per person in US

Question: How realistic is the assumption that Y does not change? Whydoes this say GDP ”per person”?

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Video: 200 Countries, 200 Years, 4 Minutes(click above)

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Refresher I

I. What is the production function?

II. If Y is not fixed, how can it increase?

III. To explain the historical facts, what does our model need?

⇒ GDP growth

⇒ GDP growth in the steady state

⇒ GDP per capita growth in the steady state

⇒ This is long-run model, what does that mean?

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Refresher II

I. The production function: Y = F (K , L)

II. Assume Constant Returns to scale (CRS)

⇒ What does this mean?

III. Important: We will denote all quantities in the Solow model inper-worker terms. Lowercase letters are quantities per worker.

⇒ Output per worker: y = YL

⇒ Capital per worker: k = KL

⇒ Consumption per worker: c = CL

⇒ Investment per worker: i = IL

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Ch. 8: Capital Accumulation

Solow Growth Model

I. The production function in per-worker terms: y = f (k)

⇒ Multiply Y , K , and L in Y = F (K , L) by 1L

⇒ y = YL = F (K

L , 1) = f (k) (can do this because of CRS)

⇒ Output per worker is determined by capital per worker; the size of theeconomy does not matter.

II. If y exhibits diminishing returns, what does that mean?

⇒ The slope shows how much extra output per worker produces whengiven an extra unit of capital

⇒ Does this sound familiar? What is it?

⇒ Can you draw this?

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The consumption function (Demand for goods)

I. Demand: y = c + i

⇒ Ignore g, t, and nx for the Solow model

⇒ Consumption function: c = c(y) = (1− s)y

– s is between zero and one

– s is savings rate, so 1 − s is the marginal propensity to consume

⇒ Investment function: i = y − c = y − (1− s)y = sy

– Can you draw this?

II. y = f (k) determines how much output the economy produces, and sdetermines c and i

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Output, Consumption, and Investment

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Growth in the capital stock and the steady state

I. What determines y?

II. k will increase over time if

⇒ K increase

⇒ L increase (ignore for a few minutes)

⇒ Technology increases (ignore for a first half)

III. How does K and, thus, k change?

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Forces that influence the capital stock

I. Investment causes K to rise

⇒ so ∆k = i = sf (y)

⇒ Can you draw this?

II. Depreciation causes K to drop

⇒ Assume δ of capital wears out each year. δ is the depreciation rate.

⇒ so ∆k = −δk⇒ Can you draw this?

III. Change in capital stock = investment − depreciation

⇒ ∆k = i − δk⇒ ∆k = sf (y)− δk

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Output, consumption, and investment

Question: What gives you the highest y? What gives you the mostdepreciation? k∗ is the steady-state level of capital, what is a ”steadystate”?

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Example

Country A and country B both have the production function Y = K 0.5L0.5

I. Does this production function have constant returns to scale?

II. What is the per-worker production function, y=f(k)?

III. Assume that neither country experiences population growth ortechnological progress and that 5 percent of capital depreciates eachyear. Assume further that country A saves 10 percent of output eachyear and country B saves 20 percent of output each year. Using youranswer from part (b) and steady state condition that investmentequals depreciation, find the steady state level of capital per workerfor each country. Then find the steady state levels of income perworker and consumption per worker.

⇒ What is δ? s?

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How to find steady state (the easy way)

I. Start with ∆k = sf (k)− δk

II. In steady state ∆k∗ = 0

III. So to find k∗, set sf (k∗)− δk∗ = 0 or sf (k∗) = δk∗

IV. Equivalently: k∗

f (k∗) = sδ

V. Finally, solve for k∗

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How saving affects growth

Question: Do you see why s is key? why crowding out is bad? BUT, is ahigher s always a good thing?

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The Golden Rule Level of Capital

I. What steady state should the policymaker choose?

⇒ The steady state with the highest level of consumption

⇒ The steady-state value of k that maximizes consumption

II. The Golden Rule level of capital (k∗gold)

⇒ Rearranging the national income accounts identity: c = y − i

⇒ Substituting in definitions of y and i : c∗ = f (k∗)− δk∗

– δk∗ = i because we are in the steady state

⇒ More capital means more output, but we’ll also need more investment(and less consumption) to replace capital.

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Steady-State Consumption

Question: What is the slope of the production function, f (k∗)? What isthe slope of the δk∗ line?

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Condition that Characterizes the Gold Rule level of capital

I. The Golden Rule is described by the equation: MPK = δ

II. How did we get this?

⇒ Consider increasing k∗ by 1.

⇒ Output increases by MPK

⇒ The depreciation for the extra unit is δ

⇒ Net effect: MPK − δ

– if MPK − δ > 0, then increases in capital increase consumption– if MPK − δ < 0, then increases in capital decrease consumption

⇒ Golden Rule: MPK − δ = 0

III. The economy does not automatically gravitate towards the GoldenRule

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Steady-State Growth Rates

K accumulation∆ % Y ?∆ % y ?

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Ch. 8: Population Growth

Steady state with population growth

I. Suppose that L grows at a constant rate n

II. Before investment increased k and depreciation decreased k

III. Now growth in L causes capital per worker to fall

IV. Change in the capital stock per worker is

⇒ ∆k = i − (δ + n)k = sf (k)− (δ + n)k

⇒ (δ + n)k define break-even investment—the amount of investmentnecessary to keep the capital stock per worker constant.

V. Steady state k∗

⇒ ∆k∗ = 0, so i∗ = δk∗ + nk∗.

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Population Growth in the Solow Model

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The effect of population growth

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The Golden Rule and population growth

I. Consumption per worker: c = y − i

II. Steady-state consumption: c∗ = f (k∗)− (δ + n)k∗

III. Using same argument as above, the level of k∗ that maximizesconsumption is one at which

⇒ MPK = δ + n or MPK − δ = n

⇒ The MPK net of depreciation equals the rate of population growth

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Steady-State Growth Rates

K accumulation L growth∆ % Y 0 ?∆ % y 0 ?

Question: Does this show that things are getting better over time?

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Video: The Magic Washing Machine(click above)

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Ch. 9: The Role of Technology and Economic Policy

Efficiency of Labor

I. Old production function: Y = F (K , L)

II. New production function: Y = F (K , LxE )

⇒ E is called efficiency of labor; reflects knowledge

– Each hour of work contributes more to the production function

⇒ LxE measures the effective number of workers

– It takes into account the L and the efficiency of each worker E .

– Technological progress is analogous to increases in L.

III. Assume that the efficiency of labor E grows at a constant rate g .

⇒ g is called the rate of labor-augmenting technological progress.

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Steady State with Technological Progress

I. Instead of per-worker, we will use per-effective-worker notation.

⇒ Change in Notation Warning!!!

⇒ y = Y /(LxE ), k = K/(LxE ), etc.

⇒ We again write y = f (k)

II. ∆k = sf (k)− (δ + n + g)k

⇒ (δ + n + g)k is break-even investment.

III. Inclusion of technological progress does not substantially alter ouranalysis of the steady state

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Technological Progress and the Solow Growth Model

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Steady-State Growth Rates

K accum. L growth tech growth∆ % Total Output 0 n n + g∆ % Output per worker 0 0 g∆ % Output per effective worker – – 0∆ % Capital per effective worker – – 0

According to the Solow model, only technological progress can explainsustained growth and persistently rising living standards.

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The Golden Rule and technological progress

I. Consumption per effective worker: c = y − i

II. Steady-state consumption per effective worker:

⇒ c∗ = f (k∗)− (δ + n + g)k∗

III. Steady-state consumption is maximized if

⇒ MPK = δ + n + g or MPK − δ = n + g

⇒ The MPK net of depreciation equals the rate of total output, n + g .

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Example II

Draw a well-labeled graph that illustrates the steady state of the Solowmodel with population growth. Use the graph to find what happens tosteady-state capital per worker and income per worker in response to eachof the following exogenous changes.

I. A change in consumer preferences increases the saving rate.

II. A change in weather patterns increases the depreciation rate.

III. Better birth-control methods reduce the rate of population growth.

IV. A one-time, permanent improvement in technology increases theamount of output that can be produced from any given amount ofcapital and labor.

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